Re: What is a Löbian machine/number/combinator

[Bruno Marchal]

> On 5 May 2018, at 10:58, spudboy100 via Everything List 
> <everything-list@googlegroups.com> wrote:
> 
> 
> The Lobian Machines arise from a universe, infinite in time and extent.

They arise of the semi-computable part of the arithmetical reality (or anything 
Turing equivalent). 

A physical universe, like any notion of God, cannot make a computations felt as 
more realm or less real than another, without contradicting mechanism.




> From afar Ludwig Boltzmann looks down and laughs at the puny mortals 
> attempting to comprehend his greatness. Hugh Everett the 3rd claps Boltzmann 
> on the back and buys him a drink! The AI Tin Man says, "If I only had a 
> Brain, it would be like Boltzmanns. 

In arithmetic, this is quasi-solved. The Boltzman brains, and aberrant 
histories have plausibly the measure zero, although this is not yet entirely 
proved, to be sure, but if disproved, then mechanism is refuted, and that has 
not yet been done too!

Bruno




> 
> -Original Message-
> From: Bruno Marchal <marc...@ulb.ac.be>
> To: everything-list <everything-list@googlegroups.com>
> Sent: Sat, May 5, 2018 4:23 am
> Subject: Re: What is a Löbian machine/number/combinator
> 
> 
> On 4 May 2018, at 01:26, Quentin Anciaux <allco...@gmail.com 
> <mailto:allco...@gmail.com>> wrote:
> 
> Again the perfect example of I lost so I dodge…
> 
> Exactly. John Clark could not have provided a better illustration, indeed.
> 
> I consider a disagreement as a courtesy to pursue a conversation.
> 
> I consider mockery, insult and rhetorical dodging as “I have no argument, you 
> won the point”.
> 
> Bruno
> 
> 
> 
> 
> Le jeu. 3 mai 2018 21:36, John Clark <johnkcl...@gmail.com 
> <mailto:johnkcl...@gmail.com>> a écrit :
> 
> On Thu, May 3, 2018 at 2:01 PM, Bruno Marchal <marc...@ulb.ac.be 
> <mailto:marc...@ulb.ac.be>> wrote:
> 
> ​>> ​ You say the diary solves the referent issue because its clear the man 
> in Helsinki wrote it and he wrote it yesterday, but in one variation of the 
> thought experiment there is nobody in Helsinki today, there are people in 
> Moscow and Washington who vividly remember writing that diary but what one 
> and only one really did?
> 
> ​> ​ We assume Mechanism, so the answer is simply both, from the third person 
> point of view, and only one, for each of the first person point of view 
> obtained.
> 
> ​Counter argument #11​42
> 
> ​>>​ What did the correct answer to the question turn out to be?
> 
> ​>​ The question was the prediction of the next experience. The correct 
> answer, remaining correc,t through the experience was the prediction “I will 
> feel either W or M”, written “W v M”, keeping in mind that the question 
> concerned the experience at the first person.
> 
> So it is “W v M”.
> 
> ​Counter argument #926​  
> 
> ​>>​ Who wrote the diary?
> 
> ​> ​ The candidate of the experience.
> 
> ​And who is the ​  candidate of the experience ​? The guy who wrote the diary 
> of course.​
>  
> ​>> ​ Is the one and only one referent to the personal pronoun “I” in the 
> question the Moscow man or the Washington man?
> 
> ​> ​ Anyone. We keep only the prediction assessed by all of them. In the big 
> number iteration of that experience, the correct prediction is “white noise”.
> 
> ​Yes that's what I thought, hot air and a big noise.​
> 
> ​>> ​ It can’t be the Helsinki man because today there is no Helsinki man.
> 
> ​> ​ That contradicts the local personal identity definition that you have 
> agreed very often upon,
> 
> ​An oldie but a goodie, counter agreement #22​
> 
> ​John K Clark​  
> 
> 
> 
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Re: What is a Löbian machine/number/combinator

[spudboy100 via Everything List]

The Lobian Machines arise from a universe, infinite in time and extent. From 
afar Ludwig Boltzmann looks down and laughs at the puny mortals attempting to 
comprehend his greatness. Hugh Everett the 3rd claps Boltzmann on the back and 
buys him a drink! The AI Tin Man says, "If I only had a Brain, it would be like 
Boltzmanns. 


-Original Message-
From: Bruno Marchal <marc...@ulb.ac.be>
To: everything-list <everything-list@googlegroups.com>
Sent: Sat, May 5, 2018 4:23 am
Subject: Re: What is a Löbian machine/number/combinator





On 4 May 2018, at 01:26, Quentin Anciaux <allco...@gmail.com> wrote:


Again the perfect example of I lost so I dodge…



Exactly. John Clark could not have provided a better illustration, indeed.


I consider a disagreement as a courtesy to pursue a conversation.


I consider mockery, insult and rhetorical dodging as “I have no argument, you 
won the point”.


Bruno








Le jeu. 3 mai 2018 21:36, John Clark <johnkcl...@gmail.com> a écrit :





On Thu, May 3, 2018 at 2:01 PM, Bruno Marchal <marc...@ulb.ac.be> wrote:




​>> ​
You say the diary solves the referent issue because its clear the man in 
Helsinki wrote it and he wrote it yesterday, but in one variation of the 
thought experiment there is nobody in Helsinki today, there are people in 
Moscow and Washington who vividly remember writing that diary but what one and 
only one really did?



​> ​
We assume Mechanism, so the answer is simply both, from the third person point 
of view, and only one, for each of the first person point of view obtained.





​Counter argument #11​42




 
​>>​
What did the correct answer to the question turn out to be?







​>​
The question was the prediction of the next experience. The correct answer, 
remaining correc,t through the experience was the prediction “I will feel 
either W or M”, written “W v M”, keeping in mind that the question concerned 
the experience at the first person.


So it is “W v M”.





​Counter argument #926​
 





​>>​
Who wrote the diary?







​> ​
The candidate of the experience.





​And who is the ​
 candidate of the experience
​? The guy who wrote the diary of course.​

 


​>> ​
Is the one and only one referent to the personal pronoun “I” in the question 
the Moscow man or the Washington man? 



​> ​
Anyone. We keep only the prediction assessed by all of them. In the big number 
iteration of that experience, the correct prediction is “white noise”.





​Yes that's what I thought, hot air and a big noise.​




​>> ​
It can’t be the Helsinki man because today there is no Helsinki man.





​> ​
That contradicts the local personal identity definition that you have agreed 
very often upon,




​An oldie but a goodie, counter agreement #22​




​John K Clark​
 










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Re: What is a Löbian machine/number/combinator

[Bruno Marchal]

> On 4 May 2018, at 01:26, Quentin Anciaux  wrote:
> 
> Again the perfect example of I lost so I dodge…

Exactly. John Clark could not have provided a better illustration, indeed.

I consider a disagreement as a courtesy to pursue a conversation.

I consider mockery, insult and rhetorical dodging as “I have no argument, you 
won the point”.

Bruno



> 
> Le jeu. 3 mai 2018 21:36, John Clark  > a écrit :
> 
> On Thu, May 3, 2018 at 2:01 PM, Bruno Marchal  > wrote:
> 
> ​>> ​You say the diary solves the referent issue because its clear the man in 
> Helsinki wrote it and he wrote it yesterday, but in one variation of the 
> thought experiment there is nobody in Helsinki today, there are people in 
> Moscow and Washington who vividly remember writing that diary but what one 
> and only one really did?
> 
> ​> ​We assume Mechanism, so the answer is simply both, from the third person 
> point of view, and only one, for each of the first person point of view 
> obtained.
> 
> ​Counter argument #11​42
> 
> ​>>​What did the correct answer to the question turn out to be?
> 
> ​>​The question was the prediction of the next experience. The correct 
> answer, remaining correc,t through the experience was the prediction “I will 
> feel either W or M”, written “W v M”, keeping in mind that the question 
> concerned the experience at the first person.
> 
> So it is “W v M”.
> 
> ​Counter argument #926​ 
> 
> ​>>​Who wrote the diary?
> 
> ​> ​The candidate of the experience.
> 
> ​And who is the ​ candidate of the experience​? The guy who wrote the diary 
> of course.​
>  
> ​>> ​Is the one and only one referent to the personal pronoun “I” in the 
> question the Moscow man or the Washington man?
> 
> ​> ​Anyone. We keep only the prediction assessed by all of them. In the big 
> number iteration of that experience, the correct prediction is “white noise”.
> 
> ​Yes that's what I thought, hot air and a big noise.​
> 
> ​>> ​It can’t be the Helsinki man because today there is no Helsinki man.
> 
> ​> ​That contradicts the local personal identity definition that you have 
> agreed very often upon,
> 
> ​An oldie but a goodie, counter agreement #22​
> 
> ​John K Clark​ 
> 
> 
> 
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Re: What is a Löbian machine/number/combinator

[John Clark]

On Thu, May 3, 2018 at 7:26 PM, Quentin Anciaux  wrote:

​> ​
> Again the perfect example of I lost so I dodge...
>

After Bruno trots out the exact same argument he has 42 times before over
the last decade and I have pointed out 42 times exactly precisely why his
argument is dead wrong (and is charitable to call it a argument at all) I
figure that is quite sufficient. If an opinion is not based on logic then
logic can not destroy it and #43 is unlikely to do any good.
​

 John K Clark​

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Re: What is a Löbian machine/number/combinator

[Quentin Anciaux]

Again the perfect example of I lost so I dodge...

Le jeu. 3 mai 2018 21:36, John Clark  a écrit :

>
> On Thu, May 3, 2018 at 2:01 PM, Bruno Marchal  wrote:
>
> ​>> ​
>>> You say the diary solves the referent issue because its clear the man in
>>> Helsinki wrote it and he wrote it yesterday, but in one variation of the
>>> thought experiment there is nobody in Helsinki today, there are people in
>>> Moscow and Washington who vividly remember writing that diary but what one
>>> and only one really did?
>>
>>
>> ​> ​
>> We assume Mechanism, so the answer is simply both, from the third person
>> point of view, and only one, for each of the first person point of view
>> obtained.
>>
>
> ​Counter argument #11​42
>
> ​>>​
>>> What did the correct answer to the question turn out to be?
>>
>>
>> ​>​
>> The question was the prediction of the next experience. The correct
>> answer, remaining correc,t through the experience was the prediction “I
>> will feel either W or M”, written “W v M”, keeping in mind that the
>> question concerned the experience at the first person.
>>
>> So it is “W v M”.
>>
>
> ​Counter argument #926​
>
>
> ​>>​
>>> Who wrote the diary?
>>
>>
>> ​> ​
>> The candidate of the experience.
>>
>
> ​And who is the ​
>  candidate of the experience
> ​? The guy who wrote the diary of course.​
>
>
>> ​>> ​
>>> Is the one and only one referent to the personal pronoun “I” in the
>>> question the Moscow man or the Washington man?
>>
>>
>> ​> ​
>> Anyone. We keep only the prediction assessed by all of them. In the big
>> number iteration of that experience, the correct prediction is “white
>> noise”.
>>
>
> ​Yes that's what I thought, hot air and a big noise.​
>
>>
> ​>> ​
>>> It can’t be the Helsinki man because today there is no Helsinki man.
>>
>>
>> ​> ​
>> That contradicts the local personal identity definition that you have
>> agreed very often upon,
>>
>
> ​An oldie but a goodie, counter agreement #22​
>
> ​John K Clark​
>
>
>
>> --
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Re: What is a Löbian machine/number/combinator

[John Clark]

On Thu, May 3, 2018 at 2:01 PM, Bruno Marchal  wrote:

​>> ​
>> You say the diary solves the referent issue because its clear the man in
>> Helsinki wrote it and he wrote it yesterday, but in one variation of the
>> thought experiment there is nobody in Helsinki today, there are people in
>> Moscow and Washington who vividly remember writing that diary but what one
>> and only one really did?
>
>
> ​> ​
> We assume Mechanism, so the answer is simply both, from the third person
> point of view, and only one, for each of the first person point of view
> obtained.
>

​Counter argument #11​42

​>>​
>> What did the correct answer to the question turn out to be?
>
>
> ​>​
> The question was the prediction of the next experience. The correct
> answer, remaining correc,t through the experience was the prediction “I
> will feel either W or M”, written “W v M”, keeping in mind that the
> question concerned the experience at the first person.
>
> So it is “W v M”.
>

​Counter argument #926​


​>>​
>> Who wrote the diary?
>
>
> ​> ​
> The candidate of the experience.
>

​And who is the ​
 candidate of the experience
​? The guy who wrote the diary of course.​


> ​>> ​
>> Is the one and only one referent to the personal pronoun “I” in the
>> question the Moscow man or the Washington man?
>
>
> ​> ​
> Anyone. We keep only the prediction assessed by all of them. In the big
> number iteration of that experience, the correct prediction is “white
> noise”.
>

​Yes that's what I thought, hot air and a big noise.​

>
​>> ​
>> It can’t be the Helsinki man because today there is no Helsinki man.
>
>
> ​> ​
> That contradicts the local personal identity definition that you have
> agreed very often upon,
>

​An oldie but a goodie, counter agreement #22​

​John K Clark​



>

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Re: What is a Löbian machine/number/combinator

[Bruno Marchal]

> On 1 May 2018, at 20:44, Brent Meeker  wrote:
> 
> 
> 
> On 5/1/2018 8:52 AM, John Clark wrote:
>> 
>> 
>> On Mon, Apr 30, 2018 at 10:10 PM, Brent Meeker > > wrote:
>> 
>> ​> ​ Yes that's hard, perhaps meaningless, question to answer (and I don't 
>> think it's the question Bruno wants answered).  
>> 
>> ​ Then its not an experiment, its not a thought experiment, its not even a 
>> question its just a sequence of words with a question mark at the end. ​ 
> 
> 
> It's a rhetorical question to illustrate a consequence of Everett's relative 
> state.


No. It is a conceptual question needed to clarify to make sense of Everett but 
also of mechanism. In arithmetic we are duplicated all the times, and the wave 
(not just its apparent collapse) must be deduced from the statistical sum on 
all experiences of computational into account. 

If not, it means you invoke a God to select a computations, in a non computable 
way, nor FPI recoverable way, Which is already suspect. But that can be tested, 
and such a God does not seem to be (yet) detected, thank to QM (without 
collapse).

Bruno


> 
> Brent
> 
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Re: What is a Löbian machine/number/combinator

[Bruno Marchal]

> On 30 Apr 2018, at 18:09, John Clark  wrote:
> 
> You say the diary solves the referent issue because its clear the man in 
> Helsinki wrote it and he wrote it yesterday, but in one variation of the 
> thought experiment there is nobody in Helsinki today, there are people in 
> Moscow and Washington who vividly remember writing that diary but what one 
> and only one really did?


We assume Mechanism, so the answer is simply both, from the third person point 
of view, and only one, for each of the first person point of view obtained.




> What did the correct answer to the question turn out to be?


The question was the prediction of the next experience. The correct answer, 
remaining correc,t through the experience was the prediction “I will feel 
either W or M”, written “W v M”, keeping in mind that the question concerned 
the experience at the first person.

So it is “W v M”.



> Who wrote the diary?

The candidate of the experience.




> Is the one and only one referent to the personal pronoun “I” in the question 
> the Moscow man or the Washington man?

Anyone. We keep only the prediction assessed by all of them. In the big number 
iteration of that experience, the correct prediction is “white noise”.




> It can’t be the Helsinki man because today there is no Helsinki man.


That contradicts the local personal identity definition that you have agreed 
very often upon, and needed for the preceding step, to give sense that you ave 
survived the digital transplantation. In this duplicative case,  both the guy 
in M and in W are considered as digne living, survivors in the 
indexical-computationalist sense. They both are the Helsinki guy, but now in 
different contexts simultenously, which is not astonishing given that the 
Helsinki guy has been duplicated.

The personal identity is defined by the personal memories, well approximated by 
the content of the personal diary that the candidate take with him in the 
experience, and so are multiplied (from outside) although each first person 
obtained can count that it has only one diary at all times.


Bruno

“The most difficult subjects can be explained to the most slow-witted man if he 
has not formed any idea of them already; but the simplest thing cannot be made 
clear to the most intelligent man if he is firmly persuaded that he knows 
already, without a shadow of doubt, what is laid before him.” Leo Tolstoy.








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Re: What is a Löbian machine/number/combinator

[Brent Meeker]

On 5/1/2018 8:52 AM, John Clark wrote:



On Mon, Apr 30, 2018 at 10:10 PM, Brent Meeker > wrote:


​> ​
/Yes that's hard, perhaps meaningless, question to answer (and I
don't think it's the question Bruno wants answered). /


​
Then its not an experiment, its not a thought experiment, its not even 
a question its just a sequence of words with a question mark at the end.

​



It's a rhetorical question to illustrate a consequence of Everett's 
relative state.


Brent

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Re: What is a Löbian machine/number/combinator

[John Clark]

On Mon, Apr 30, 2018 at 10:10 PM, Brent Meeker  wrote:

​> ​
> *Yes that's hard, perhaps meaningless, question to answer (and I don't
> think it's the question Bruno wants answered).  *
>

​
Then its not an experiment, its not a thought experiment, its not even a
question its just a sequence of words with a question mark at the end.
​

 John K Clark​

 ​

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Re: What is a Löbian machine/number/combinator

[Brent Meeker]

On 4/30/2018 6:10 PM, John Clark wrote:
On Mon, Apr 30, 2018 at 2:56 PM, Brent Meeker >wrote:


​>/​/
/That's like saying that a man who took a plane from Helsinki to
Moscow who has a diary can't have been written by "the Helsinki
man" because he's no longer in Helsinki./


​No, its like saying if a man took a plane from Helsinki to Moscow 
*AND* at exactly the same time the same man took a plane from Helsinki 
to Washington then it would be silly to ask what one and only one city 
the man who wrote the diary ended up in. How could a person get on 2 
planes at the same time?  It's easy if you have a 
person duplicating machine.


Yes that's hard, perhaps meaningless, question to answer (and I don't 
think it's the question Bruno wants answered).   But it's not because 
the Helsinki man isn't in Helsinki anymore.


Brent

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Re: What is a Löbian machine/number/combinator

[John Clark]

On Mon, Apr 30, 2018 at 2:56 PM, Brent Meeker  wrote:

​>* ​*
> *That's like saying that a man who took a plane from Helsinki to Moscow
> who has a diary can't have been written by "the Helsinki man" because he's
> no longer in Helsinki.*
>

​No, its like saying if a man took a plane from Helsinki to Moscow *AND* at
exactly the same time the same man took a plane from Helsinki to Washington
then it would be silly to ask what one and only one city the man who wrote
the diary ended up in. How could a person get on 2 planes at the same
time?  It's easy if you have a person duplicating machine.

 John K Clark




>
>

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Re: What is a Löbian machine/number/combinator

[Brent Meeker]

On 4/30/2018 9:09 AM, John Clark wrote:
You say the diary solves the referent issue because its clear the man 
in Helsinki wrote it and he wrote it yesterday, but in one variation 
of the thought experiment there is nobody in Helsinki today, there are 
people in Moscow and Washington who vividly remember writing that 
diary but what one and only one really did? What did the correct 
answer to the question turn out to be? Who wrote the diary? Is the one 
and only one referent to the personal pronoun “I” in the question the 
Moscow man or the Washington man? It can’t be the Helsinki man because 
today there is no Helsinki man.


That's like saying that a man who took a plane from Helsinki to Moscow 
who has a diary can't have been written by "the Helsinki man" because 
he's no longer in Helsinki.


Brent

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Re: What is a Löbian machine/number/combinator

[John Clark]

On Mon, Apr 30, 2018 at 6:00 AM, Telmo Menezes 
wrote:

>
​> ​
>
> *Sometimes you just go silent on the topic and come back to it​ ​months
> later. This also happened with other topics before, I did not forget, even
> though I didn't comment.*
>

Is that something I'm suposed to be ashamed about?

> *any device that one can use to argue (e.g. the diary), you make fun of,
> with no arguments. The diary solves the referent issue,*

HOW? I don’t dispute the man in Helsinki wrote the diary and it correctly
reflects what the man in Helsinki thoughts emotions and expectations were
yesterday, but the question was about today not yesterday, and it was not
about what the man in Helsinki *expects* to happen tomorrow but what *will*
happen tomorrow: What one and only one city will “I”, the man who is in
Helsinki right “now”, see tomorrow?

You say the diary solves the referent issue because its clear the man in
Helsinki wrote it and he wrote it yesterday, but in one variation of the
thought experiment there is nobody in Helsinki today, there are people in
Moscow and Washington who vividly remember writing that diary but what one
and only one really did? What did the correct answer to the question turn
out to be? Who wrote the diary? Is the one and only one referent to the
personal pronoun “I” in the question the Moscow man or the Washington man?
It can’t be the Helsinki man because today there is no Helsinki man.

In another variation of the thought experiment the man in Helsinki remains
intact after the copies are made in Moscow and Washington, then we can say
what the answer turned out to be, not surprisingly it turned out the man in
Helsinki saw Helsinki today just as he did yesterday and if you didn’t tell
him he wouldn’t even know copies of himself had been made. Even the
Washington man and Moscow man agree that today the Helsinki man is seeing
Helsinki. At least in this variation the thought experiment is not
gibberish but I’ll be damned if I can find any deep philosophical insights
to be gained from it. And I don’t see what the diary does except to add yet
more wheels within a wheels to a already pointless exercise.

> *> personal pronouns are part of language. You don't get to dictate the
> rules on how people​ ​communicate.*


True, but I do get to criticize logicians who don’t use language logically
and throw around personal pronouns exactly as they do in everyday life even
though they're talking about hyper-exotic thought experiments with personal
pronouns duplicating machines. I tell Bruno if my complaints are baseless
then shut me up by simply using proper nouns instead of personal pronouns,
but of course he can’t do that, so instead he starts talking about “*THE*
1p”, but in a world that contains 1p duplicating machines there is no such
thing as *THE* 1p there is only *A* 1p.

> *There are endless acronyms and technical lingo in all of the scientific
> fields that I am familiar with*

Yeah but in real scientific papers the author of the paper didn't invent
them all, and they are used in places other than this one tiny list, and
they don’t have pompous sounding terms like "first person indeterminacy”
which just means I don’t know what’s going to happen next.

​ ​
John K Clark

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Re: What is a Löbian machine/number/combinator

[Telmo Menezes]

On 23 April 2018 at 16:14, John Clark  wrote:
> On Mon, Apr 23, 2018 at 7:09 AM, Telmo Menezes 
> wrote:
>
>> > The above is about how Bruno keeps answering your questions, and then
>> > you either pretend that he didn't
>
>
> Like the time just a few days ago when Bruno said all my objections were
> already answered in a post that could be "easily" found in the archives, and
> when I asked exactly where it was he said he couldn't be expected to find it
> because he has sent thousands of posts?

There is no point in pointing you to the answers once again. This has
been tried before. This first moment you realize that you may lose the
argument, you switch to direct insults, or to making fun or some word,
etc. Sometimes you just go silent on the topic and come back to it
months later. This also happened with other topics before, I did not
forget, even though I didn't comment.

>> > or make fun of his mode of expression, with your "pee pees" and
>> > "homemade terms"
>
> That is not Ad Hominem that is a statement of fact, his pee pee notation

Ok, I have been wasting my time. If you really believe that the above
sentences represent "statements of fact", then you are too uneducated
about what serious scientific debate is. You would need to learn some
basic things before there being a point in continuing with this. I
wonder how Bruno can be so patient.

> with its circular definitions and personal pronouns with no unique referent

As per above, this has been debated to death. You simply ignore all
the answers. It could be that I am wrong and you are right, that is
always a possibility, but you are not serious. You are a bully,
because any device that one can use to argue (e.g. the diary), you
make fun of, with no arguments.

The diary solves the referent issue, and personal pronouns are part of
language. You don't get to dictate the rules on how people
communicate.

> is homemade,

What does this even mean? I know what you want it to mean: you want to
evoke the image of Bruno as an isolated crank thinking of crazy ideas
alone in his home. It's just another insult and bully tactic.

What is the opposite of a homemade idea? An idea created by committee?
Under permission from a higher authority? Following certain rules and
regulations? Trickling down from the organization chart of some agency
or corporation? All ideas worth talking about are homemade, in any
sense that I can imagine.

> and so are his endless acronyms

There are endless acronyms and technical lingo in all of the
scientific fields that I am familiar with. They are there always for
the same reason: because people keep debating around the same
concepts, so they create shorthand notations. This doesn't even mean
that they agree with each other, it just means that they are willing
to make an effort to communicate. Many acronyms appeared naturally on
this mailing list before you arrived. Russell wrote a book that
mentions some of them -- this is how I found the list. Then you arrive
one day and you decide you get to make the rules?

> that he seems to expect any
> scientifically literate person should know when in fact they are seen on
> this very tiny list and nowhere else on the planet. Why would somebody who
> had a clear idea that was very good muddy things up by doing that? They
> wouldn't, therefore the idea must not be clear and it must not be very good.
>>
>> > You frequently brag about not reading after the first line,
>
> If you find a blunder in a proof only a fool would keep reading because a
> proof builds on what comes before so everything after that point is pure
> nonsense.

You did not find a blunder, you attempted to invent one.

> As for bragging,... I don't claim to be a genius but I do claim
> not to be a fool.

In my experience, wise people tend to ask themselves if they are being foolish.

Telmo.

>  John K Clark
>
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Re: What is a Löbian machine/number/combinator

[Bruno Marchal]

> On 27 Apr 2018, at 21:56, John Clark  wrote:
> 
> 
> On Fri, Apr 27, 2018 at 11:52 AM, Bruno Marchal  > wrote:
> 
>  ​> ​That step is very simple,
> 
> ​Yes, simple as in stupid.​ 
> 
> ​> ​accessible by very young people without any knowledge
> 
> Because very young people with no knowledge never stopped to think exactly 
> what personal pronouns mean in a world that contains people duplicating 
> machines. 
>  
> ​> ​it is made by defining the first person discourse made by a robot, say, 
> and which (the diary)
> 
> The only thing stupider than that step is the diary.
>  
> ​> ​Of course, I ahem to rehash this from time to time,
>  
> Why don't we make it easy on ourselves and give numbers to our old ideas that 
> we sent to the list years ago? That way you could for example say "argument 
> #11392" and I could just say "counter-argument #11393". Think of all the wear 
> and tear on out typing fingers we could save!
> 
> ​ John K Clark​


Insulting is not a valid argument. It is the “argument" of those having no 
argument.

Bruno


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Re: What is a Löbian machine/number/combinator

[Russell Standish]

On Fri, Apr 27, 2018 at 03:56:46PM -0400, John Clark wrote:
> 
> Why don't we make it easy on ourselves and give numbers to our old ideas
> that we sent to the list years ago? That way you could for example say
> "argument #11392" and I could just say "counter-argument #11393". Think of
> all the wear and tear on out typing fingers we could save!
> 

If #11392 could be turned into a URL that points to the actual
argument in question, that would be a fantastic idea! Unfortunately,
it is a lot of work in practice - I provided a number of such links
into the archive in my book.


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Principal, High Performance Coders
Visiting Senior Research Fellowhpco...@hpcoders.com.au
Economics, Kingston University http://www.hpcoders.com.au


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Re: What is a Löbian machine/number/combinator

[John Clark]

On Fri, Apr 27, 2018 at 11:52 AM, Bruno Marchal  wrote:


> ​> ​
> *That step is very simple,*
>

​Yes, simple as in stupid.​


​>* ​*
> *accessible by very young people without any knowledge*


Because very young people with no knowledge never stopped to think exactly
what personal pronouns mean in a world that contains people duplicating
machines.


> ​> ​
> it is made by defining the first person discourse made by a robot, say,
> and which (the diary)
>

The only thing stupider than that step is the diary.


> ​> ​
> Of course, I ahem to rehash this from time to time,
>

Why don't we make it easy on ourselves and give numbers to our old ideas
that we sent to the list years ago? That way you could for example say
"argument #11392" and I could just say "counter-argument #11393". Think of
all the wear and tear on out typing fingers we could save!

​ John K Clark​

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Re: What is a Löbian machine/number/combinator

[Bruno Marchal]

> On 26 Apr 2018, at 19:53, John Clark  wrote:
> 
> 
> 
> On Thu, Apr 26, 2018 at 4:18 AM, Bruno Marchal  > wrote:
> 
> ​> ​If you have an argument against step 3 just show it.
> 
> ​FOR GODS SAKE You want to go back years to day one when this entire 
> idiotic conversation started and rehash it all over again, but I'd rather 
> watch paint dry.

That is what you tell us the last time you were debunked, and it is normal. 
That step is very simple, and like M. Jones said, accessible by very young 
people without any knowledge. When it is made by defining the first person 
discourse made by a robot, say, and which (the diary) enter the 
annihilation/copy box, the verification is entirely third person describable. 
That made you oscillate between trivial and non-sensical. But when saying 
trivial, you did not explain why you did not move on the next step (4).

I have not the time for now. But I intend to motivate and explain this to the 
new people. We will see if they found what is “wrong”.

Of course, I ahem to rehash this from time to time, as it is the key of the 
whole reversal between physics and number psychology or theology, especially 
for those who are not familiar with mathematical logic (the second part can use 
the first part as only a motivation).

Bruno



> 
>  John K Clark
> 
> 
>  
> 
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Re: What is a Löbian machine/number/combinator

[Lawrence Crowell]

On Thursday, April 26, 2018 at 3:09:03 AM UTC-5, Bruno Marchal wrote:
>
>
> On 22 Apr 2018, at 18:04, Lawrence Crowell  > wrote:
>
> With a T a theory that admits diagonalization and for Bew(x) a formula 
> with a free x,.we have 
>
> ├_T S ↔ Bew(gn(S)).
>
> The Löb theorem involves the diagonalization in T D(x,y) such that for any 
> D(x,y) = k is the Gödel number diag(x) = k and y = k. This corresponds I 
> think to the φ_u(x,y). It is some algebra to show this leads to the 
> equation above.
>
> The  Löb theorem if├_T Bew(gn(S)) → S then├_T S has parallels with the 
> modal logical
>
> □(□S → S) →  □S,
>
> which is a way of saying that if □S → S then S. 
>
>
> It is a way of saying that if □S → S is *provable*, then S is provable. 
>

>
> This is a fancy way of just saying that if a statement S is provable then 
> S holds. 
>
>
> ?
>
> The Löb formula says the contrary. It says for example with S = f (false), 
> that if the machine is consistent (~provable(f), i.e []f -> f), then f is 
> provable. So if the machine is could prove []f -> f it would prove f and be 
> inconsistent.
>
>
>
>
Yes, that is now S is interpreted. 

I do not have time to go into this discussion right now. I will try to get 
back in a day or so.

LC
 

>
>
> In part this corroborates with what you write. I would say the axiom of 
> reflection, if I recall the name for it,  □S → S is usually thought of as 
> an axiom. 
>
>
>
> It is an axiom of the soul (SAGrz) and of the Noùs (G*), but the machine 
> cannot prove it. That is why we can apply the idea of Theaetetus. As 
> typically []p -> p is not provable, it makes sense to define knowledge by 
> “[]p & p”, like in Plato. That gives a modal logic of knowledge, but by 
> Tarski (and variants), that cannot be defined by the machine, which is 
> nice, as it confirms Brouwer theory of the mental.
>
>
>
>
> In the  Löb theorem we appear to have instances where maybe this might not 
> hold.
>
>
> Not maybe. Certainly. Typical cases []f -> f is not provable. []<>[]f -> 
> <>[]f is not provable, etc.
>
>
>
> If we think of the complement, with ¬ = NOT, is
>
> ¬□S → ¬□(□S → S) 
>
> equal to
>
> ¬□¬¬S → ¬□¬¬(□S → S) 
>
> or for ¬□¬ = ◊, non necessarily not = possibly, we then have
>
> ◊¬S → ◊¬(□S → S) or
>
> ◊¬S → ◊(¬S → ¬□S)  
>
>
> (that line will be false when we do the sigma_1 restriction!)
>
>
>
> ◊¬S → ◊(¬S → ◊¬S)
>
>
> OK. That is almost the dual presentation of Löb’s formula, but it will not 
> work on the sigma_1 (semi-computable) restriction.
>
> Here, out of that restriction, you could use ~S instead of S, so that you 
> have  ◊S → ◊(S → ◊S)   
>
>
> with the conclusion that ¬S → (¬S → ◊¬S). The ◊ = possibly means we have 
> an open door of sorts. We do not have the falsity of S implying logically 
> some proof thereof.
>
>
>
> This means that incompleteness entails the platonic nuances []p & p, []p & 
> <>p, … That plays a key role in the derivation of physics from arithmetic 
> (as imposed by Digital Mechanism).
>
> Bruno
>
>
>
>
>
> LC
>
> On Wednesday, April 18, 2018 at 12:11:35 PM UTC-5, Bruno Marchal wrote:
>>
>> Somewhere: (and I copy my answer, as some people asked me this in this 
>> list too).
>>
>>
>>
>> What are Lobian numbers? Can you give a reference? I know little bit 
>> about Godel’s work.
>>
>>
>>
>> Consider any Turing universal machinery, for example the programming 
>> language c++. 
>>
>> N is the set of natural numbers.
>>
>> It is known that the enumeration of all programs computing a (perhaps not 
>> everywhere defined) function from N to N exists, and so we get a list of 
>> all partial computable function phi_i from N to N. (i.e. phi_0, phi_1, 
>> phi_2, …), by enumerating the program with one natural number argument) 
>> written in C++, in their lexico-graphical order (length, and alphabetical 
>> for the programs with the same length).
>>
>> We can define a universal number as a number u such that phI_u(x, y) = 
>> phi_x(y). We say that u implements x on y. (It is a constructive definition 
>> of a computer in the language of the computer).
>>
>> Now, once we have a universal number, we can transform/extend it into a 
>> theory, which is the first order logical specification of how u operates. 
>> That is a standard mapping from, say, c++ to a Turing universal logical 
>> theory. 
>>
>> I assume we have done that, so now I say that a universal number is 
>> Löbian when it has enough induction axioms (added to its logical 
>> specification) so that it can prove enough of some special formula. 
>>
>> If “[]” represents the provability predicate (Gödel 1931)of some first 
>> order Turing universal theory/number, Löbian means that it can prove p -> 
>> []p for all p equivalent with a semi-computable predicate known as sigma_1 
>> predicate). In fact “p -> []p” is equivalent with Turing universality, and 
>> if a Universal can prove this for all p sigma_1, it will not only be Turing 
>> universal, but it will know (in some 

Re: What is a Löbian machine/number/combinator

[John Clark]

On Thu, Apr 26, 2018 at 4:18 AM, Bruno Marchal  wrote:

​> *​*
> *If you have an argument against step 3 just show it.*
>

​*FOR GODS SAKE *
You want to go back years to day one when this entire idiotic conversation
started and rehash it all over again, but I'd rather watch paint dry.

 John K Clark

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Re: What is a Löbian machine/number/combinator

[Bruno Marchal]

> On 23 Apr 2018, at 16:14, John Clark  wrote:
> 
> On Mon, Apr 23, 2018 at 7:09 AM, Telmo Menezes  > wrote:
> 
> > The above is about how Bruno keeps answering your questions, and then you 
> > either pretend that he didn't
>  
> Like the time just a few days ago when Bruno said all my objections were 
> already answered in a post that could be "easily" found in the archives, and 
> when I asked exactly where it was he said he couldn't be expected to find it 
> because he has sent thousands of posts?


If you have an argument against step 3 just show it. All those everybody has 
seen dismissed the use of the diaries and the 1P/3p distinction systematically, 
and that has been debunked by many participants in the list, but you keep 
calling that by name, which is an invalid way to proceed (to say the least).

Bruno



> 
> > or make fun of his mode of expression, with your "pee pees" and "homemade 
> > terms"
> That is not Ad Hominem that is a statement of fact, his pee pee notation with 
> its circular definitions and personal pronouns with no unique referent is 
> homemade, and so are his endless acronyms that he seems to expect any 
> scientifically literate person should know when in fact they are seen on this 
> very tiny list and nowhere else on the planet. Why would somebody who had a 
> clear idea that was very good muddy things up by doing that? They wouldn't, 
> therefore the idea must not be clear and it must not be very good. 
> 
> > You frequently brag about not reading after the first line,
> If you find a blunder in a proof only a fool would keep reading because a 
> proof builds on what comes before so everything after that point is pure 
> nonsense.  As for bragging,... I don't claim to be a genius but I do claim 
> not to be a fool.
> 
>  John K Clark 
> 
> 
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Re: What is a Löbian machine/number/combinator

[Bruno Marchal]

> On 22 Apr 2018, at 19:28, John Clark  wrote:
> 
> On Sun, Apr 22, 2018 at 12:21 AM, Russell Standish  > wrote:
> 
> >> How can I determine if that particular Turing Machine is doing something 
> >> fundamentally different from what every other Turing Machine is doing?
> 
> > I would say that it is a machine that proves Loeb's theorem.
> A machine that could show that Loeb’s theorem was consistent with 
> Zermelo–Fraenkel set theory plus the axiom of choice and its negation was not 
> but could do nothing else would display no more general intelligence than a 
> chess program or a checkers program or even a tic tac toe program. 
> 
> 

?

Löb’s theorem is a scheme of theorem of arithmetic, or ZF. That has to be as 
consistent than arithmetic or set theory.


> > Not all Turing machines are capable of that, even universal machines absent 
> > the right software.
> Without the right software a universal machine is not capable of doing 
> anything,
> 

?

A universal machine is the right software able to do any computation. But of 
course not all proof. It need the scheme of induction formulas. RA and PA are 
Turing universal, but only PA is Löbian (it can prove its own universality and 
the consequences of it).


> with the right software it can calculate anything that can be calculated 
> including loeb’s theorem.
> 
> 


That does not make sense. Usual confusion compute/prove.

Bruno

> 
>  >​>​ There is no way I can ever know if Hod Lipson 's robots are self aware, 
> I don't even know if Hod Lipson is self aware, all I know for sure is that 
> both behave intelligently. 
>  
> > His argument is that his robot is self-aware, for some operational 
> > definition of self-aware.
> If his definition is operational it must involve intelligent behavior, I 
> don’t see how it could be otherwise.
> 
> 
> ​ ​John K Clark
> 
> 
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Re: What is a Löbian machine/number/combinator

[Bruno Marchal]

> On 22 Apr 2018, at 18:04, Lawrence Crowell  
> wrote:
> 
> With a T a theory that admits diagonalization and for Bew(x) a formula with a 
> free x,.we have 
> 
> ├_T S ↔ Bew(gn(S)).
> 
> The Löb theorem involves the diagonalization in T D(x,y) such that for any 
> D(x,y) = k is the Gödel number diag(x) = k and y = k. This corresponds I 
> think to the φ_u(x,y). It is some algebra to show this leads to the equation 
> above.
> 
> The  Löb theorem if├_T Bew(gn(S)) → S then├_T S has parallels with the modal 
> logical
> 
> □(□S → S) →  □S,
> 
> which is a way of saying that if □S → S then S.

It is a way of saying that if □S → S is *provable*, then S is provable. 


> This is a fancy way of just saying that if a statement S is provable then S 
> holds. 

?

The Löb formula says the contrary. It says for example with S = f (false), that 
if the machine is consistent (~provable(f), i.e []f -> f), then f is provable. 
So if the machine is could prove []f -> f it would prove f and be inconsistent.




> 
> In part this corroborates with what you write. I would say the axiom of 
> reflection, if I recall the name for it,  □S → S is usually thought of as an 
> axiom.


It is an axiom of the soul (SAGrz) and of the Noùs (G*), but the machine cannot 
prove it. That is why we can apply the idea of Theaetetus. As typically []p -> 
p is not provable, it makes sense to define knowledge by “[]p & p”, like in 
Plato. That gives a modal logic of knowledge, but by Tarski (and variants), 
that cannot be defined by the machine, which is nice, as it confirms Brouwer 
theory of the mental.




> In the  Löb theorem we appear to have instances where maybe this might not 
> hold.

Not maybe. Certainly. Typical cases []f -> f is not provable. []<>[]f -> <>[]f 
is not provable, etc.



> If we think of the complement, with ¬ = NOT, is
> 
> ¬□S → ¬□(□S → S) 
> 
> equal to
> 
> ¬□¬¬S → ¬□¬¬(□S → S) 

> or for ¬□¬ = ◊, non necessarily not = possibly, we then have
> 
> ◊¬S → ◊¬(□S → S) or
> 
> ◊¬S → ◊(¬S → ¬□S)  
> 

(that line will be false when we do the sigma_1 restriction!)



> ◊¬S → ◊(¬S → ◊¬S)

OK. That is almost the dual presentation of Löb’s formula, but it will not work 
on the sigma_1 (semi-computable) restriction.

Here, out of that restriction, you could use ~S instead of S, so that you have  
◊S → ◊(S → ◊S)   

> 
> with the conclusion that ¬S → (¬S → ◊¬S). The ◊ = possibly means we have an 
> open door of sorts. We do not have the falsity of S implying logically some 
> proof thereof.


This means that incompleteness entails the platonic nuances []p & p, []p & <>p, 
… That plays a key role in the derivation of physics from arithmetic (as 
imposed by Digital Mechanism).

Bruno




> 
> LC
> 
> On Wednesday, April 18, 2018 at 12:11:35 PM UTC-5, Bruno Marchal wrote:
> Somewhere: (and I copy my answer, as some people asked me this in this list 
> too).
> 
> 
>> 
>> What are Lobian numbers? Can you give a reference? I know little bit about 
>> Godel’s work.
> 
> 
> Consider any Turing universal machinery, for example the programming language 
> c++. 
> 
> N is the set of natural numbers.
> 
> It is known that the enumeration of all programs computing a (perhaps not 
> everywhere defined) function from N to N exists, and so we get a list of all 
> partial computable function phi_i from N to N. (i.e. phi_0, phi_1, phi_2, …), 
> by enumerating the program with one natural number argument) written in C++, 
> in their lexico-graphical order (length, and alphabetical for the programs 
> with the same length).
> 
> We can define a universal number as a number u such that phI_u(x, y) = 
> phi_x(y). We say that u implements x on y. (It is a constructive definition 
> of a computer in the language of the computer).
> 
> Now, once we have a universal number, we can transform/extend it into a 
> theory, which is the first order logical specification of how u operates. 
> That is a standard mapping from, say, c++ to a Turing universal logical 
> theory. 
> 
> I assume we have done that, so now I say that a universal number is Löbian 
> when it has enough induction axioms (added to its logical specification) so 
> that it can prove enough of some special formula. 
> 
> If “[]” represents the provability predicate (Gödel 1931)of some first order 
> Turing universal theory/number, Löbian means that it can prove p -> []p for 
> all p equivalent with a semi-computable predicate known as sigma_1 
> predicate). In fact “p -> []p” is equivalent with Turing universality, and if 
> a Universal can prove this for all p sigma_1, it will not only be Turing 
> universal, but it will know (in some technical sense) that it is Turing 
> Universal.
> 
> “[]” itself is sigma_1, which entails that []p -> [][]p is provable.
> 
> Those corresponds to what is called “sufficiently rich theories” (for proving 
> their own incompleteness theorem).
> 
> Löbianity appears when you add to:
> 
> 0 ≠ s(x)
> s(x) = s(y) -> x = y

Re: What is a Löbian machine/number/combinator

[John Clark]

On Mon, Apr 23, 2018 at 7:09 AM, Telmo Menezes  wrote:

> *The above is about how Bruno keeps answering your questions, and then
> you either pretend that he didn't*


Like the time just a few days ago when Bruno said all my objections were
already answered in a post that could be "easily" found in the archives,
and when I asked exactly where it was he said he couldn't be expected to
find it because he has sent thousands of posts?

> *or make fun of his mode of expression, with your "pee pees" and
> "homemade terms"*

That is not Ad Hominem that is a statement of fact, his pee pee notation
with its circular definitions and personal pronouns with no unique referent
is homemade, and so are his endless acronyms that he seems to expect any
scientifically literate person should know when in fact they are seen on
this very tiny list and nowhere else on the planet. Why would somebody who
had a clear idea that was very good muddy things up by doing that? They
wouldn't, therefore the idea must not be clear and it must not be very
good.

> > *You frequently brag about not reading after the first line,*

If you find a blunder in a proof only a fool would keep reading because a
proof builds on what comes before so everything after that point is pure
nonsense.  As for bragging,... I don't claim to be a genius but I do claim
not to be a fool.

 John K Clark

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Re: What is a Löbian machine/number/combinator

[Telmo Menezes]

On 22 April 2018 at 19:06, John Clark  wrote:
> On Sun, Apr 22, 2018 at 3:21 AM, Telmo Menezes 
> wrote:
>>>
>>> >>  Maybe I've got my Latin wrong, please explain to me again what "ad
>>> >> hominem" means.
>>
>>
>> > It means to use personal attacks against the author of an argument
>> > instead of addressing the substance of the argument.
>
> Like this?
>
> "I think you make a basic logic mistake. It is true that some brilliant
> people are assholes, but being an asshole does not make you brilliant.”

No. "Ad Hominem" refers to ignoring an argument and instead going for
the personal attack. I never ignored your arguments, I replied to all
of them. The above is about how Bruno keeps answering your questions,
and then you either pretend that he didn't or make fun of his mode of
expression, with your "pee pees" and "homemade terms" and all the
rest. You frequently brag about not reading after the first line, and
then you spare no unfounded criticism. You lie frequently about him
not answering you, as Quentin pointed out before. My problem with you
is not that you disagree with me, it's that you sabotage communication
on purpose.

> I have never called anybody on this list an asshole, and before the era of
> Donald Trump I very rarely used that word in any context even in private
> conversations. It’s true I did use some colorful language in reference to
> Quentin but only after he called me a liar nearly every day for the better
> part of a year.

For me it took seven years of witnessing your bully strategies.

Telmo.

>   John K Clark
>
>
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Re: What is a Löbian machine/number/combinator

[John Clark]

On Sun, Apr 22, 2018 at 12:21 AM, Russell Standish  wrote:

> >> How can I determine if that particular Turing Machine is doing
>> something fundamentally different from what every other Turing Machine is
>> doing?
>
>
> >* I would say that it is a machine that proves Loeb's theorem.*

A machine that could show that Loeb’s theorem was consistent
with Zermelo–Fraenkel set theory plus the axiom of choice and its negation
was not but could do nothing else would display no more general
intelligence than a chess program or a checkers program or even a tic tac
toe program.

> *> Not all Turing machines are capable of that, even universal machines
> absent the right software.*

Without the right software a universal machine is not capable of doing
anything, with the right software it can calculate anything that can be
calculated including loeb’s theorem.


 >
>> ​>​
>> There is no way I can ever know if Hod Lipson 's robots are self aware, I
>> don't even know if Hod Lipson is self aware, all I know for sure is that
>> both behave intelligently.
>
>

> *His argument is that his robot is self-aware, for some operational
> definition of self-aware.*

If his definition is operational it must involve intelligent behavior, I
don’t see how it could be otherwise.

​ ​
John K Clark

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Re: What is a Löbian machine/number/combinator

[John Clark]

On Sun, Apr 22, 2018 at 3:21 AM, Telmo Menezes  wrote:

> >>  Maybe I've got my Latin wrong, please explain to me again what "ad
>> hominem" means.
>
>
> > *It means to use personal attacks against the author of an argument
> instead of addressing the substance of the argument.*

Like this?

*"I think you make a basic logic mistake. It is true that some brilliant
people are assholes, but being an asshole does not make you brilliant.”*

I have never called anybody on this list an asshole, and before the era of
Donald Trump I very rarely used that word in any context even in private
conversations. It’s true I did use some colorful language in reference to
Quentin but only after he called me a liar nearly every day for the better
part of a year.

  John K Clark

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Re: What is a Löbian machine/number/combinator

[Lawrence Crowell]

With a T a theory that admits diagonalization and for Bew(x) a formula with 
a free x,.we have 

├_T S ↔ Bew(gn(S)).

The Löb theorem involves the diagonalization in T D(x,y) such that for any 
D(x,y) = k is the Gödel number diag(x) = k and y = k. This corresponds I 
think to the φ_u(x,y). It is some algebra to show this leads to the 
equation above.

The  Löb theorem if├_T Bew(gn(S)) → S then├_T S has parallels with the 
modal logical

□(□S → S) →  □S,

which is a way of saying that if □S → S then S. This is a fancy way of just 
saying that if a statement S is provable then S holds. 

In part this corroborates with what you write. I would say the axiom of 
reflection, if I recall the name for it,  □S → S is usually thought of as 
an axiom. In the  Löb theorem we appear to have instances where maybe this 
might not hold. If we think of the complement, with ¬ = NOT, is

¬□S → ¬□(□S → S) 

equal to

¬□¬¬S → ¬□¬¬(□S → S) 

or for ¬□¬ = ◊, non necessarily not = possibly, we then have

◊¬S → ◊¬(□S → S) or

◊¬S → ◊(¬S → ¬□S) or 

◊¬S → ◊(¬S → ◊¬S)

with the conclusion that ¬S → (¬S → ◊¬S). The ◊ = possibly means we have an 
open door of sorts. We do not have the falsity of S implying logically some 
proof thereof.

LC

On Wednesday, April 18, 2018 at 12:11:35 PM UTC-5, Bruno Marchal wrote:
>
> Somewhere: (and I copy my answer, as some people asked me this in this 
> list too).
>
>
>
> What are Lobian numbers? Can you give a reference? I know little bit about 
> Godel’s work.
>
>
>
> Consider any Turing universal machinery, for example the programming 
> language c++. 
>
> N is the set of natural numbers.
>
> It is known that the enumeration of all programs computing a (perhaps not 
> everywhere defined) function from N to N exists, and so we get a list of 
> all partial computable function phi_i from N to N. (i.e. phi_0, phi_1, 
> phi_2, …), by enumerating the program with one natural number argument) 
> written in C++, in their lexico-graphical order (length, and alphabetical 
> for the programs with the same length).
>
> We can define a universal number as a number u such that phI_u(x, y) = 
> phi_x(y). We say that u implements x on y. (It is a constructive definition 
> of a computer in the language of the computer).
>
> Now, once we have a universal number, we can transform/extend it into a 
> theory, which is the first order logical specification of how u operates. 
> That is a standard mapping from, say, c++ to a Turing universal logical 
> theory. 
>
> I assume we have done that, so now I say that a universal number is Löbian 
> when it has enough induction axioms (added to its logical specification) so 
> that it can prove enough of some special formula. 
>
> If “[]” represents the provability predicate (Gödel 1931)of some first 
> order Turing universal theory/number, Löbian means that it can prove p -> 
> []p for all p equivalent with a semi-computable predicate known as sigma_1 
> predicate). In fact “p -> []p” is equivalent with Turing universality, and 
> if a Universal can prove this for all p sigma_1, it will not only be Turing 
> universal, but it will know (in some technical sense) that it is Turing 
> Universal.
>
> “[]” itself is sigma_1, which entails that []p -> [][]p is provable.
>
> Those corresponds to what is called “sufficiently rich theories” (for 
> proving their own incompleteness theorem).
>
> Löbianity appears when you add to:
>
> 0 ≠ s(x)
> s(x) = s(y) -> x = y
> x = 0 v Ey(x = s(y))
> x+0 = x
> x+s(y) = s(x+y)
> x*0=0
> x*s(y)=(x*y)+x,
>
> The induction axioms:
>
> (F(0) & Ax(F(x) -> F(s(x))) -> AxF(x), with F(x) being a formula in the 
> arithmetical language (with "0, s, +, *)
>
>
> F being a formula belonging to some set of formula. If you limit F to the 
> recursive, sigma_0, formula, you don’t get Löbianity, unless you add the 
> exponentiation axioms.
>
> You can (and I will) limit p to the sigma_1 sentences, the semi-computable 
> predicate/function. That is enough to get Löbianity, and inherit, in the 
> “ideal” sound case the “theology” of number/machine/combinator… beings.
>
> With p sigma_1 Universality means that p_>[]p is true, and Löbianity is 
> when the machine/number proves p -> []p for all p (sigma_1).
>
> []p -> p, although true (by definition of sound machine/number) remains 
> unprovable in general. Typically the Löbian machine cannot prove []f -> f.
>
>
> Peano is a Löbian theory/program/idea/machine/word Universal).
>
> ZF too, much more “crazy machine” which believes in the axiom of infinity, 
> but then get doubt about the choice axioms!
> (As I stay in very elementary arithmetic (no induction axioms) I still 
> studies the web of Löbian dreams realised in the non Löbian reality.
>
>
> Provability is relative, but computability is absolute. Sigma_1 
> completeness, that is the truth of p -> []p, for p sigma_1, is Turing 
> universal.
> Löbianity is when the machine believes in enough induction axioms to prove 
> p -> []p for each p sigma_1. 
>
> It 

Re: What is a Löbian machine/number/combinator

[Bruno Marchal]

> On 22 Apr 2018, at 09:21, Telmo Menezes  wrote:
> 
> On 22 April 2018 at 01:47, John Clark  wrote:
>> On Sat, Apr 21, 2018 at 5:27 PM, Telmo Menezes 
>> wrote:
>> 
>>> 
 
>>> As Russell said, an approximation of the Löbian machine can probably be
>>> derived from Bruno's post in Prolog.
>> 
>> 
>> Then a "Löbian machine" is just a particular Turing Machine and there is 
>> nothing fundamentally new about it.
> 
> If computationalism is true, then human brains are just particular
> Turing Machines. Better tell neuroscientists to cease their
> investigations, turns out their entire field is uninteresting. Come
> on...
> 
>>> I am complaining about
>>> 
>>> personal insults. For example, in the sentence above you insult both
>>> me and Bruno without providing anything of substance.
>> 
>> 
>> Mr.Snowflake, I'll let others decide if I am a bully or not as you claim,
>> but I maintain it is a fact you can't handle scientific criticism, and it it
>> is true then is a statement with substance.
> 
> I have had papers rejected and it doesn't feel good. I had to cry
> myself to sleep with a bucket of ice cream. What can I say?
> 
>>> you used the classical bully technique ofmaking fun of his mode of 
>>> expression with "ad hominem".
>> 
>> 
>> I think it would be fair to say nobody on this list has received more
>> personal insults than I have, but I have never once used that ridiculous
>> Latin phrase.
> 
> Perhaps because nobody on this mailing list has dished out as many
> personal insults as you have?
> 
 
>>> I think you make a basic logic mistake. It is true that some brilliant
>>> 
>>> people are assholes, but being an asshole does not make you brilliant.
>> 
>> 
>> Maybe I've got my Latin wrong, please explain to me again what "ad hominem"
>> means.
> 
> It means to use personal attacks against the author of an argument
> instead of addressing the substance of the argument. It's exactly what
> you do to Bruno all the time.
> 

Many thanks Telmo. The bad guys are usually not the main problem, but those who 
see and say nothing, are fundamentally worst, be it on cannabis, jews, sexual 
or moral harassement.

Best!

Bruno



> Telmo.
> 
>> John K Clark
>> 
>> 
>> 
>> 
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Re: What is a Löbian machine/number/combinator

[Bruno Marchal]

> On 22 Apr 2018, at 06:21, Russell Standish  wrote:
> 
> On Sat, Apr 21, 2018 at 08:08:50PM -0400, John Clark wrote:
>> On Sat, Apr 21, 2018 at 6:15 PM, Russell Standish 
>> wrote:
>> 
>> ​> ​
>>> *Yes, of course a Loebian machine is a type of Turing machine.*
>> 
>> 
>> How can I determine if that particular Turing Machine is doing something
>> fundamentally different from what every other Turing Machine is doing?
>> 
> 
> I would say that it is a machine that proves Loeb's theorem. Not all
> Turing machines are capable of that, even universal machines absent
> the right software. But I may have misunderstood this :).

Not at all. That is correct. 

The main difference can be sum up by saying that a machine, having a 
believability predicate, noted “[]”

1) is universal if p -> []p is true for them on all p sigma_1. (And that can be 
proved equivalent with “identifiable” with  a number u such that phi_u(x, y) = 
phi_x(y). 
For example “p -> []p” is true for RA = [], and indeed RA can compute all 
phi_x(y) for any enumeration of the phi_i.

2) is a Löbian machine if it is a universal machine which can prove its own 
universality and the consequences, like its own incompleteness, in particular 
Gödel and Löb theorems, and much more.
That means that is is a believer-machine “[]", for which not only p -> []p is 
true, but the machine is rich enough (in term of beliefs) to be able to prove 
it.

The main ingredient to become Löbian, is in believing enough induction axioms.

p sigma_1 means that p has the shape ExP(x, y), with P decidable (sigma_0). If 
you can convince yourself that you can search for a number having some 
decidable property P, and find it if it exists, with no bounding time of 
research, than you are Löbian. Note that all Löbian entity can know that they 
are Löbian, bit none can known that they are consistent, and none can define 
their own soundness.

Universal machine have logical limitation, and Löbian machine are the one 
knowing that they are universal, and the limitations this entails.

The price to pay for being universal, as I have explained a lot some years ago, 
is that you need the ability to search numbers which might not exist, and this 
never stop, without knowing it for sure.

Machines are like kids, we can forbid them to ever escape some collection of 
total computable function (security), but then they cannot be universal 
(liberty). Löbian machine are aware of that tension between security and 
liberty, from the start.



> 
>> 
>>> ​> *​*
>>> 
>>> 
>>> *The question I want to ask is has Hod Lipson built a Loebian machine in
>>> physical matter?*
>> 
>> 
>> ​There is no way I can ever know if ​
>> Hod Lipson
>> ​'s robots are self aware, I don't even know if ​
>> Hod Lipson
>> ​ is self aware, all I know for sure is that both behave intelligently. ​
>> 
> 
> His argument is that his robot is self-aware, for some operational
> definition of self-aware. Of course this claim is bound to be
> controversial.  Regardless, I'm curious as to the relationship between
> that and Loebianity.

>From the video, it is hard to say. If they can find induction rules, which are 
>often the base of learning, and planning in AI, then there will be Löbian if 
>they can reason and talk in classical logic.

Bruno



> 
> 
> -- 
> 
> 
> Dr Russell StandishPhone 0425 253119 (mobile)
> Principal, High Performance Coders
> Visiting Senior Research Fellowhpco...@hpcoders.com.au
> Economics, Kingston University http://www.hpcoders.com.au
> 
> 
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Re: What is a Löbian machine/number/combinator

[Bruno Marchal]

> On 22 Apr 2018, at 02:08, John Clark  wrote:
> 
> On Sat, Apr 21, 2018 at 6:15 PM, Russell Standish  > wrote:
> 
> ​> ​Yes, of course a Loebian machine is a type of Turing machine.
> 
> How can I determine if that particular Turing Machine is doing something 
> fundamentally different from what every other Turing Machine is doing? 

By reading my papers, or my posts, perhaps by consulting some textbook in logic 
before (like Davis, Mendelson, etc.).

You seem to confuse machine and set of beliefs that we can associate to machine 
when they know some (first order) logic, which I put in the definition of the 
observer (provided to exist in arithmetic).

Bruno

>  
> ​> ​The question I want to ask is has Hod Lipson built a Loebian machine
> in physical matter?
> 
> ​There is no way I can ever know if ​Hod Lipson​'s robots are self aware, I 
> don't even know if ​Hod Lipson​ is self aware, all I know for sure is that 
> both behave intelligently. ​
> 
> John K Clark
> 
> 
>  
> 
> 
>  
> 
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Re: What is a Löbian machine/number/combinator

[Bruno Marchal]

> On 22 Apr 2018, at 01:47, John Clark  wrote:
> 
> On Sat, Apr 21, 2018 at 5:27 PM, Telmo Menezes  > wrote:
>  
> ​> ​As​ ​​R​ussell said, an approximation of the Löbian machine can probably 
> be
> derived from Bruno's post in Prolog.
> 
> Then a ​"​Löbian machine​"​ is just a particular Turing Machine and there is 
> nothing fundamentally new about it.

?

Then the human are just a particular machine and there is nothing fundamental 
new about this.

All effective set of beliefs (theories) are machines, but not all machines are 
theories.

Provability is the key notion, and there are as much provability notion than 
there are machine. And those are the one endowed with 8 modes of self-reference 
needed to extract physics.

Computability is absolute, an all notions are equivalent, with respect to 
computability, but not on provability.




>   
>  ​> ​I am complaining about​ personal insults. For example, in the sentence 
> above you insult both
> me and Bruno without providing anything of substance.
> 
> ​Mr.Snowflake, I'll let others decide if I am a bully or not as you claim,

People will believe that you are not a bully when you will convince just one 
person that step 3 is invalid.
You can’t, because the argument is quite simple and almost immediate, once we 
don’t change the definition given, and keep using the 3p/1p distinction.

Bruno




> but I maintain it is a fact you can't handle scientific criticism, and it it 
> is true then is a statement with substance.
>  
> you used the classical bully technique​ of​ making fun of his mode of 
> expression with "ad hominem".
> 
> I think it would be fair to say  nobody on this list has received more 
> personal insults than I have, but I have never once used that ridiculous 
> Latin phrase.
> 
> ​> ​I think you make a basic logic mistake. It is true that some brilliant​ 
> people are assholes, but being an asshole does not make you brilliant.
>  
>  Maybe I've got my Latin wrong, please explain to me again what "ad hominem" 
> means.
> 
> ​ John K Clark ​
> 
>  
> 
> 
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Re: What is a Löbian machine/number/combinator

[Bruno Marchal]

> On 22 Apr 2018, at 00:56, John Clark  wrote:
> 
> On Sat, Apr 21, 2018 at 5:10 PM, Telmo Menezes  > wrote:
> 
> >  Turing machines don’t need infinite tape, they need sufficient tape,​  if 
> > you start to run out of tape then add more,
> 
> ​> ​And for the general case there will be instances where you always need 
> more.
> 
> Any calculation will ALWAYS require only a finite amount of tape, if more 
> tape is ALWAYS required then the problem can not be calculated​.​
>  
> ​> ​Non-Turing universal machines can perform some computations. Even​ useful 
> ones, for sure.
>  
> Well..., I admit none of the 64 possible one state Turing Machine is 
> universal and none of the 20,736 possible two state Turing Machines is 
> either, and I admit even a one state machine could perform useful 
> calculations, but if you know how to make a one state machine then it is a 
> trivial matter to make a N state machine, and that is universal. And a one 
> state machine is as simple as things get, anything simpler can't calculate 
> anything. 
> 
> ​> ​Computations realized in the physical world will always stop,
> 
> ​I agree they will always stop but they will not always produce an answer.

In which case we say they do not stop. If they are stopped by an asteroid, or 
the Big Crunch, the first person associated to them still feel to continue, as 
its mind is associated with all computations in arithmetic, where the non 
stopping genuinely do not stop. 

To say all machines stop, you have to talk of very special case of machines.



> 
>  ​>​ If you apply to Turing the same demands that​ you apply to Bruno, you 
> can only conclude that Turing was a moron for
> working on mathematical models that correspond to machines that cannot​ exist.
> 
> The difference is a Turing Machine in the real physical world can very often 
> make calculations, often enough to create a trillion dollar industry​​, and 
> Turing told us exactly how to build such a device, but Bruno's "Löbian 
> machine" can NEVER make a calculation in the real physical world because 
> Bruno has no idea how to make one.

See my answer in my other post of today. It is really an easy exercise to 
implement a Löbian machine.The combinators, like RA are Turing universal, but 
not Löbian. But implemented on prolog, you need only to add the induction 
axioms: like

P(K) and P(S) and [(P(x) & P(y)) -> P((x y))] -> for all x, y P(x, y)

(Induction axioms written for the combinators, for a change).



>  
> ​> ​These machines are finite approximations of the machine that Turing​ 
> defined,
> 
> A Turing Machine exists in the real physical world

You cannot invoke god or reality, or real, or truth, in a scientific discourse. 
You assume a “real physical world”. I do not, and then shows that such an 
assumption, in company of Mechanism, leads to a contradiction.



> that can calculate 2+2 and that machine has no need to be infinite and the 
> answer it produces is exact not approximate. But Bruno can't even tell how to 
> build a "finite approximation" of a Löbian machine in the real physical world.


Why do you say things like that, which is utterly ridiculous (and ad hominem)?. 
Why not ask politely “how would you implement a Löbian machine” in the physical 
world (real or apparent), and that is easy to answer, and indeed is part of the 
easy exercise of my course. When you buy a physical computer, the hard part of 
Löbianity is already implemented. To make it into a set of beliefs, you need 
only to reimplemented in first order logic, already done if you use Prolog, and 
then to add the induction axioms, either as a meta-rule, or as a higher order 
axiom.

Bruno




> 
> John K Clark
> 
> 
> 
> 
>  
> 
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Re: What is a Löbian machine/number/combinator

[Bruno Marchal]

> On 22 Apr 2018, at 00:15, Russell Standish  wrote:
> 
> On Sat, Apr 21, 2018 at 11:00:32AM -0400, John Clark wrote:
>>> *I suspect he does know how to write a "Loebian machine" in Lisp or​
>>> ​Prolog*
>> 
>> 
>> If so then a "Loebian machine" is just a type of Turing Machine and Bruno
>> has not discovered anything fundamental that Turing didn't know about 82
>> years ago
>> ​.
> 
> Yes, of course a Loebian machine is a type of Turing machine. However,
> I doubt that Turing knew about it 82 years ago, given that Loeb's
> theorem was only published in 1953.

1955. (Einstein died that year, and I born!)


> It is possible that Turing knew
> about it before his death in 1954, although I rather think that
> unlikely, given what was going on in Turing's life then.

Löbianity is already in Gödel’s Principia-Mathematica PM, and Gödel oversaw it 
by its remark that his incompleteness can be formalised in his system PM. 

Gödel’s insight was proved rigorously, but “uglily” by Hilbert and Bernays in 
1937.

Then Löb, in 1955, proved that arithmetic is weakly Löbian (close for the rule 
[]p -> p ==> p, and that the “rich” machine (those like PM which can prove 
their own universality and its price of incompleteness) proves its formal 
counterpart []([]p -> p) -> []p. (Löb’s formula).

Then in 1976, Solovay proved that the Löb’s formula axiomatises completely the 
logic of self-reference, bot for the provable part of the machine, and for the 
true part of the machine. G has Löb’s as main axiom (unique when based on 
normal (Kripkean) modal logic. G* has all theorems of G as axioms, including 
Löb’s formula, but is not normal, as it lacks the necessitation rule.


> 
> The question I want to ask is has Hod Lipson built a Loebian machine
> in physical matter?
> 
> https://www.ted.com/talks/hod_lipson_builds_self_aware_robots

If such machine [] are universal (p -> []p is true for them), and can prove 
their universality (p -> []p is provable by them) then they are Löbian.

All human beings are physical incarnation of Löbian’s machine, but the theology 
applies only the sound machine, which soundness is beyond the human mind to 
even be defined.

Very often, I call Löbian machine MODEST machine, as Löb’s formula, like 
Gödel’s particular case (it is Löb with p replaced by f) makes the machine 
quite modest with respect to truth. They prove correctness with respect to p 
ONLY WHEN they prove p. 

Cheers,

Bruno



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> 
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Re: What is a Löbian machine/number/combinator

[Bruno Marchal]

> On 21 Apr 2018, at 10:10, Telmo Menezes  wrote:
> 
> On 20 April 2018 at 19:04, John Clark  wrote:
>> I never got past the first line of Bruno’s post because he said:
>> 
>> "Consider any Turing universal machinery, for example the programming
>> language c++”
>> 
>> C++ is Turing complete but is not a Turing machine because machines are
>> physical objects made of atoms but C++ is not nor is any language. As for
>> Löbian machines that is yet another term that Bruno made up and is seen on
>> this list but nowhere else.
> 
> You sound like one of the peer-reviewers who rejected Turing's paper.
> It's almost uncanny. He said:
> 
> "This is a bizarre paper. It begins by defining a computing device
> absolutely unlike anything I have seen, then proceeds to show—I
> haven't quite followed the needlessly complicated formalism [...]"
> 
> In his defense, at least he understood that it was meant to be a
> formalism, and not the plans to build an actual device.
> 
>> And Turing explained exactly precisely how to
>> make one of his machines in the real physical world
> 
> Nope. Turing machines have infinite tapes.

Better to put it in the environment. When you implement/incarnate a universal 
machine/number, to install a finite code in a finite machine, and it is 
important for it to be finite or RE to be able to be arithmetised.

Of course, the universal machine when incarnated will either complain of lack 
of memory, or use the wall of the cave to pursue the computation! But the 
universal (Lôbian) machine is finite (or RE).



> They cannot possibly be
> created in the physical world. They were proposed by Turing as an
> *abstract* model of computation,

Inspired by the human computer. The physical computations are only physical 
implementation of them. (If we except Babbage's dream).




> and he was upfront about it. Turing
> created this model to answer theoretical questions, not to propose
> some device. C++ is itself an abstract model and it is Turing
> universal, but it does not make sense to say that my physical computer
> is Turing universal because it does not have infinite memory, nor
> could it.





> You fundamentally miss the point of theoretical computer
> science.

Yes. Clark missed it, but here with due respect, you miss the fact that a 
finite physical computer is an exact complete incarnation of a universal FINITE 
number. It is important, as if it was infinite, we would not have a finite 
Gödel number implementing  “[]” in arithmetic, and no arithmetization. 

Keep Wolfram’s challenge (some years ago) in mind; to find the smallest 
universal Turing machine, i.e. the smallest (finite!) set u of quadruplets, 
which makes the machine u computing ph_i(x) for all is and x given as an input.


My general definition of universal number is only, when given a universal 
machinery, that is a computable enumeration of all partial recursive functions 
phi_0, phi_1, phi_2 …, u is a universal machine if u is such that phI_u(x, y) = 
phi_x(y). u is a precise number, mimicking the Turing machine x on the input y.
It is a FINITE set of quadruplets mimicking any set of quadruplets in, say, the 
enumeration of set of quadruplet.

The infinite tape is only an help for the intuition. It plays the role of the 
diary in which a human computer keeps track of its intermediate result. It is 
the “memory space”, and it is usually locally finite, but not part of the 
interpreter code, which is the universal numbers/machine (that the doctor will 
put on an hard disk).






> 
>> but Bruno has no idea
>> how to even start to build one of his machines, which means he doesn’t
>> understand how it works
> 
> Let us know where we can get our hand on some infinite capacity hard
> drives. I'm sick of paying through the nose for backups.
> 
>> or even exactly what it is he’s talking about.
> 
> You're a bully.


Yes, Clark is a bully, and it just pursue the work of bullies which aggravate 
their case to hide the bullying. Like in Brussels, the bullies continue the 
bullies to hide the bullying. None of those people have ver study the work, nor 
even accept to talk with me, even just in private. They have never read the 
work, and that is what they try to hide. But there is no problem with genuine 
scientists, even if sometimes they get some metaphysical vertigo, which is 
normal in this dogmatic Aristotelian era concerning the fundamental science. 
But even this is only a pretext to hide the bullies, which in Brussels has 
preceded my thesis for long. The thesis is more the result of the bullying than 
its cause.

Bruno



> 
> Telmo.
> 
>> 
>> 
>> John K Clark
>> 
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Re: What is a Löbian machine/number/combinator

[Bruno Marchal]

> On 21 Apr 2018, at 09:21, Russell Standish  wrote:
> 
> On Fri, Apr 20, 2018 at 01:04:30PM -0400, John Clark wrote:
>> I never got past the first line of Bruno’s post because he said:
>> 
>> "*Consider any Turing universal machinery, for example the programming
>> language c++*”
>> 
>> C++ is Turing complete but is not a Turing machine because machines are
>> physical objects made of atoms but C++ is not nor is any language.
> 
> Nor is a Turing machine for that matter. It is an abstract model of
> computation.

Yes, it is an term in a theory of machine and computation. It is mathematically 
(arithmetically) precise.

For a platonist, a “physical computation” is a physical “model” of the real 
things which is immaterial.

I would not use abstract, because I find a set of quadruplets much less 
abstract that a physical computer approximation of it, which is an even more 
abstract and complex number relations (quantum field?, atoms? …).




> For Bruno, the term "machine" means such an abstract model.


Better to avoid “model" and use “theory” . For logicians and painters, the 
“model” is the real thing that we approximate with finite things, like 
machines, brains, theories, or paintings.




> 
>> As
>> for Löbian machines that is yet another term that Bruno made up and is seen
>> on this list but nowhere else. And Turing explained exactly precisely how
>> to make one of his machines in the real physical world but Bruno has no
>> idea how to even start to build one of his machines, which means he doesn’t
>> understand how it works or even exactly what it is he’s talking about.
>> 
> 
> I suspect he does know how to write a "Loebian machine" in Lisp or
> Prolog (say),

Yes. As I said to Clark, that is even a common easy exercise in my teaching. 
Few students have difficulties, except for minor mistakes.



> but I wanted to press him a bit on this. If he can do
> this much, then it is a relatively trivial matter to install a lisp
> interpreter on a PC, run the program and reify it as a physical machine.


Yes, a lisp interpreter, when install on a PC, provides a physical 
implementation Lisp, which is immaterial.

Here computationalism can be compared to the usual “mind = software” and 
“matter = hardware”, but that distinction can be shown relative in general, and 
then “hardware” can be shown to be a (lawful) psychological appearance (and a 
delusion when reified).(assuming of course indexical digital mechanism).

Bruno


> 
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Re: What is a Löbian machine/number/combinator

[Bruno Marchal]

> On 20 Apr 2018, at 19:04, John Clark  wrote:
> 
> I never got past the first line of Bruno’s post because he said:
> 
> "Consider any Turing universal machinery, for example the programming 
> language c++” 
> 
> C++ is Turing complete but is not a Turing machine because machines are 
> physical objects made of atoms but C++ is not nor is any language.
> 

That is not correct. There are many definition of Turing machine. 

Let me give you the most standard definition of Turing machine. You need two 
infinite set of symbols:

Q = {q_1, q_2, q_3, q_4, …}.  (Called set of internal configurations}

S = {S_0, S_1, S_2, S_3, …}. (Called set of tape symbol}

Along with the symbols R and L. (Right and Left symbols)

A quadruplet is then define by a sequence of fours symbols having the shapes

q_i S_j S_k q_r

or q_i S_j R q_r
or q_i S_j L q_r

(Davis makes the general relativized theory at once, and so add q_i S_j q_k 
q_l, which is used to handle Turing’s Oracle, but I will ignore this here).

Now a Turing machine is defined to be a finite set of quadruples. That’s all. 
It is set, so the order of the quadruples is irrelevant.

If there is not two different quadruples with the same beginning q_i S_j, then 
we say that the machine is deterministic. If not it is non deterministic.


The intuitive interpretation of q_i S_j S_k q_r is that if the machine is in 
the internal configuration q_i, in front of the symbol S_j, and if some 
quadruple in the machine contains that quadruplet, the next step of the 
computation will be obtained by overwriting S_k (on S_j) and getting the state 
q_r, or if q_i S_j R q_r was in the machine (which is a set of such 
quadruplets) moving to the right of the" tape” and getting the internal 
configuration q_r. (And similarly, for L = Left).

Then we can define computation by finite sequence of “instantaneous tape 
description”, which are finite set of tape symbols, + a symbols of internal 
configuration, like S_4 S_5 q_22 S_2 S_6, which means that the machine is in 
front of S_2, in configuration q_22, and if q_22 S_2 begins one of its 
quadruplet, the machine “acts” accordingly, and if not, the machine stops.

But the terms “tape”, “configuration” are just intuitive help, and not part of 
the definition.

Note that most definition use only two tape symbols S_0 (called “blank”) and 
S_1 (written 1).
Note that Turing’s original definition use quintuplet, and he allowed the 
machine to “move” and overwrite symbols simultaneously. But it is better to 
reason on the quadruplets (more general). As you can guess (and try to prove) a 
the quadruplet-TM can emulate the quintuplet -TM.


Important remark: a universal Turing machine is a Turing machine, and thus a 
FINITE set of quadruplets. There is nothing infinite in the universal machine. 
It is an interpreter of Turing formalism coded in one (universal) program, that 
is here a FINITE set of quadruplets. That machine, assuming only the two 
symboles “.” (for blank, S_0), and 1 for S_1.

The universal machine is thus a finite set of quadruplet, and it starts on

q_1 1.

And it interprets the first block “1” as some Gödel 
number of a Turing machine (set of quadruplet) acting on the input represented 
by the second bloc. Obviously it has q_1 1 as a beginning of some of its 
quadruplet to avoid stopping at the start!

The key point is that the universal machine is a FINITE object.

Turing’s talk on an infinite tape is only an aid to the intuition. It is better 
to consider the tape as being an environment, mental or physical, or even as a 
special oracle.




> As for Löbian machines that is yet another term that Bruno made up and is 
> seen on this list but nowhere else.
> 

It is a slightly more precise version of the “enough rich” 
machine/theory/set-of-beliefs notion, and it can be defined by for all p 
sigma_1, the machine can prove “p -> []p”, with “[]p” being the usual tedious 
arithmetical definition of provability.

As I explained recently, with p sigma_1,

The truth of all p -> []p is equivalent with “[]” (the machine’s arithmetical 
provability predicate) being a universal machine, in this sense or equivalent.

Then a machine “[]" is Löbian if she can prove “p -> []p”

As “[]” is itself sigma_1, (this should be obvious!!! Please tell me if you 
(anyone) don’t see this), we get 
[]p -> [][]p, so Löbianity entails the axiom 4 (called self-awareness by 
Smullyan).

A pure K4 reasoner has to go the Löb’s Island to become Löbian, but any enough 
rich machine can prove Gödel’s diagonal lemma, making them “born in the Löb’s 
Island” so to speak.

The typical example are any sound machine believing in enough axioms of 
arithmetic, like PA, ZF, and many others (including us, as far as we are 
correct and agree with PA, say).



> And Turing explained exactly precisely how to make one of his machines in the 
> real physical world
> 

He is even the first to build 

Re: What is a Löbian machine/number/combinator

[Bruno Marchal]

> On 20 Apr 2018, at 00:38, Russell Standish  wrote:
> 
> On Wed, Apr 18, 2018 at 07:11:33PM +0200, Bruno Marchal wrote:
>> Somewhere: (and I copy my answer, as some people asked me this in this list 
>> too).
>> 
>> 
>>> 
>>> What are Lobian numbers? Can you give a reference? I know little bit about 
>>> Godel’s work.
>> 
>> 
>> Consider any Turing universal machinery, for example the programming 
>> language c++. 
>> 
>> N is the set of natural numbers.
>> 
>> It is known that the enumeration of all programs computing a (perhaps not 
>> everywhere defined) function from N to N exists, and so we get a list of all 
>> partial computable function phi_i from N to N. (i.e. phi_0, phi_1, phi_2, 
>> …), by enumerating the program with one natural number argument) written in 
>> C++, in their lexico-graphical order (length, and alphabetical for the 
>> programs with the same length).
>> 
>> We can define a universal number as a number u such that phI_u(x, y) = 
>> phi_x(y). We say that u implements x on y. (It is a constructive definition 
>> of a computer in the language of the computer).
> 
> Some niggles: You haven't defined φᵢ(x,y). You need some sort of
> composition operator ∘ (perhaps x∘y is the concatenation of the bit
> representation of the number), and define φᵢ(x,y)=φᵢ(x∘y)

You need a computable bijection  between NXN and N, in case you want the 
universal function to be contained in the enumeration of the one-variable 
functions.

In that case we would write that u , the universal machine/number is such that 

 phi_u() = phi_x(y). (U emulates the number/machine x 
on the input y).

This provides homogeneity, but it is not necessary, as we can consider the many 
enumerations of one-variable, two-variables, … semi-computable functions, and 
use a two variable functions (program, input) for the universal functions.

With the combinators, each combinators in the enumeration (K, S, (K, K), …) can 
be seen as a function of zero variables!

And with Davis’ definition of Turing machines, all the enumerations (of 
one-variable, two variables, …functions) are all identical. You decide if you 
give one or two or three arguments to the machine.

The homogeneity is required for having a good notion of extensional recursive 
equivalence, but I don’t use them, as I require only intensional equivalence 
(based on programs behaviour). 

> 
>> 
>> Now, once we have a universal number, we can transform/extend it into a 
>> theory, which is the first order logical specification of how u operates. 
>> That is a standard mapping from, say, c++ to a Turing universal logical 
>> theory. 
>> 
> 
> I assume that is possible. How would one go about this in practice?

By axiomatising in first order logic the terms of c++, and writing enough 
axioms to get the Turing universality.




> 
>> I assume we have done that, so now I say that a universal number is Löbian 
>> when it has enough induction axioms (added to its logical specification) so 
>> that it can prove enough of some special formula. 
>> 
> 
> Isn't it true that the actual set of universal numbers rather depends
> on one chosen enumeration? So universality is not a property of the
> numbers per se?

Indeed. It is an intensional notion, but as I have explained sometimes that we 
have an intensional Church-Turing thesis (which followed from the usual 
extensional one).

But, once we fix the “base” (choosing between arithmetic, combinators, c++, 
etc.), universality becomes an intrinsic property of numbers. That is why we 
can stay entirely in Robinson Arithmetic (ontologically) to get the internal 
complete phenomenology of the more rich Löbian number that RA emulates.

Bruno




> 
> -- 
> 
> 
> Dr Russell StandishPhone 0425 253119 (mobile)
> Principal, High Performance Coders
> Visiting Senior Research Fellowhpco...@hpcoders.com.au
> Economics, Kingston University http://www.hpcoders.com.au
> 
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Re: What is a Löbian machine/number/combinator

[Telmo Menezes]

On 22 April 2018 at 01:47, John Clark  wrote:
> On Sat, Apr 21, 2018 at 5:27 PM, Telmo Menezes 
> wrote:
>
>>
>> >
>> As Russell said, an approximation of the Löbian machine can probably be
>> derived from Bruno's post in Prolog.
>
>
>  Then a "Löbian machine" is just a particular Turing Machine and there is 
> nothing fundamentally new about it.

If computationalism is true, then human brains are just particular
Turing Machines. Better tell neuroscientists to cease their
investigations, turns out their entire field is uninteresting. Come
on...

>> I am complaining about
>>
>> personal insults. For example, in the sentence above you insult both
>> me and Bruno without providing anything of substance.
>
>
> Mr.Snowflake, I'll let others decide if I am a bully or not as you claim,
> but I maintain it is a fact you can't handle scientific criticism, and it it
> is true then is a statement with substance.

I have had papers rejected and it doesn't feel good. I had to cry
myself to sleep with a bucket of ice cream. What can I say?

>> you used the classical bully technique ofmaking fun of his mode of 
>> expression with "ad hominem".
>
>
> I think it would be fair to say nobody on this list has received more
> personal insults than I have, but I have never once used that ridiculous
> Latin phrase.

Perhaps because nobody on this mailing list has dished out as many
personal insults as you have?

>> >
>> I think you make a basic logic mistake. It is true that some brilliant
>>
>> people are assholes, but being an asshole does not make you brilliant.
>
>
>  Maybe I've got my Latin wrong, please explain to me again what "ad hominem"
> means.

It means to use personal attacks against the author of an argument
instead of addressing the substance of the argument. It's exactly what
you do to Bruno all the time.

Telmo.

> John K Clark
>
>
>
>
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Re: What is a Löbian machine/number/combinator

[Russell Standish]

On Sat, Apr 21, 2018 at 08:08:50PM -0400, John Clark wrote:
> On Sat, Apr 21, 2018 at 6:15 PM, Russell Standish 
> wrote:
> 
> ​> ​
> > *Yes, of course a Loebian machine is a type of Turing machine.*
> 
> 
> How can I determine if that particular Turing Machine is doing something
> fundamentally different from what every other Turing Machine is doing?
> 

I would say that it is a machine that proves Loeb's theorem. Not all
Turing machines are capable of that, even universal machines absent
the right software. But I may have misunderstood this :).

> 
> > ​> *​*
> >
> >
> > *The question I want to ask is has Hod Lipson built a Loebian machine in
> > physical matter?*
> 
> 
> ​There is no way I can ever know if ​
> Hod Lipson
> ​'s robots are self aware, I don't even know if ​
> Hod Lipson
> ​ is self aware, all I know for sure is that both behave intelligently. ​
>

His argument is that his robot is self-aware, for some operational
definition of self-aware. Of course this claim is bound to be
controversial.  Regardless, I'm curious as to the relationship between
that and Loebianity.


-- 


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Principal, High Performance Coders
Visiting Senior Research Fellowhpco...@hpcoders.com.au
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Re: What is a Löbian machine/number/combinator

[John Clark]

On Sat, Apr 21, 2018 at 6:15 PM, Russell Standish 
wrote:

​> ​
> *Yes, of course a Loebian machine is a type of Turing machine.*


How can I determine if that particular Turing Machine is doing something
fundamentally different from what every other Turing Machine is doing?


> ​> *​*
>
>
> *The question I want to ask is has Hod Lipson built a Loebian machine in
> physical matter?*


​There is no way I can ever know if ​
Hod Lipson
​'s robots are self aware, I don't even know if ​
Hod Lipson
​ is self aware, all I know for sure is that both behave intelligently. ​

John K Clark

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Re: What is a Löbian machine/number/combinator

[John Clark]

On Sat, Apr 21, 2018 at 5:27 PM, Telmo Menezes 
wrote:


> ​> ​
>
> *As​ ​​R​ussell said, an approximation of the Löbian machine can probably
> be derived from Bruno's post in Prolog.*


Then a
​"​
Löbian machine
​"​
is just a particular Turing Machine and there is nothing fundamentally new
about it.


>
> ​> ​
> I am complaining about
> ​
> personal insults. For example, in the sentence above you insult both
> me and Bruno without providing anything of substance.


​
Mr.Snowflake, I'll let others decide if I am a bully or not as you claim,
but I maintain it is a fact you can't handle scientific criticism, and it
it is true then is a statement with substance.


> you used the classical bully technique
> ​
> of
> ​
> making fun of his mode of expression with "ad hominem".


I think it would be fair to say  nobody on this list has received more
personal insults than I have, but I have never once used that ridiculous
Latin phrase.

​> ​
> *I think you make a basic logic mistake. It is true that some
> brilliant​ people are assholes, but being an asshole does not make you
> brilliant.*
>

 Maybe I've got my Latin wrong, please explain to me again what "ad
hominem" means.

​ John K Clark ​

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Re: What is a Löbian machine/number/combinator

[John Clark]

On Sat, Apr 21, 2018 at 5:10 PM, Telmo Menezes 
wrote:

>  Turing machines don’t need infinite tape, they need sufficient tape,
>> ​
>>  if you start to run out of tape then add more,
>
>
> *​> ​And for the general case there will be instances where you always
> need more.*


Any calculation will ALWAYS require only a *finite* amount of tape, if more
tape is ALWAYS required then the problem can not be calculated
​.​


> ​> ​
> Non-Turing universal machines can perform some computations. Even
> ​
> useful ones, for sure.
>

Well..., I admit none of the 64 possible one state Turing Machine is
universal and none of the 20,736 possible two state Turing Machines is
either, and I admit even a one state machine could perform useful
calculations, but if you know how to make a one state machine then it is a
trivial matter to make a N state machine, and that is universal. And a one
state machine is as simple as things get, anything simpler can't calculate
anything.

​> ​
> Computations realized in the physical world will always stop,


​I agree they will always stop but they will not always produce an answer.


> ​>​
> * If you apply to Turing the same demands that​ you apply to Bruno, you
> can only conclude that Turing was a moron for*

* working on mathematical models that correspond to machines that
> cannot​ exist.*


The difference is a Turing Machine in the real physical world can very
often make calculations, often enough to create a trillion dollar industry
​​
, and Turing told us exactly how to build such a device, but Bruno's
"Löbian machine" can NEVER make a calculation in the real physical world
because Bruno has no idea how to make one.


> ​> ​
> These machines are finite approximations of the machine that Turing
> ​
> defined,


A Turing Machine exists in the real physical world that can calculate 2+2
and that machine has no need to be infinite and the answer it produces is
exact not approximate. But Bruno can't even tell how to build a "finite
approximation" of a Löbian machine in the real physical world.

John K Clark

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Re: What is a Löbian machine/number/combinator

[Russell Standish]

On Sat, Apr 21, 2018 at 11:00:32AM -0400, John Clark wrote:
> > *I suspect he does know how to write a "Loebian machine" in Lisp or​
> > ​Prolog*
> 
> 
> If so then a "Loebian machine" is just a type of Turing Machine and Bruno
> has not discovered anything fundamental that Turing didn't know about 82
> years ago
> ​.

Yes, of course a Loebian machine is a type of Turing machine. However,
I doubt that Turing knew about it 82 years ago, given that Loeb's
theorem was only published in 1953. It is possible that Turing knew
about it before his death in 1954, although I rather think that
unlikely, given what was going on in Turing's life then.

The question I want to ask is has Hod Lipson built a Loebian machine
in physical matter?

https://www.ted.com/talks/hod_lipson_builds_self_aware_robots

Cheers.


-- 


Dr Russell StandishPhone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Senior Research Fellowhpco...@hpcoders.com.au
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Re: What is a Löbian machine/number/combinator

[Telmo Menezes]

[I messed up and sent the unfinished email. Here's the rest...]

On 21 April 2018 at 23:10, Telmo Menezes  wrote:
> On 21 April 2018 at 16:44, John Clark  wrote:
>> On Sat, Apr 21, 2018 at 4:10 AM, Telmo Menezes 
>> wrote:
>>
 >
 >
 And Turing explained exactly precisely how to make one of his machines in
 the real physical world
>>>
>>>
>>> >
>>> Nope. Turing machines have infinite tapes.
>>
>>
>> Nope. Turing machines don’t need infinite tape, they need sufficient tape,
>> if you start to run out of tape then add more,

And for the general case there will be instances where you always need more.

>>  but after any finite number
>> of operations only a finite amount of tape is needed.
>> And with calculating
>> something like the Busy Beaver number only those machines that halt after a
>> finite number of operations count,

Non-Turing universal machines can perform some computations. Even
useful ones, for sure.

>> in fact in ANY successful calculation the
>> machine will eventually halt.
>

while(true) {
if (temperature < 21) {
heater = true;
}
else {
heater = false;
}
}

>> Yes some machines will never halt (like the
>> Turing Machine programed to find the 7918th Busy Beaver number) and so you
>> will keep adding tape forever, but that is the very definition of
>> non-computability. But even in that case at any given time the machine only
>> has or needs a finite amount of tape.

The non-computability of the Entscheidungsproblem is about the
impossibility of having a computation that will tell you in finite
time if an arbitrary other computation will ever stop or not.
Computations realized in the physical world will always stop, because
of physical limitations. If you apply to Turing the same demands that
you apply to Bruno, you can only conclude that Turing was a moron for
working on mathematical models that correspond to machines that cannot
exist. In fact, for you the Turing Machine is not a machine because it
cannot be physically realized.

>>
>>>
>>> >
>>> They were proposed by Turing as an
>>> *abstract* model of computation,
>>> and he was upfront about it.
>>
>>
>> Turing never claimed there were not far more complicated ways for an
>> engineer to make a computer, ways that worked faster and were far more
>> practical but were more difficult to understand. But he did show that any
>> computer could be reduced to his very simple machine, and people have
>> actually built real physical machines that work exactly as Turing said they
>> would.

These machines are finite approximations of the machine that Turing
defined, and which is not a machine according to your criterion
because it cannot be built.

>> When Bruno does more than just write mathematical symbols on a piece
>> of paper and makes a working physical model of a "Löbian machine " (and I
>> don't care if its ridiculously slow and impractical ) I'll retract
>> everything I said and place Bruno’s name next to Turing’s on my list of
>> greats. All I want is to see a working model of a physical "Löbian machine "
>> that is the equivalent to this model Turing Machine:

This is not a working model of the Turing Machine, it is a finite
approximation. It's cute, but it adds nothing to Turing's results. As
Russell said, an approximation of the Löbian machine can probably be
derived from Bruno's post in Prolog. He provided a Lisp implementation
of the Universal Dovetailer. It's an interesting exercise, but so
what?

>>
>> https://www.youtube.com/watch?v=E3keLeMwfHY
>>
>>
>>> >
>>> You're a bully.
>>
>> And you are a delicate snowflake who can't handle scientific criticism, and
>> a fool too if you think Bruno has said anything profound.

Maybe I am a delicate snowflake, but that is besides the point. I am
not complaining about scientific criticism, I am complaining about
personal insults. For example, in the sentence above you insult both
me and Bruno without providing anything of substance. The last time
Bruno pointed this out to you, you used the classical bully technique
of making fun of his mode of expression with "ad hominem". Throughout
the years you never tire of insulting people who remain polite when
talking to you. Maybe you will really require some extra centuries of
artificial life extension to learn some basic kindness.

I think you make a basic logic mistake. It is true that some brilliant
people are assholes, but being an asshole does not make you brilliant.

Telmo.

>> John K Clark
>>
>>
>>
>>
>> --
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Re: What is a Löbian machine/number/combinator

[Telmo Menezes]

On 21 April 2018 at 16:44, John Clark  wrote:
> On Sat, Apr 21, 2018 at 4:10 AM, Telmo Menezes 
> wrote:
>
>>> >
>>> >
>>> And Turing explained exactly precisely how to make one of his machines in
>>> the real physical world
>>
>>
>> >
>> Nope. Turing machines have infinite tapes.
>
>
> Nope. Turing machines don’t need infinite tape, they need sufficient tape,
> if you start to run out of tape then add more,

And for the general case there will be instances where you always need more.

>  but after any finite number
> of operations only a finite amount of tape is needed.
> And with calculating
> something like the Busy Beaver number only those machines that halt after a
> finite number of operations count,

Non-Turing universal machines can perform some computations. Even
useful ones, for sure.

> in fact in ANY successful calculation the
> machine will eventually halt.

while(true) {
 if (temperature < 21) {
}
}

> Yes some machines will never halt (like the
> Turing Machine programed to find the 7918th Busy Beaver number) and so you
> will keep adding tape forever, but that is the very definition of
> non-computability. But even in that case at any given time the machine only
> has or needs a finite amount of tape.

The non-computability of the Entscheidungsproblem is about the
impossibility of having a computation that will tell you in finite
time if an arbitrary other computation will ever stop or not.
Computations realized in the physical world will always stop, because
of physical limitations. If you apply to Turing the same demands that
you apply to Bruno, you can only conclude that Turing was a moron for
working on mathematical models that correspond to machines that cannot
exist. In fact, for you the Turing Machine is not a machine because it
cannot be physically realized.

>
>>
>> >
>> They were proposed by Turing as an
>> *abstract* model of computation,
>> and he was upfront about it.
>
>
> Turing never claimed there were not far more complicated ways for an
> engineer to make a computer, ways that worked faster and were far more
> practical but were more difficult to understand. But he did show that any
> computer could be reduced to his very simple machine, and people have
> actually built real physical machines that work exactly as Turing said they
> would.

These machines are finite approximations of the machine that Turing
defined, and which is not a machine according to your criterium
because it cannot be built.

> When Bruno does more than just write mathematical symbols on a piece
> of paper and makes a working physical model of a "Löbian machine " (and I
> don't care if its ridiculously slow and impractical ) I'll retract
> everything I said and place Bruno’s name next to Turing’s on my list of
> greats. All I want is to see a working model of a physical "Löbian machine "
> that is the equivalent to this model Turing Machine:
>
>
> https://www.youtube.com/watch?v=E3keLeMwfHY
>
>
>> >
>> You're a bully.
>
> And you are a delicate snowflake who can't handle scientific criticism, and
> a fool too if you think Bruno has said anything profound.
>
> John K Clark
>
>
>
>
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Re: What is a Löbian machine/number/combinator

[John Clark]

On Sat, Apr 21, 2018 at 3:21 AM, Russell Standish 
wrote:

>
​>>​
>>  C++ is Turing complete but is not a Turing machine because machines are
>>  physical objects made of atoms but C++ is not nor is any language.
>
>
> *​> ​Nor is a Turing machine for that matter.*


​
This Turing machine certainly seems to be made of matter, atoms in
particular
​:​


https://www.youtube.com/watch?v=E3keLeMwfHY


​> *​*
> *I suspect he does know how to write a "Loebian machine" in Lisp or​
> ​Prolog*


If so then a "Loebian machine" is just a type of Turing Machine and Bruno
has not discovered anything fundamental that Turing didn't know about 82
years ago
​.

 John K Clark​


​

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Re: What is a Löbian machine/number/combinator

[John Clark]

On Sat, Apr 21, 2018 at 4:10 AM, Telmo Menezes 
wrote:

>
>> ​>​
>> And Turing explained exactly precisely how to make one of his machines in
>> the real physical world
>
>
> *​> ​Nope. Turing machines have infinite tapes.*


Nope. Turing machines don’t need infinite tape, they need sufficient tape,
if you start to run out of tape then add more, but after any finite number
of operations only a finite amount of tape is needed. And with calculating
something like the Busy Beaver number only those machines that halt after a
finite number of operations count, in fact in ANY successful calculation
the machine will eventually halt. Yes some machines will never halt (like
the Turing Machine programed to find the 7918th Busy Beaver number) and so
you will keep adding tape forever, but that is the very definition of
non-computability. But even in that case at any given time the machine only
has or needs a finite amount of tape.



> ​> ​
> *They were proposed by Turing as an​ ​*abstract* model of computation,​
> ​and he was upfront about it.*


Turing never claimed there were not far more complicated ways for an
engineer to make a computer, ways that worked faster and were far more
practical but were more difficult to understand. But he did show that any
computer could be reduced to his very simple machine, and people have
actually built real physical machines that work exactly as Turing said they
would. When Bruno does more than just write mathematical symbols on a piece
of paper and makes a working physical model of a "Löbian machine " (and I
don't care if its ridiculously slow and impractical ) I'll retract
everything I said and place Bruno’s name next to Turing’s on my list of
greats. All I want is to see a working model of a physical "Löbian machine
" that is the equivalent to this model Turing Machine:

https://www.youtube.com/watch?v=E3keLeMwfHY


​> ​
> *You're a bully.*

And you are a delicate snowflake who can't handle scientific criticism, and
a fool too if you think Bruno has said anything profound.

​John K Clark​

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Re: What is a Löbian machine/number/combinator

[Telmo Menezes]

On 20 April 2018 at 19:04, John Clark  wrote:
> I never got past the first line of Bruno’s post because he said:
>
> "Consider any Turing universal machinery, for example the programming
> language c++”
>
> C++ is Turing complete but is not a Turing machine because machines are
> physical objects made of atoms but C++ is not nor is any language. As for
> Löbian machines that is yet another term that Bruno made up and is seen on
> this list but nowhere else.

You sound like one of the peer-reviewers who rejected Turing's paper.
It's almost uncanny. He said:

"This is a bizarre paper. It begins by defining a computing device
absolutely unlike anything I have seen, then proceeds to show—I
haven't quite followed the needlessly complicated formalism [...]"

In his defense, at least he understood that it was meant to be a
formalism, and not the plans to build an actual device.

> And Turing explained exactly precisely how to
> make one of his machines in the real physical world

Nope. Turing machines have infinite tapes. They cannot possibly be
created in the physical world. They were proposed by Turing as an
*abstract* model of computation, and he was upfront about it. Turing
created this model to answer theoretical questions, not to propose
some device. C++ is itself an abstract model and it is Turing
universal, but it does not make sense to say that my physical computer
is Turing universal because it does not have infinite memory, nor
could it. You fundamentally miss the point of theoretical computer
science.

> but Bruno has no idea
> how to even start to build one of his machines, which means he doesn’t
> understand how it works

Let us know where we can get our hand on some infinite capacity hard
drives. I'm sick of paying through the nose for backups.

> or even exactly what it is he’s talking about.

You're a bully.

Telmo.

>
>
> John K Clark
>
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Re: What is a Löbian machine/number/combinator

[Russell Standish]

On Fri, Apr 20, 2018 at 01:04:30PM -0400, John Clark wrote:
> I never got past the first line of Bruno’s post because he said:
> 
> "*Consider any Turing universal machinery, for example the programming
> language c++*”
> 
> C++ is Turing complete but is not a Turing machine because machines are
> physical objects made of atoms but C++ is not nor is any language.

Nor is a Turing machine for that matter. It is an abstract model of
computation. For Bruno, the term "machine" means such an abstract model.

> As
> for Löbian machines that is yet another term that Bruno made up and is seen
> on this list but nowhere else. And Turing explained exactly precisely how
> to make one of his machines in the real physical world but Bruno has no
> idea how to even start to build one of his machines, which means he doesn’t
> understand how it works or even exactly what it is he’s talking about.
>

I suspect he does know how to write a "Loebian machine" in Lisp or
Prolog (say), but I wanted to press him a bit on this. If he can do
this much, then it is a relatively trivial matter to install a lisp
interpreter on a PC, run the program and reify it as a physical machine.
 
-- 


Dr Russell StandishPhone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Senior Research Fellowhpco...@hpcoders.com.au
Economics, Kingston University http://www.hpcoders.com.au


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Re: What is a Löbian machine/number/combinator

[John Clark]

I never got past the first line of Bruno’s post because he said:

"*Consider any Turing universal machinery, for example the programming
language c++*”

C++ is Turing complete but is not a Turing machine because machines are
physical objects made of atoms but C++ is not nor is any language. As
for Löbian machines that is yet another term that Bruno made up and is seen
on this list but nowhere else. And Turing explained exactly precisely how
to make one of his machines in the real physical world but Bruno has no
idea how to even start to build one of his machines, which means he doesn’t
understand how it works or even exactly what it is he’s talking about.

​  ​
John K Clark

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Re: What is a Löbian machine/number/combinator

[Russell Standish]

On Wed, Apr 18, 2018 at 07:11:33PM +0200, Bruno Marchal wrote:
> Somewhere: (and I copy my answer, as some people asked me this in this list 
> too).
> 
> 
> > 
> > What are Lobian numbers? Can you give a reference? I know little bit about 
> > Godel’s work.
> 
> 
> Consider any Turing universal machinery, for example the programming language 
> c++. 
> 
> N is the set of natural numbers.
> 
> It is known that the enumeration of all programs computing a (perhaps not 
> everywhere defined) function from N to N exists, and so we get a list of all 
> partial computable function phi_i from N to N. (i.e. phi_0, phi_1, phi_2, …), 
> by enumerating the program with one natural number argument) written in C++, 
> in their lexico-graphical order (length, and alphabetical for the programs 
> with the same length).
> 
> We can define a universal number as a number u such that phI_u(x, y) = 
> phi_x(y). We say that u implements x on y. (It is a constructive definition 
> of a computer in the language of the computer).

Some niggles: You haven't defined φᵢ(x,y). You need some sort of
composition operator ∘ (perhaps x∘y is the concatenation of the bit
representation of the number), and define φᵢ(x,y)=φᵢ(x∘y)

> 
> Now, once we have a universal number, we can transform/extend it into a 
> theory, which is the first order logical specification of how u operates. 
> That is a standard mapping from, say, c++ to a Turing universal logical 
> theory. 
>

I assume that is possible. How would one go about this in practice?

> I assume we have done that, so now I say that a universal number is Löbian 
> when it has enough induction axioms (added to its logical specification) so 
> that it can prove enough of some special formula. 
> 

Isn't it true that the actual set of universal numbers rather depends
on one chosen enumeration? So universality is not a property of the
numbers per se?

-- 


Dr Russell StandishPhone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Senior Research Fellowhpco...@hpcoders.com.au
Economics, Kingston University http://www.hpcoders.com.au


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Re: What is a Löbian machine/number/combinator

[Bruno Marchal]

> On 19 Apr 2018, at 02:57, Lawrence Crowell  
> wrote:
> 
> I will try to get back to this with comments, but right now I am too tired 
> out.

OK. You need to rest, we have all  time,

Bruno


> 
> LC
> 
> On Wednesday, April 18, 2018 at 12:11:35 PM UTC-5, Bruno Marchal wrote:
> Somewhere: (and I copy my answer, as some people asked me this in this list 
> too).
> 
> 
>> 
>> What are Lobian numbers? Can you give a reference? I know little bit about 
>> Godel’s work.
> 
> 
> Consider any Turing universal machinery, for example the programming language 
> c++. 
> 
> N is the set of natural numbers.
> 
> It is known that the enumeration of all programs computing a (perhaps not 
> everywhere defined) function from N to N exists, and so we get a list of all 
> partial computable function phi_i from N to N. (i.e. phi_0, phi_1, phi_2, …), 
> by enumerating the program with one natural number argument) written in C++, 
> in their lexico-graphical order (length, and alphabetical for the programs 
> with the same length).
> 
> We can define a universal number as a number u such that phI_u(x, y) = 
> phi_x(y). We say that u implements x on y. (It is a constructive definition 
> of a computer in the language of the computer).
> 
> Now, once we have a universal number, we can transform/extend it into a 
> theory, which is the first order logical specification of how u operates. 
> That is a standard mapping from, say, c++ to a Turing universal logical 
> theory. 
> 
> I assume we have done that, so now I say that a universal number is Löbian 
> when it has enough induction axioms (added to its logical specification) so 
> that it can prove enough of some special formula. 
> 
> If “[]” represents the provability predicate (Gödel 1931)of some first order 
> Turing universal theory/number, Löbian means that it can prove p -> []p for 
> all p equivalent with a semi-computable predicate known as sigma_1 
> predicate). In fact “p -> []p” is equivalent with Turing universality, and if 
> a Universal can prove this for all p sigma_1, it will not only be Turing 
> universal, but it will know (in some technical sense) that it is Turing 
> Universal.
> 
> “[]” itself is sigma_1, which entails that []p -> [][]p is provable.
> 
> Those corresponds to what is called “sufficiently rich theories” (for proving 
> their own incompleteness theorem).
> 
> Löbianity appears when you add to:
> 
> 0 ≠ s(x)
> s(x) = s(y) -> x = y
> x = 0 v Ey(x = s(y))
> x+0 = x
> x+s(y) = s(x+y)
> x*0=0
> x*s(y)=(x*y)+x,
> 
> The induction axioms:
> 
> (F(0) & Ax(F(x) -> F(s(x))) -> AxF(x), with F(x) being a formula in the 
> arithmetical language (with "0, s, +, *)
> 
> 
> F being a formula belonging to some set of formula. If you limit F to the 
> recursive, sigma_0, formula, you don’t get Löbianity, unless you add the 
> exponentiation axioms.
> 
> You can (and I will) limit p to the sigma_1 sentences, the semi-computable 
> predicate/function. That is enough to get Löbianity, and inherit, in the 
> “ideal” sound case the “theology” of number/machine/combinator… beings.
> 
> With p sigma_1 Universality means that p_>[]p is true, and Löbianity is when 
> the machine/number proves p -> []p for all p (sigma_1).
> 
> []p -> p, although true (by definition of sound machine/number) remains 
> unprovable in general. Typically the Löbian machine cannot prove []f -> f.
> 
> 
> Peano is a Löbian theory/program/idea/machine/word Universal).
> 
> ZF too, much more “crazy machine” which believes in the axiom of infinity, 
> but then get doubt about the choice axioms!
> (As I stay in very elementary arithmetic (no induction axioms) I still 
> studies the web of Löbian dreams realised in the non Löbian reality.
> 
> 
> Provability is relative, but computability is absolute. Sigma_1 completeness, 
> that is the truth of p -> []p, for p sigma_1, is Turing universal.
> Löbianity is when the machine believes in enough induction axioms to prove p 
> -> []p for each p sigma_1. 
> 
> It obeys to the modal logics of self-reference G and G*, which helps to 
> summarise the “theology” of the finite universal 
> number/machine/combinator/ universal system >.
> 
> Best,
> 
> Bruno
> 
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Re: What is a Löbian machine/number/combinator

[Lawrence Crowell]

I will try to get back to this with comments, but right now I am too tired 
out.

LC

On Wednesday, April 18, 2018 at 12:11:35 PM UTC-5, Bruno Marchal wrote:
>
> Somewhere: (and I copy my answer, as some people asked me this in this 
> list too).
>
>
>
> What are Lobian numbers? Can you give a reference? I know little bit about 
> Godel’s work.
>
>
>
> Consider any Turing universal machinery, for example the programming 
> language c++. 
>
> N is the set of natural numbers.
>
> It is known that the enumeration of all programs computing a (perhaps not 
> everywhere defined) function from N to N exists, and so we get a list of 
> all partial computable function phi_i from N to N. (i.e. phi_0, phi_1, 
> phi_2, …), by enumerating the program with one natural number argument) 
> written in C++, in their lexico-graphical order (length, and alphabetical 
> for the programs with the same length).
>
> We can define a universal number as a number u such that phI_u(x, y) = 
> phi_x(y). We say that u implements x on y. (It is a constructive definition 
> of a computer in the language of the computer).
>
> Now, once we have a universal number, we can transform/extend it into a 
> theory, which is the first order logical specification of how u operates. 
> That is a standard mapping from, say, c++ to a Turing universal logical 
> theory. 
>
> I assume we have done that, so now I say that a universal number is Löbian 
> when it has enough induction axioms (added to its logical specification) so 
> that it can prove enough of some special formula. 
>
> If “[]” represents the provability predicate (Gödel 1931)of some first 
> order Turing universal theory/number, Löbian means that it can prove p -> 
> []p for all p equivalent with a semi-computable predicate known as sigma_1 
> predicate). In fact “p -> []p” is equivalent with Turing universality, and 
> if a Universal can prove this for all p sigma_1, it will not only be Turing 
> universal, but it will know (in some technical sense) that it is Turing 
> Universal.
>
> “[]” itself is sigma_1, which entails that []p -> [][]p is provable.
>
> Those corresponds to what is called “sufficiently rich theories” (for 
> proving their own incompleteness theorem).
>
> Löbianity appears when you add to:
>
> 0 ≠ s(x)
> s(x) = s(y) -> x = y
> x = 0 v Ey(x = s(y))
> x+0 = x
> x+s(y) = s(x+y)
> x*0=0
> x*s(y)=(x*y)+x,
>
> The induction axioms:
>
> (F(0) & Ax(F(x) -> F(s(x))) -> AxF(x), with F(x) being a formula in the 
> arithmetical language (with "0, s, +, *)
>
>
> F being a formula belonging to some set of formula. If you limit F to the 
> recursive, sigma_0, formula, you don’t get Löbianity, unless you add the 
> exponentiation axioms.
>
> You can (and I will) limit p to the sigma_1 sentences, the semi-computable 
> predicate/function. That is enough to get Löbianity, and inherit, in the 
> “ideal” sound case the “theology” of number/machine/combinator… beings.
>
> With p sigma_1 Universality means that p_>[]p is true, and Löbianity is 
> when the machine/number proves p -> []p for all p (sigma_1).
>
> []p -> p, although true (by definition of sound machine/number) remains 
> unprovable in general. Typically the Löbian machine cannot prove []f -> f.
>
>
> Peano is a Löbian theory/program/idea/machine/word Universal).
>
> ZF too, much more “crazy machine” which believes in the axiom of infinity, 
> but then get doubt about the choice axioms!
> (As I stay in very elementary arithmetic (no induction axioms) I still 
> studies the web of Löbian dreams realised in the non Löbian reality.
>
>
> Provability is relative, but computability is absolute. Sigma_1 
> completeness, that is the truth of p -> []p, for p sigma_1, is Turing 
> universal.
> Löbianity is when the machine believes in enough induction axioms to prove 
> p -> []p for each p sigma_1. 
>
> It obeys to the modal logics of self-reference G and G*, which helps to 
> summarise the “theology” of the finite universal 
> number/machine/combinator/ universal system >.
>
> Best,
>
> Bruno
>

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What is a Löbian machine/number/combinator

[Bruno Marchal]

Somewhere: (and I copy my answer, as some people asked me this in this list 
too).


> 
> What are Lobian numbers? Can you give a reference? I know little bit about 
> Godel’s work.


Consider any Turing universal machinery, for example the programming language 
c++. 

N is the set of natural numbers.

It is known that the enumeration of all programs computing a (perhaps not 
everywhere defined) function from N to N exists, and so we get a list of all 
partial computable function phi_i from N to N. (i.e. phi_0, phi_1, phi_2, …), 
by enumerating the program with one natural number argument) written in C++, in 
their lexico-graphical order (length, and alphabetical for the programs with 
the same length).

We can define a universal number as a number u such that phI_u(x, y) = 
phi_x(y). We say that u implements x on y. (It is a constructive definition of 
a computer in the language of the computer).

Now, once we have a universal number, we can transform/extend it into a theory, 
which is the first order logical specification of how u operates. That is a 
standard mapping from, say, c++ to a Turing universal logical theory. 

I assume we have done that, so now I say that a universal number is Löbian when 
it has enough induction axioms (added to its logical specification) so that it 
can prove enough of some special formula. 

If “[]” represents the provability predicate (Gödel 1931)of some first order 
Turing universal theory/number, Löbian means that it can prove p -> []p for all 
p equivalent with a semi-computable predicate known as sigma_1 predicate). In 
fact “p -> []p” is equivalent with Turing universality, and if a Universal can 
prove this for all p sigma_1, it will not only be Turing universal, but it will 
know (in some technical sense) that it is Turing Universal.

“[]” itself is sigma_1, which entails that []p -> [][]p is provable.

Those corresponds to what is called “sufficiently rich theories” (for proving 
their own incompleteness theorem).

Löbianity appears when you add to:

0 ≠ s(x)
s(x) = s(y) -> x = y
x = 0 v Ey(x = s(y))
x+0 = x
x+s(y) = s(x+y)
x*0=0
x*s(y)=(x*y)+x,

The induction axioms:

(F(0) & Ax(F(x) -> F(s(x))) -> AxF(x), with F(x) being a formula in the 
arithmetical language (with "0, s, +, *)


F being a formula belonging to some set of formula. If you limit F to the 
recursive, sigma_0, formula, you don’t get Löbianity, unless you add the 
exponentiation axioms.

You can (and I will) limit p to the sigma_1 sentences, the semi-computable 
predicate/function. That is enough to get Löbianity, and inherit, in the 
“ideal” sound case the “theology” of number/machine/combinator… beings.

With p sigma_1 Universality means that p_>[]p is true, and Löbianity is when 
the machine/number proves p -> []p for all p (sigma_1).

[]p -> p, although true (by definition of sound machine/number) remains 
unprovable in general. Typically the Löbian machine cannot prove []f -> f.


Peano is a Löbian theory/program/idea/machine/word []p, for p sigma_1, is Turing universal.
Löbianity is when the machine believes in enough induction axioms to prove p -> 
[]p for each p sigma_1. 

It obeys to the modal logics of self-reference G and G*, which helps to 
summarise the “theology” of the finite universal 
number/machine/combinator/.

Best,

Bruno

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