Re: The seven step-Mathematical preliminaries 2

[m.a.]

Bruno,
   Thanks for the encouragement. I intend to follow your instructions 
and it's a relief to know that some of my answers were correct. However, I will 
be away for two weeks and unable to work on the lessons. I'll try to make up 
for it when I return. Best,   


marty a.


  - Original Message - 
  From: Bruno Marchal 
  To: everything-list@googlegroups.com 
  Sent: Friday, June 05, 2009 10:03 AM
  Subject: Re: The seven step-Mathematical preliminaries 2


  Hi Marty,


  On 05 Jun 2009, at 00:30, m.a. wrote:






Bruno,
   I don't have dyslexia 






  Good news.










but my keyboard doesn't contain either the UNION symbol or the INTERSECTION 
symbol 




  Nor do mine!








(unless I want to go into an INSERT pull down menu every time I use those 
symbols). 




  Like I have to do too.








I don't need you to switch to English symbols, but I would like to see the 
English equivalents of the symbols you use (so that I can use them). 




  I gave them.








I would also like a reference table defining each term in both your symbols 
and their English equivalents which I could look back to when I get confused. 




  I suggest you do this by yourself. It is a good exercise and it will help you 
not only in the understanding, but in the memorizing. Then you submit it to the 
list, and I can verify the understanding. 










Please include examples.


  Up to now, I did it for any notions introduced. Just ask me one or two or 
(name your number) examples more in case you have a doubt. If I send too much 
posts, and if there are too long, people will dismiss them. try to ask explicit 
question, like you did, actually.






I tend to be somewhat careless when dealing  with very fine distinctions 




  This means that a lot of work is awaiting for you. It is normal. Everyone can 
understand what I explain, but some have more work to do.








and may type the wrong symbol while intending to type the correct one. 




  That is unimportant. I am used to do typo errors too. One of my favorite book 
on self-reference (the one by Smorynski) contains an average of two or three 
typo error per page. Of course, once a typo error is found, it is better to 
correct it. 










Also, I must admit that the lessons are going too fast for me and are 
moving ahead before I've mastered the previous material.




  We have all the time, and up to now I did not proceed without having the 
answer of all exercises. You make no faults in the first set of seven exercise, 
and that is why I have quickly proceed to the second round. For that one, you 
make just one error, + the dismiss of a paragraph on "UNION".  To slow me down 
it is enough to tell me things like "I don't understand what you mean by this 
or that" and you quote the unclear passage. If you can't do an exercise, just 
wait for some other (Kim?) to propose a solution. Or try to guess one and 
submit, or just ask. I will not proceed to new matters before I am sure you 
grasp all what has been already presented.
  What is possible is that you understand, but fail ti memorize. This will lead 
to problems later. So you have to make your own summary and be sure you can 
easily revise the definition.








If I'm requesting too much simplification, please let me know because I'm 
quite well adjusted to my math disabilities and won't take offence at all. 
Thanks,  marty a.






  I think that there is no problem at all. I am just waiting for explicit 
question from the second round. You can ask any question, and slow me down as 
much as you want so that we proceed at your own rhythm. 
  Don't ask me to slow down in any abstract way. You are the one who have to 
slow me down by pointing on what you don't understand in a post.
  take it easy, and take all your time. Don't try to understand the more 
advanced replies I give to people who have a bigger baggage.


  You did show me that you have understood the notion of set, and the notion of 
intersection of sets. Have you a problem with the notion of union of sets? If 
that is the case, just quote the passage of my post that you don't understand, 
or the example that I gave, and I will explain. Try to keep those post in some 
well ranged place so as to re-access them easily.


  I ask this to Kim too, and any one interested: just let me know what you 
don't understand, so that I can explain, give other examples, etc. 


  Take it easy, you seem quite good, you suffer just of a problem of 
familiarity with notations. You read the post too quickly, I suspect also.


  Are you OK? I can understand you could be afraid of the amount of work, but 
given that we have all the time, there is no exams, nor deadline, I am not sure 
there is any probl

Re: The seven step-Mathematical preliminaries

[A. Wolf]

> From what you said earlier, BIGGEST={0,1,...,BIGGEST-1}.  Then
> BIGGEST+1={0,1,...,BIGGEST-1} union {BIGGEST} = {0,1,...,BIGGEST}.
> Why would {0,1,...BIGGEST} not be a natural number while
> {0,1,...,BIGGEST-1} is?

If {0, 1, ... , BIGGEST-1} is a natural number, then {0,1,...,BIGGEST} is 
too, and then so is {0, 1, ... , BIGGEST+1}, etc.  There's no such thing as 
a largest natural number: that's the whole point of the construction.  The 
set of all natural numbers is an infinite set, unbounded above.  The set N 
has no largest element within it: it is the set of all finite ordinals.  N 
(usually called omega when treated as an ordinal) has no predecessor, 
because it is formed by taking the limit of all the ordinals below it, *not* 
by applying the successor function "x+ = x U {x}".  This is the way 
well-ordering works...it's not symmetric.  So any set described {a, b, ... , 
z} in the standard way is not N.

N is not the successor of any natural number; rather, it contains them all. 
This allows us to talk about (and prove things about) all natural numbers. 
This isn't an arbitrary mathematical choice.  Without infinite sets, we 
would be unable to rigorously prove things by induction, which is necessary 
for a wide array of basic arithmetical proofs.  This is because a finite set 
of natural numbers cannot be closed under successor (or addition or 
multiplication, for that matter).  If you relied on only finitely many 
numbers, your functions could take natural numbers and hand you back 
something that isn't a number at all.  This makes even basic math untenable. 
Taking the closure of {} under successor is the solution.

(There are non-standard models of the natural numbers that contain numbers 
other than the elements of N, but these are not well-ordered.)

Anna


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Re: Cognitive Theoretic Model of the Universe

[Bruno Marchal]

On 05 Jun 2009, at 14:23, ronaldheld wrote:

>
> Bruno:
> I understand a little better. is there a citition for a version of
> Church Thesis that all algorithm can be written in
> FORTRAN?


The original Church Thesis, (also due to Post, Turing, Markov, Kleene,  
and others independently)

is this:

A function is computable if and only if it is programmable in LAMDA  
CALCULUS.

Then it is an easy but tedious exercise of programing to show that you  
can simulate LAMDA CALCULUS with FORTRAN, and that you can simulate  
FORTRAN with LAMBDA CALCULUS. So they compute the same functions.

And the same is true with LISP, or JAVA, or ALGOL, or C++, etc... in  
the place of FORTRAN.

A thorough introduction to Church thesis, and I would say one far  
deeper than usual, is integrally part of the seventh step of UDA. So  
we will come back on this soon or later. Church thesis is really the  
key and the motor of both UDA and AUDA. I have discovered that it is  
rarely well understood, even by many "experts". Like Gödel's theorem,  
Church's thesis is often deformed or misused.

Bruno


http://iridia.ulb.ac.be/~marchal/




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Re: The seven step-Mathematical preliminaries 2

[Bruno Marchal]

Hi Marty,

On 05 Jun 2009, at 00:30, m.a. wrote:



> Bruno,
>I don't have dyslexia



Good news.





> but my keyboard doesn't contain either the UNION symbol or the  
> INTERSECTION symbol


Nor do mine!




> (unless I want to go into an INSERT pull down menu every time I use  
> those symbols).


Like I have to do too.




> I don't need you to switch to English symbols, but I would like to  
> see the English equivalents of the symbols you use (so that I can  
> use them).


I gave them.




> I would also like a reference table defining each term in both your  
> symbols and their English equivalents which I could look back to  
> when I get confused.


I suggest you do this by yourself. It is a good exercise and it will  
help you not only in the understanding, but in the memorizing. Then  
you submit it to the list, and I can verify the understanding.





> Please include examples.

Up to now, I did it for any notions introduced. Just ask me one or two  
or (name your number) examples more in case you have a doubt. If I  
send too much posts, and if there are too long, people will dismiss  
them. try to ask explicit question, like you did, actually.



> I tend to be somewhat careless when dealing  with very fine  
> distinctions


This means that a lot of work is awaiting for you. It is normal.  
Everyone can understand what I explain, but some have more work to do.




> and may type the wrong symbol while intending to type the correct one.


That is unimportant. I am used to do typo errors too. One of my  
favorite book on self-reference (the one by Smorynski) contains an  
average of two or three typo error per page. Of course, once a typo  
error is found, it is better to correct it.





> Also, I must admit that the lessons are going too fast for me and  
> are moving ahead before I've mastered the previous material.


We have all the time, and up to now I did not proceed without having  
the answer of all exercises. You make no faults in the first set of  
seven exercise, and that is why I have quickly proceed to the second  
round. For that one, you make just one error, + the dismiss of a  
paragraph on "UNION".  To slow me down it is enough to tell me things  
like "I don't understand what you mean by this or that" and you quote  
the unclear passage. If you can't do an exercise, just wait for some  
other (Kim?) to propose a solution. Or try to guess one and submit, or  
just ask. I will not proceed to new matters before I am sure you grasp  
all what has been already presented.
What is possible is that you understand, but fail ti memorize. This  
will lead to problems later. So you have to make your own summary and  
be sure you can easily revise the definition.




> If I'm requesting too much simplification, please let me know  
> because I'm quite well adjusted to my math disabilities and won't  
> take offence at all. Thanks,  marty a.



I think that there is no problem at all. I am just waiting for  
explicit question from the second round. You can ask any question, and  
slow me down as much as you want so that we proceed at your own rhythm.
Don't ask me to slow down in any abstract way. You are the one who  
have to slow me down by pointing on what you don't understand in a post.
take it easy, and take all your time. Don't try to understand the more  
advanced replies I give to people who have a bigger baggage.

You did show me that you have understood the notion of set, and the  
notion of intersection of sets. Have you a problem with the notion of  
union of sets? If that is the case, just quote the passage of my post  
that you don't understand, or the example that I gave, and I will  
explain. Try to keep those post in some well ranged place so as to re- 
access them easily.

I ask this to Kim too, and any one interested: just let me know what  
you don't understand, so that I can explain, give other examples, etc.

Take it easy, you seem quite good, you suffer just of a problem of  
familiarity with notations. You read the post too quickly, I suspect  
also.

Are you OK? I can understand you could be afraid of the amount of  
work, but given that we have all the time, there is no exams, nor  
deadline, I am not sure there is any problem. Of course, things of  
life (like holidays, taxes, etc.) can slow us down too, but this is  
not a real problem. Of course you can realize you don't want really to  
learn all this: in that case you tell me, and we can stop, or make a  
pause, etc.

I choose the path (given that the goal is given: explaining the real  
stuff in the UDA-step seven), and you can accelerate me, slow me down,  
halt, etc. as you wish. OK?


Just tell me if you have a problem with the two statements quoted  
below. I think we could make post with fewer examples, and fewer  
exercises, perhaps. Don't be ashamed by any question you want to ask.  
There is no shame in questioning anything.

>
> Examples
> {1, 2, 3} ∩  {3, 4, 5} = {3}
> {1, 2, 3} ∪

Re: The seven step-Mathematical preliminaries

[Brian Tenneson]

Torgny Tholerus wrote:
> Brian Tenneson skrev:
>   
>> On Thu, Jun 4, 2009 at 8:27 AM, Torgny Tholerus > > wrote:
>>
>>
>> Brian Tenneson skrev:
>> >
>> >
>> > Torgny Tholerus wrote:
>> >> It is impossible to create a set where the successor of every
>> element is
>> >> inside the set, there must always be an element where the
>> successor of
>> >> that element is outside the set.
>> >>
>> > I disagree.  Can you prove this?
>> > Once again, I think the debate ultimately is about whether or not to
>> > adopt the axiom of infinity.
>> > I think everyone can agree without that axiom, you cannot "build" or
>> > "construct" an infinite set.
>> > There's nothing right or wrong with adopting any axioms.  What
>> results
>> > is either interesting or not, relevant or not.
>>
>> How do you handle the Russell paradox with the set of all sets
>> that does
>> not contain itself?  Does that set contain itself or not?
>>
>>  
>> If we're talking about ZFC set theory, then the axiom of foundation 
>> prohibits sets from being elements of themselves.
>> I think we agree that in ZFC, there is no set of all sets.
>> 
>
> But there is a set of all sets.  You can construct it by taking all 
> sets, and from them doing a new set, the set of all sets.  But note, 
> this set will not contain itself, because that set did not exist before.
>   
If that set does not contain itself then it is not a set of all sets.

>   
>>  
>>
>>
>>
>> My answer is that that set does not contain itself, because no set can
>> contain itself.  So the set of all sets that does not contain
>> itself, is
>> the same as the set of all sets.  And that set does not contain
>> itself.
>> This set is a set, but it does not contain itself.  It is exactly the
>> same with the natural numbers, *BIGGEST+1 is a natural number, but it
>> does not belong to the set of all natural numbers.  *The set of
>> all sets
>> is a set, but it does not belong to the set of all sets.
>>
>> How can BIGGEST+1 be a natural number but not belong to the set of all 
>> natural numbers?
>> 
>
> One way to represent natural number as sets is:
>
> 0 = {}
> 1 = {0} = {{}}
> 2 = {0, 1} = 1 union {1} = {{}, {{}}}
> 3 = {0, 1, 2} = 2 union {2} = ...
> . . .
> n+1 = {0, 1, 2, ..., n} = n union {n}
> . . .
>
> Here you can then define that a is less then b if and only if a belongs 
> to b.
>
> With this notation you get the set N of all natural numbers as {0, 1, 2, 
> ...}.  But the remarkable thing is that N is exactly the same as 
> BIGGEST+1.  BIGGEST+1 is a set with the same structure as all the other 
> natural numbers, so it is then a natural number.  But BIGGEST+1 is not a 
> member of N, the set of all natural numbers.  BIGGEST+1 is bigger than 
> all natural numbers, because all natural numbers belongs to BIGGEST+1.
>   
Right, so n+1 is a natural number whenever n is. 
>   
>>  
>>
>>
>> >
>> >> What the largest number is depends on how you define "natural
>> number".
>> >> One possible definition is that N contains all explicit numbers
>> >> expressed by a human being, or will be expressed by a human
>> being in the
>> >> future.  Amongst all those explicit numbers there will be one
>> that is
>> >> the largest.  But this "largest number" is not an explicit number.
>> >>
>> >>
>> > This raises a deeper question which is this: is mathematics
>> dependent
>> > on humanity or is mathematics independent of humanity?
>> > I wonder what would happen to that human being who finally expresses
>> > the largest number in the future.  What happens to him when he wakes
>> > up the next day and considers adding one to yesterday's number?
>>
>> This is no problem.  If he adds one to the explicit number he
>> expressed
>> yesterday, then this new number is an explicit number, and the number
>> expressed yesterday was not the largest number.  Both 17 and 17+1 are
>> explicit numbers.
>>
>> This goes back to my earlier comment that it's hard for me to believe 
>> that the following statement is false:
>> every natural number has a natural number successor
>> We -must- be talking about different things, then, when we use the 
>> phrase natural number.
>> I can't say your definition of natural numbers is right and mine is 
>> wrong, or vice versa.  I do wonder what advantages there are to the 
>> ultrafinitist approach compared to the math I'm familiar with. 
>> 
>
> The biggest advantage is that everything is finite, and you can then 
> really know that the mathematical theory you get is consistent, it does 
> not contain any contradictions.
>
>   
 From what you said earlier, BIGGEST={0,1,...,BIGGEST-1}.  Then 
BIGGEST+1={0,1,...,BIGGEST-1} union {BIGGEST} = {0,1,...,BIGGEST}.
Why would {0,1,...BIGGEST} not be a natural number while 
{0,1,...,BIGGEST

Re: Cognitive Theoretic Model of the Universe

[Bruno Marchal]

On 04 Jun 2009, at 21:23, Brent Meeker wrote:

>
> Bruno Marchal wrote:
>> ...
>>> Bruno Marchal wrote:
>>
>> The whole point of logic is to consider the "Peano's axioms" as a
>> mathematical object itself, which is studied mathematically in the
>> usual informal (yet rigorous and typically mathematica) way.
>>
>> PA, and PA+GOLDBACH are different mathematical objects. They are
>> different theories, or different machines.
>>
>> Now if GOLDBACH is provable by PA, then PA and PA+GOLDBACH shed the
>> same light on the same arithmetical truth. In that case I will
>> identify PA and PA+GOLDBACH, in many contexts, because most of the
>> time I identify a theory with its set of theorems. Like I identify a
>> person with its set of (possible) beliefs.
>>
>> If GOLDBACH is *true, but not provable* by PA, then PA and PA 
>> +GOLDBACH
>> still talk on the same reality, but PA+GOLDBACH will shed more light
>> on it, by proving more theorems on the numbers and numbers relations
>> than PA. I do no more identify them, and they have different set of
>> theorems.
>>
>> If GOLDBACH is false. Well GOLBACH is PI_1, that is, its negation is
>> SIGMA_1, that is, it has the shape "it exist a number such that it
>> verify this decidable property". Indeed the negation of Goldbach
>> conjecture is "it exists a number bigger than 2 which is not the sum
>> of two primes". This, if true, is verifiable already by the much
>> weaker RA (Robinson arithmetic). So, if GOLDBACH is false PA +
>> GOLDBACH is inconsistent. That is a mathematical object quite
>> different from PA!
>
> So what then is the status of the natural numbers?  Are there many
> different objects in Platonia which we loosely refer to as "the  
> natural
> numbers" or is there only one such object and the Goldbach  
> conjecture is
> either true of false of this object?

Nobody can answer this question in your place.
But if you believe that the principle of excluded middle can be  
applied to closed arithmetical sentences, like 99,999% of the  
mathematician, then you have to believe that the Goldbach conjecture  
is either true or false.
Even intuitionist will admit that Goldabch conjecture is true or  
false, given its Sigma_1 character. This means that, about the (true- 
or-false) nature of GOLDBACH is doubtable only for an ultrafinitist.
BTW, Goldbach conjecture asserts that all female (even) numbers can be  
written as a sum of two primes, except the number two. (I forget the  
word "even" in my enunciation above!).





>
>>
>> Here, you would have taken the twin primes conjecture, and things
>> would have been different, and more complex.
>
> Because, even if it is false, it cannot be proven false by  
> exhibiting an
> example?

Yes. And this entails that both PA+TPC and PA + (~TPC) could be  
consistent, yet one of those theory has to be unsound, or if you  
prefer has to enunciate false arithmetical statements (yet consistent  
with PA).
"Sound" is relative to the usual understanding of the natural numbers  
which is presupposed in any work in mathematical logic or computer  
science, like it is presupposed in any part of any physical theory.  
That usual meaning is taught in primary school without any trouble.
In model theory, this notion of soundness can be made more precise,  
through the notion of standard model of PA for example, but this  
presupposes, in the meta-theory, an understanding of that usual notion  
of numbers.
Nobody doubts the consistency and soundness of the theories like RA  
and PA. (Even Torgny, who fakes that he doubts them for a  
philosophical purpose unrelated to our discussion, like he fakes to be  
a faking zombie, etc. This is clear from older post by Torgny).


>
>
>>
>> Note that a theory of set like ZF shed even much more large light on
>> arithmetical truth, (and is still incomplete on arithmetic, by  
>> Gödel ...).
>> Incidentally it can be shown that ZF and ZFC, although they shed
>> different light on the mathematical truth in general, does shed
>> exactly the same light on arithmetical truth. They prove the same
>> arithmetical theorems. On the numbers, the axiom of choice add
>> nothing. This is quite unlike the ladder of infinity axioms.
>>
>> I would say it is and will be particularly important to distinguish
>> chatting beings like RA, PA, ZF, ZFC, etc... and what those beings  
>> are
>> talking about.
>>
>> Bruno
>
> Do you mean PA talks about the natural numbers but PA+theorems is a
> different mathematical object than N?


I am not sure I understand what you mean. PA is an (immaterial)  
machine, or a program if you want. I guess that, by PA+theorems, you  
mean the set of theorems of PA. In some context we can identify PA and  
PA+theorems, because the context makes things unambiguous. But  
strictly speaking those are different mathematical object: PA is  
finite (well, as I defined it usually), But PA+theorems is infinite.  
Both talk about N, and both are different of N. Indeed PA is a finite  
(or infinite

RE: The seven step-Mathematical preliminaries

[Jesse Mazer]

> Date: Fri, 5 Jun 2009 08:33:47 +0200
> From: tor...@dsv.su.se
> To: everything-list@googlegroups.com
> Subject: Re: The seven step-Mathematical preliminaries
> 
> 
> Brian Tenneson skrev:
>>
>>
>> On Thu, Jun 4, 2009 at 8:27 AM, Torgny Tholerus > > wrote:
>>
>>
>> Brian Tenneson skrev:
>>>
>>>
>>> Torgny Tholerus wrote:
>>>> It is impossible to create a set where the successor of every
>> element is
>>>> inside the set, there must always be an element where the
>> successor of
>>>> that element is outside the set.
>>>>
>>> I disagree.  Can you prove this?
>>> Once again, I think the debate ultimately is about whether or not to
>>> adopt the axiom of infinity.
>>> I think everyone can agree without that axiom, you cannot "build" or
>>> "construct" an infinite set.
>>> There's nothing right or wrong with adopting any axioms.  What
>> results
>>> is either interesting or not, relevant or not.
>>
>> How do you handle the Russell paradox with the set of all sets
>> that does
>> not contain itself?  Does that set contain itself or not?
>>
>>  
>> If we're talking about ZFC set theory, then the axiom of foundation 
>> prohibits sets from being elements of themselves.
>> I think we agree that in ZFC, there is no set of all sets.
> 
> But there is a set of all sets.  You can construct it by taking all 
> sets, and from them doing a new set, the set of all sets.  But note, 
> this set will not contain itself, because that set did not exist before.
> 
>>  
>>
>>
>>
>> My answer is that that set does not contain itself, because no set can
>> contain itself.  So the set of all sets that does not contain
>> itself, is
>> the same as the set of all sets.  And that set does not contain
>> itself.
>> This set is a set, but it does not contain itself.  It is exactly the
>> same with the natural numbers, *BIGGEST+1 is a natural number, but it
>> does not belong to the set of all natural numbers.  *The set of
>> all sets
>> is a set, but it does not belong to the set of all sets.
>>
>> How can BIGGEST+1 be a natural number but not belong to the set of all 
>> natural numbers?
> 
> One way to represent natural number as sets is:
> 
> 0 = {}
> 1 = {0} = {{}}
> 2 = {0, 1} = 1 union {1} = {{}, {{}}}
> 3 = {0, 1, 2} = 2 union {2} = ...
> . . .
> n+1 = {0, 1, 2, ..., n} = n union {n}
> . . .
> 
> Here you can then define that a is less then b if and only if a belongs 
> to b.
> 
> With this notation you get the set N of all natural numbers as {0, 1, 2, 
> ...}.  But the remarkable thing is that N is exactly the same as 
> BIGGEST+1.  BIGGEST+1 is a set with the same structure as all the other 
> natural numbers, so it is then a natural number.  But BIGGEST+1 is not a 
> member of N, the set of all natural numbers.
Here you're just contradicting yourself. If you say BIGGEST+1 "is then a 
natural number", that just proves that the set N was not in fact the set "of 
all natural numbers". The alternative would be to say BIGGEST+1 is *not* a 
natural number, but then you need to provide a definition of "natural number" 
that would explain why this is the case.
> The biggest advantage is that everything is finite, and you can then 
> really know that the mathematical theory you get is consistent, it does 
> not contain any contradictions.

Even if you define "natural number" in such a way that there are only a finite 
number of them (which you haven't actually done, you've just asserted it 
without providing any specific definition), you still could have an infinite 
number of *propositions* about them if you allow each proposition to contain an 
unlimited number of AND and OR operators. For example, even if I say that the 
only natural numbers are 1,2,3, I can still make arbitrarily long propositions 
like ((3>1) AND (2>1)) OR (3>1)) AND ((2>3) OR (3>1)) AND ((2>3) OR ((1>3) OR 
((2>1) OR ((1>3) OR (3>1). Of course a non-finitist would be able to prove 
that these infinite number of propositions are consistent, but I don't know if 
an ultrafinitist would (likewise a non-finitist can accept a proof that 
something like the Peano axioms are consistent based on an understanding of 
their application to a model dealing with rows of dots, even if the Peano 
axioms cannot be used to formally prove their own consistency).
Jesse
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Re: Cognitive Theoretic Model of the Universe

[Quentin Anciaux]

Well as FORTRAN is a turing complete language, then you can.

As long as the programming language is universal/turing complete you can.

http://en.wikipedia.org/wiki/Turing_completeness

Regards,
Quentin

2009/6/5 ronaldheld :
>
> Bruno:
>  I understand a little better. is there a citition for a version of
> Church Thesis that all algorithm can be written in
> FORTRAN?
>                         Ronald
>
>
> On Jun 4, 10:49 am, Bruno Marchal  wrote:
>> Hi Ronald,
>>
>> On 02 Jun 2009, at 16:45, ronaldheld wrote:
>>
>>
>>
>> > Bruno:
>> >   Since I program in Fortran, I am uncertain how to interpret things.
>>
>> I was alluding to old, and less old, disputes again programmers, about
>> which programming language to prefer.
>> It is a version of Church Thesis that all algorithm can be written in
>> FORTRAN. But this does not mean that it is relevant to define an
>> algorithm by a fortran program. I thought this was obvious, and I was
>> using that "known" confusion to point on a similar confusion in Set
>> Theory, like Langan can be said to perform.
>>
>> In Set Theorist, we still find often the error consisting in defining
>> a mathematical object by a set. I have done that error in my youth.
>> What you can do, indeed, is to *represent* (almost all) mathematical
>> objects by sets. Langan seems to make that mistake.
>>
>> The point is just that we have to distinguish a mathematical object
>> and the representation of that object in some mathematical theory.
>>
>> I will have the opportunity to give a precise example in the 7th
>> thread later.
>>
>> In usual mathematical practice, this mistake is really not important,
>> yet, in logic it is more important to take into account that
>> distinction, and then in cognitive science it is *very* important.
>> Crucial, I would say. The error consisting in identifying
>> consciousness and brain state belongs to that family, for example. To
>> confuse a person and its body belongs to that family of error too.
>>
>> All such error are of the form of the confusion between the Moon and
>> the finger which point to the moon, or the confusion between a map and
>> the territory.
>>
>> I have nothing against the use of FORTRAN. On the contrary I have a
>> big respect for that old venerable high level programming language :)
>>
>> Bruno
>>
>> http://iridia.ulb.ac.be/~marchal/
> >
>



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Re: Cognitive Theoretic Model of the Universe

[ronaldheld]

Bruno:
 I understand a little better. is there a citition for a version of
Church Thesis that all algorithm can be written in
FORTRAN?
 Ronald


On Jun 4, 10:49 am, Bruno Marchal  wrote:
> Hi Ronald,
>
> On 02 Jun 2009, at 16:45, ronaldheld wrote:
>
>
>
> > Bruno:
> >   Since I program in Fortran, I am uncertain how to interpret things.
>
> I was alluding to old, and less old, disputes again programmers, about  
> which programming language to prefer.
> It is a version of Church Thesis that all algorithm can be written in  
> FORTRAN. But this does not mean that it is relevant to define an  
> algorithm by a fortran program. I thought this was obvious, and I was  
> using that "known" confusion to point on a similar confusion in Set  
> Theory, like Langan can be said to perform.
>
> In Set Theorist, we still find often the error consisting in defining  
> a mathematical object by a set. I have done that error in my youth.
> What you can do, indeed, is to *represent* (almost all) mathematical  
> objects by sets. Langan seems to make that mistake.
>
> The point is just that we have to distinguish a mathematical object  
> and the representation of that object in some mathematical theory.
>
> I will have the opportunity to give a precise example in the 7th  
> thread later.
>
> In usual mathematical practice, this mistake is really not important,  
> yet, in logic it is more important to take into account that  
> distinction, and then in cognitive science it is *very* important.  
> Crucial, I would say. The error consisting in identifying  
> consciousness and brain state belongs to that family, for example. To  
> confuse a person and its body belongs to that family of error too.
>
> All such error are of the form of the confusion between the Moon and  
> the finger which point to the moon, or the confusion between a map and  
> the territory.
>
> I have nothing against the use of FORTRAN. On the contrary I have a  
> big respect for that old venerable high level programming language :)
>
> Bruno
>
> http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries

[Quentin Anciaux]

2009/6/5 Torgny Tholerus :
>
> Kory Heath skrev:
>> On Jun 4, 2009, at 8:27 AM, Torgny Tholerus wrote:
>>
>>> How do you handle the Russell paradox with the set of all sets that
>>> does
>>> not contain itself?  Does that set contain itself or not?
>>>
>>> My answer is that that set does not contain itself, because no set can
>>> contain itself.  So the set of all sets that does not contain
>>> itself, is
>>> the same as the set of all sets.  And that set does not contain
>>> itself.
>>> This set is a set, but it does not contain itself.  It is exactly the
>>> same with the natural numbers, BIGGEST+1 is a natural number, but it
>>> does not belong to the set of all natural numbers.  The set of all
>>> sets
>>> is a set, but it does not belong to the set of all sets.
>>>
>>
>> So you're saying that the set of all sets doesn't contain all sets.
>> How is that any less paradoxical than the Russell paradox you're
>> trying to avoid?
>>
>
> The secret is the little word "all".  To be able to use that word, you
> have to define it.

I call that secret bullshit, and to understand that word (bullshit),
you have to define it.

Sorry but I think we're talking in english here, all means all not
what you decide it means.

Quentin.

> You can define it by saying: "By 'all sets' I mean
> that set and that set and that set and ...".  When you have made that
> definition, you are then able to create a new set, the set of all sets.
> But you must be carefull with what you do with that set.  That set does
> not contain itself, because it was not included in your definition of
> "all sets".
>
> If you call the set of all sets for A, then you have:
>
> For all x such that x is a set, then x belongs to A.
> A is a set.
>
> But it is illegal to substitute A for x, so you can not deduce:
>
> A is a set, then A belongs to A.
>
> This deductuion is illegal, because A is not included in the definition
> of "all x".
>
> --
> Torgny Tholerus
>
> >
>



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Re: The seven step-Mathematical preliminaries

[Torgny Tholerus]

Kory Heath skrev:
> On Jun 4, 2009, at 8:27 AM, Torgny Tholerus wrote:
>   
>> How do you handle the Russell paradox with the set of all sets that  
>> does
>> not contain itself?  Does that set contain itself or not?
>>
>> My answer is that that set does not contain itself, because no set can
>> contain itself.  So the set of all sets that does not contain  
>> itself, is
>> the same as the set of all sets.  And that set does not contain  
>> itself.
>> This set is a set, but it does not contain itself.  It is exactly the
>> same with the natural numbers, BIGGEST+1 is a natural number, but it
>> does not belong to the set of all natural numbers.  The set of all  
>> sets
>> is a set, but it does not belong to the set of all sets.
>> 
>
> So you're saying that the set of all sets doesn't contain all sets.  
> How is that any less paradoxical than the Russell paradox you're  
> trying to avoid?
>   

The secret is the little word "all".  To be able to use that word, you 
have to define it.  You can define it by saying: "By 'all sets' I mean 
that set and that set and that set and ...".  When you have made that 
definition, you are then able to create a new set, the set of all sets.  
But you must be carefull with what you do with that set.  That set does 
not contain itself, because it was not included in your definition of 
"all sets".

If you call the set of all sets for A, then you have:

For all x such that x is a set, then x belongs to A.
A is a set.

But it is illegal to substitute A for x, so you can not deduce:

A is a set, then A belongs to A.

This deductuion is illegal, because A is not included in the definition 
of "all x".

-- 
Torgny Tholerus

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