Re: The seven step-Mathematical preliminaries
On Wed Jun 3 0:39 , Bruno Marchal marc...@ulb.ac.be sent:
Hi Kim, Hi Marty and others,
So it is perhaps time to do some math.
It is
Obviously this is a not a course in math, but it is an explanation
from scratch of the seven step of the universal dovetailer argument.
It is a shortcut, and most probably we will make some digression from
time to time, but let us try not to digress too much.
Kim, you are OK with this?
I am
I have to take into account the problem you
did have with math, and which makes this lesson a bit challenging for
me, and I guess for you too.
Hopefully my innocence will allow me to bypass the pedantry and orthodoxies of
the field and allow a
shortcut to a high level of understanding of the UDA. Only a complete neophyte
would have the gall to
say something like that!
I begin with the very useful and elementary notion of set, as
explained in what is called naive set theory, and which is the base
of almost all part of math.
= begin
===
1) SET
Informal definition: a set is a collection of object, called elements,
with the idea that it, the collection or set, can be considered itself
as an object. It is a many seen as a one, if you want. If the set is
not to big, we can describe it exhaustively by listing the elements,
if the set is bigger, we can describe it by some other way. Usually we
use accolades {, followed by the elements, separated by commas, and
then }, in the exhaustive description of a set.
Example/exercise:
1) The set of odd natural numbers which are little than 10. This is a
well defined, and not to big set, so we can describe it exhaustively by
{1, 3, 5, 7, 9}. In this case we say that 7 belongs to {1, 3, 5, 7, 9}.
Exercise 1: does the number 24 belongs to the set {1, 3, 5, 7, 9}?
No
2) the set of even natural number which are little than 13. It is {0,
2, 4, 6, 8, 10, 12}. OK? Some people can have a difficulty which is
not related to the notion of set, for example they can ask themselves
if zero (0) is really an even number. We will come back to this.
3) The set of odd natural numbers which are little than 100. This set
is already too big to describe exhaustively. We will freely describe
such a set by a quasi exhaustion like {1, 3, 5, 7, 9, 11, ... 95, 97,
99}.
Exercise 2: does the number 93 belongs to the set of odd natural
numbers which are little than 100, that is: does 93 belongs to {1, 3,
5, 7, 9, 11, ... 95, 97, 99}?
Yes
4) The set of all natural numbers. This set is hard to define, yet I
hope you agree we can describe it by the infinite quasi exhaustion by
{0, 1, 2, 3, ...}.
Exercise 3: does the number 666 belongs to the set of natural numbers,
that is does 666 belongs to {0, 1, 2, 3, ...}.
Yes
Exercice 4: does the real number square-root(2) belongs to {0, 1, 2,
3, ...}?
No idea what square-root(2) means. When I said I was innumerate I wasn't
kidding! I could of course look
it up or ask my mathematics teacher friends but I just know your explanation
will make theirs seem trite.
5) When a set is too big or cumbersome, mathematician like to give
them a name. They will usually say: let S be the set {14, 345, 78}.
Then we can say that 14 belongs to S, for example.
Exercise 5: does 345 belongs to S?
Clearly, yes
A set is entirely defined by its elements. Put in another way, we will
say that two sets are equal if they have the same elements.
Exercise 6. Let S be the set {0, 1, 45} and let M be the set described
by {45, 0, 1}. Is it true or false that S is equal to M?
True - unless integer position within a given sequence in a set plays a role. I
will guess that it does not
Exercise 7. Let S be the set {666} and M be the set {6, 6, 6}. Is is
true or false that S is equal to M?
False - the commas separate each natural number
Seven exercises are enough. Are you ready to answer them.
Done - apart from the square root question
I hope you
don't find them too much easy, because I intend to proceed in a way
such that all exercise will be as easy, despite we will climb toward
very much deeper notion. Feel free to ask question, comments, etc. I
will try to adapt myself.
Very excited about doing this. If you can make it all as approachable as this I
am over the moon!
Next: we will see some operation on sets (union, intersection), and
the notion of subset. If all this work, I will build a latex document,
and make it the standard reference for the seventh step for the non
mathematician, or for the beginners in mathematics.
What a wonderful idea!
Kim
Bruno
http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries
On 02 Jun 2009, at 22:00, Brent Meeker wrote:
Bruno Marchal wrote:
...
A set is entirely defined by its elements. Put in another way, we
will
say that two sets are equal if they have the same elements.
Exercise 6. Let S be the set {0, 1, 45} and let M be the set
described
by {45, 0, 1}. Is it true or false that S is equal to M?
Exercise 7. Let S be the set {666} and M be the set {6, 6, 6}. Is is
true or false that S is equal to M?
But there are no duplicates in sets; so {6,6,6} is either not a set
(instead it's a triple) or it's just strange notation for {6}. Right?
Right. I was not well inspired with this exercise. At least it should
have been given AFTER having said that a set is determined only by its
elements. So {6, 6, 6} is really the same as the set {6}. Something
like {6, 6, 6} is usually called a bag, and will never been used.
Of course {666} is different from {6}.
Apology. This will probably happen again, when I am distracted, so if
an exercise seems weird or senseless, please don't panic!
Bruno
http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries
On 02 Jun 2009, at 21:41, James Rose wrote: What is the definition of a machine? I have a sense that there is an intuitive one but not an explicit one, appropriate to the discussions here. It is part of the goal of the seven step thread to define what is a mathematical machine. Part of this will consist in showing that such a notion is everything but obvious. Here Brian was anticipating on what will follow, actually. I have no problem with anticipation like that, to sustain the motivation, but of course they does not belong to the pedagogical sequences, and people should not worry if they don't understand them. Bruno James - Original Message From: Bruno Marchal marc...@ulb.ac.be To: everything-list@googlegroups.com Sent: Tuesday, June 2, 2009 12:29:47 PM Subject: Re: The seven step-Mathematical preliminaries The beauty of all this, Brian, is that the correct (arithmetically) universal machine will never been able to answer the question are you a machine?, but she (it) will be able to bet she is a (unknown) machine. She will never know which one, and she will refute all theories saying which machine she could be, unless she decided to identify herself with the virgin, never programmed, universal one. There is a way to attribute a first person view to a machine, but then, from that first person view, the machine will be correct in saying I am not a machine. The consequence of computationalism are so much counterintuitive that even machines cannot really believe in comp. Yet, those machine which believe in the numbers and induction will be able to explain exactly all this. Machines can prove that if they are a correct machine, then they cannot believe that they are a correct machine. It is related to the incompleteness phenomenon and the logic of self- reference which is exploited in the AUDA. Actually it works also for Turing hypermachines, and a vast collection of machine extensions, and even self-aware structures completly unrelated to machine, which unfortunately needs a lot of model theory to be described (like truth in all transitive model of ZF, if this rings a bell). But here we anticipate a lot. Hope this can open your appetite. Bruno On 02 Jun 2009, at 21:08, Brian Tenneson wrote: Thanks for the links. I'll look over them and hopefully I'll understand what I see. At least if I have questions I can ask though maybe not in this thread. I don't yet know precisely what you mean by a machine but I do have superficial knowledge of Turing machines; I'm assuming there is a resemblance between the two concepts. I surmise that a machine can have an input like a question and if it halts then the question has a decidable answer, else it has no decidable answer. What about posing the following question am I a machine or the statement I am a machine and maybe some machines halt on an answer and some don't. Ie, if X is a machine, then have it attempt to compute the statement X is a machine. (I know I'm a bit fuzzy on the details.) For machines X that return X is a machine I would be inclined to think such machines possess at least some form of self-awareness, a kind of abstract self-awareness devoid of sensation (or so it would appear). -Brian Bruno Marchal wrote: On 02 Jun 2009, at 18:54, Brian Tenneson wrote: Thank you for starting this discussion. I have only joined recently and have little knowledge of your research. To see it laid out in the sequence you describe should make it clear to me what it is all about. I'm particularly interested in the interaction between consciousness and computation. In Max Tegmark's Ensemble TOE paper he alludes to a self-aware structure. I take structure to be an object of study in logic (model theory, in particular) but am not at all sure how consciousness, which I envision self-awareness to be deeply tied to, connects to mathematics. It seems you're going to build up to a statement such as consciousness is computable OR consciousness is not computable, or something about consciousness, at least. In UDA, I avoid the use of consciousness. I just use the hypothesis that consciousness, or first person experience remains unchanged for a functional substitution made at the correct comp substitution level (this is the comp hypothesis). Then the UD Argument is supposed to show, that physicalism cannot be maintained and that physics is a branch of computer science, or even just number theory. In AUDA, I refine the constructive feature of UDA to begin the extraction of physics. You can read my paper here, and print the UD slides, because I currently refer often to the steps of that reasoning: http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHALAbstract.html I have written a better one, but I must still put it in my webapge. It seems to me that Tegmark is a bit fuzzy about the way he attaches the first
Re: The seven step-Mathematical preliminaries
Excellent!
Kim, are you OK with Marty's answers?
Does someone have a (non philosophical) problem?
I will be busy right now (9h22 am). This afternoon I will send the
next seven exercises.
Bruno
On 02 Jun 2009, at 21:57, m.a. wrote:
Bruno,
I appreciate the simplicity of the examples. My answers
follow the questions.marty a.
- Original Message -
From: Bruno Marchal marc...@ulb.ac.be
= begin
===
1) SET
Informal definition: a set is a collection of object, called
elements,
with the idea that it, the collection or set, can be considered
itself
as an object. It is a many seen as a one, if you want. If the set is
not to big, we can describe it exhaustively by listing the elements,
if the set is bigger, we can describe it by some other way.
Usually we
use accolades {, followed by the elements, separated by commas,
and
then }, in the exhaustive description of a set.
Example/exercise:
1) The set of odd natural numbers which are little than 10. This
is a
well defined, and not to big set, so we can describe it
exhaustively by
{1, 3, 5, 7, 9}. In this case we say that 7 belongs to {1, 3, 5,
7, 9}.
Exercise 1: does the number 24 belongs to the set {1, 3, 5, 7,
9}?NO
2) the set of even natural number which are little than 13. It is
{0,
2, 4, 6, 8, 10, 12}. OK? Some people can have a difficulty which is
not related to the notion of set, for example they can ask
themselves
if zero (0) is really an even number. We will come back to this.
3) The set of odd natural numbers which are little than 100. This
set
is already too big to describe exhaustively. We will freely describe
such a set by a quasi exhaustion like {1, 3, 5, 7, 9, 11, ... 95,
97,
99}.
Exercise 2: does the number 93 belongs to the set of odd natural
numbers which are little than 100, that is: does 93 belongs to {1,
3,
5, 7, 9, 11, ... 95, 97,
99}? YES
4) The set of all natural numbers. This set is hard to define, yet I
hope you agree we can describe it by the infinite quasi exhaustion
by
{0, 1, 2, 3, ...}.
Exercise 3: does the number 666 belongs to the set of natural
numbers,
that is does 666 belongs to {0, 1, 2,
3, ...}. YES
Exercice 4: does the real number square-root(2) belongs to {0, 1, 2,
3
, ...}?
NO
(a guess)
5) When a set is too big or cumbersome, mathematician like to give
them a name. They will usually say: let S be the set {14, 345, 78}.
Then we can say that 14 belongs to S, for example.
Exercise 5: does 345 belongs to
S?YES
A set is entirely defined by its elements. Put in another way, we
will
say that two sets are equal if they have the same elements.
Exercise 6. Let S be the set {0, 1, 45} and let M be the set
described
by {45, 0, 1}. Is it true or false that S is equal to
M? YES
Exercise 7. Let S be the set {666} and M be the set {6, 6, 6}. Is is
true or false that S is equal to
M?NO
Seven exercises are enough. Are you ready to answer them. I hope you
don't find them too much easy, because I intend to proceed in a way
such that all exercise will be as easy, despite we will climb toward
very much deeper notion. Feel free to ask question, comments, etc. I
will try to adapt
myself
. SO
FAR SO GOOD
Next: we will see some operation on sets (union, intersection), and
the notion of subset. If all this work, I will build a latex
document,
and make it the standard reference for the seventh step for the non
mathematician, or for the beginners in mathematics.
Bruno
http://iridia.ulb.ac.be/~marchal/
http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries
Bruno Marchal skrev:
On 02 Jun 2009, at 19:43, Torgny Tholerus wrote:
Bruno Marchal skrev:
4) The set of all natural numbers. This set is hard to define, yet I
hope you agree we can describe it by the infinite quasi exhaustion by
{0, 1, 2, 3, ...}.
Let N be the biggest number in the set {0, 1, 2, 3, ...}.
Exercise: does the number N+1 belongs to the set of natural numbers,
that is does N+1 belongs to {0, 1, 2, 3, ...}?
Yes. N+1 belongs to {0, 1, 2, 3, ...}.
This follows from classical logic and the fact that the proposition N
be the biggest number in the set {0, 1, 2, 3, ...} is always false.
And false implies all propositions.
No, you are wrong. The answer is No.
Proof:
Define biggest number as:
a is the biggest number in the set S if and only if for every element e
in S you have e a or e = a.
Now assume that N+1 belongs to the set of natural numbers.
Then you have N+1 N or N+1 = N.
But this is a contradiction. So the assumption must be false. So we
have proved that N+1 does not belongs to the set of natural numbers.
--
Torgny Tholerus
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Re: The seven step-Mathematical preliminaries
2009/6/3 Torgny Tholerus tor...@dsv.su.se:
Bruno Marchal skrev:
On 02 Jun 2009, at 19:43, Torgny Tholerus wrote:
Bruno Marchal skrev:
4) The set of all natural numbers. This set is hard to define, yet I
hope you agree we can describe it by the infinite quasi exhaustion by
{0, 1, 2, 3, ...}.
Let N be the biggest number in the set {0, 1, 2, 3, ...}.
Exercise: does the number N+1 belongs to the set of natural numbers,
that is does N+1 belongs to {0, 1, 2, 3, ...}?
Yes. N+1 belongs to {0, 1, 2, 3, ...}.
This follows from classical logic and the fact that the proposition N
be the biggest number in the set {0, 1, 2, 3, ...} is always false.
And false implies all propositions.
No, you are wrong. The answer is No.
Proof:
Define biggest number as:
a is the biggest number in the set S if and only if for every element e
in S you have e a or e = a.
Now assume that N+1 belongs to the set of natural numbers.
Then you have N+1 N or N+1 = N.
But this is a contradiction. So the assumption must be false. So we
have proved that N+1 does not belongs to the set of natural numbers.
Hi,
No, what you've demonstrated is that there is no biggest number (you
falsified the hypothesis which is there exists a biggest number). You
did a demonstration par l'absurde (in french, don't know how it is
called in english). And you have shown a contradiction, which implies
that your assumption is wrong (there exists a biggest number), not
that this number is not in the set.
Regards,
Quentin
--
Torgny Tholerus
--
All those moments will be lost in time, like tears in rain.
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Re: The seven step-Mathematical preliminaries
Quentin Anciaux skrev:
2009/6/3 Torgny Tholerus tor...@dsv.su.se:
Bruno Marchal skrev:
On 02 Jun 2009, at 19:43, Torgny Tholerus wrote:
Bruno Marchal skrev:
4) The set of all natural numbers. This set is hard to define, yet I
hope you agree we can describe it by the infinite quasi exhaustion by
{0, 1, 2, 3, ...}.
Let N be the biggest number in the set {0, 1, 2, 3, ...}.
Exercise: does the number N+1 belongs to the set of natural numbers,
that is does N+1 belongs to {0, 1, 2, 3, ...}?
Yes. N+1 belongs to {0, 1, 2, 3, ...}.
This follows from classical logic and the fact that the proposition N
be the biggest number in the set {0, 1, 2, 3, ...} is always false.
And false implies all propositions.
No, you are wrong. The answer is No.
Proof:
Define biggest number as:
a is the biggest number in the set S if and only if for every element e
in S you have e a or e = a.
Now assume that N+1 belongs to the set of natural numbers.
Then you have N+1 N or N+1 = N.
But this is a contradiction. So the assumption must be false. So we
have proved that N+1 does not belongs to the set of natural numbers.
Hi,
No, what you've demonstrated is that there is no biggest number (you
falsified the hypothesis which is there exists a biggest number). You
did a demonstration par l'absurde (in french, don't know how it is
called in english). And you have shown a contradiction, which implies
that your assumption is wrong (there exists a biggest number), not
that this number is not in the set.
How do you know that there is no biggest number? Have you examined all
the natural numbers? How do you prove that there is no biggest number?
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Re: The seven step-Mathematical preliminaries
2009/6/3 Torgny Tholerus tor...@dsv.su.se:
Quentin Anciaux skrev:
2009/6/3 Torgny Tholerus tor...@dsv.su.se:
Bruno Marchal skrev:
On 02 Jun 2009, at 19:43, Torgny Tholerus wrote:
Bruno Marchal skrev:
4) The set of all natural numbers. This set is hard to define, yet I
hope you agree we can describe it by the infinite quasi exhaustion by
{0, 1, 2, 3, ...}.
Let N be the biggest number in the set {0, 1, 2, 3, ...}.
Exercise: does the number N+1 belongs to the set of natural numbers,
that is does N+1 belongs to {0, 1, 2, 3, ...}?
Yes. N+1 belongs to {0, 1, 2, 3, ...}.
This follows from classical logic and the fact that the proposition N
be the biggest number in the set {0, 1, 2, 3, ...} is always false.
And false implies all propositions.
No, you are wrong. The answer is No.
Proof:
Define biggest number as:
a is the biggest number in the set S if and only if for every element e
in S you have e a or e = a.
Now assume that N+1 belongs to the set of natural numbers.
Then you have N+1 N or N+1 = N.
But this is a contradiction. So the assumption must be false. So we
have proved that N+1 does not belongs to the set of natural numbers.
Hi,
No, what you've demonstrated is that there is no biggest number (you
falsified the hypothesis which is there exists a biggest number). You
did a demonstration par l'absurde (in french, don't know how it is
called in english). And you have shown a contradiction, which implies
that your assumption is wrong (there exists a biggest number), not
that this number is not in the set.
How do you know that there is no biggest number?
You just did.
You shown that by assuming there is one it entails a contradiction.
Have you examined all
the natural numbers?
No, that's what demonstration is all about.
How do you prove that there is no biggest number?
You did it.
--
All those moments will be lost in time, like tears in rain.
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Re: The seven step-Mathematical preliminaries
Quentin Anciaux kirjoitti:
2009/6/3 Torgny Tholerus tor...@dsv.su.se:
...
How do you know that there is no biggest number?
You just did.
You shown that by assuming there is one it entails a contradiction.
Have you examined all
the natural numbers?
No, that's what demonstration is all about.
Clearly you two disagree on what {0, 1, 2, 3, ...} means.
All definitions of natural numbers I have seen imply that N+1 is a
natural number whenever N is. Then there clearly is no biggest number.
But I can see someone could have philosophical objections to the
conventional definition. I've heard of ultrafinitists, e.g., but have
not checked how they define natural numbers (if they do).
jp
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RE: The seven step-Mathematical preliminaries
Date: Wed, 3 Jun 2009 13:14:16 +0200
Subject: Re: The seven step-Mathematical preliminaries
From: allco...@gmail.com
To: everything-list@googlegroups.com
2009/6/3 Torgny Tholerus tor...@dsv.su.se:
Bruno Marchal skrev:
On 02 Jun 2009, at 19:43, Torgny Tholerus wrote:
Bruno Marchal skrev:
4) The set of all natural numbers. This set is hard to define, yet I
hope you agree we can describe it by the infinite quasi exhaustion by
{0, 1, 2, 3, ...}.
Let N be the biggest number in the set {0, 1, 2, 3, ...}.
Exercise: does the number N+1 belongs to the set of natural numbers,
that is does N+1 belongs to {0, 1, 2, 3, ...}?
Yes. N+1 belongs to {0, 1, 2, 3, ...}.
This follows from classical logic and the fact that the proposition N
be the biggest number in the set {0, 1, 2, 3, ...} is always false.
And false implies all propositions.
No, you are wrong. The answer is No.
Proof:
Define biggest number as:
a is the biggest number in the set S if and only if for every element e
in S you have e a or e = a.
Now assume that N+1 belongs to the set of natural numbers.
Then you have N+1 N or N+1 = N.
But this is a contradiction. So the assumption must be false. So we
have proved that N+1 does not belongs to the set of natural numbers.
Hi,
No, what you've demonstrated is that there is no biggest number (you
falsified the hypothesis which is there exists a biggest number). You
did a demonstration par l'absurde (in french, don't know how it is
called in english). And you have shown a contradiction, which implies
that your assumption is wrong (there exists a biggest number), not
that this number is not in the set.
The English term for this is proof by contradiction:
http://en.wikipedia.org/wiki/Proof_by_contradiction
Of course, Torgny's conclusion is a little off--he did not show the assumption
N+1 belongs to the set of natural numbers must be wrong as he suggested,
rather he showed the assumption N is the largest natural number must have
been wrong. Just by the usual definition of natural numbers, if N is a natural
number then N+1 must be one too (the page at
http://en.wikipedia.org/wiki/Recursion#Formal_definitions_of_recursion says
that natural numbers are defined in a recursive way: 'the formal definition of
natural numbers in set theory is: 1 is a natural number, and each natural
number has a successor, which is also a natural number'). If Torgny doesn't
agree, I think he needs to provide an alternate definition of natural number
where it would not be true *by definition* that N+1 is a natural number if N is.
Jesse
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Re: The seven step-Mathematical preliminaries
I don't know if Bruno is about to answer this in messages I haven't
checked yet but one can visualize the square root of 2. If you draw a
square one meter by one meter, then the length of the diagonal is the
square root of 2 meters. It is approximately 1.4. What's relevant to
Bruno's question is that the square root of two is greater than one but
less than two, according to the geometry of the diagonal: the diagonal
is more than one and less than two (a picture really helps drive this
point home).
Now since the square root of two is more than one and less than two, it
does -not- belong to the set {0,1, 2, 3, 4, ...}.
In other words, the square root of two is not a natural number.
kimjo...@ozemail.com.au wrote:
On Wed Jun 3 0:39 , Bruno Marchal marc...@ulb.ac.be sent:
Hi Kim, Hi Marty and others,
So it is perhaps time to do some math.
It is
Obviously this is a not a course in math, but it is an explanation
from scratch of the seven step of the universal dovetailer argument.
It is a shortcut, and most probably we will make some digression from
time to time, but let us try not to digress too much.
Kim, you are OK with this?
I am
I have to take into account the problem you
did have with math, and which makes this lesson a bit challenging for
me, and I guess for you too.
Hopefully my innocence will allow me to bypass the pedantry and orthodoxies
of the field and allow a
shortcut to a high level of understanding of the UDA. Only a complete
neophyte would have the gall to
say something like that!
I begin with the very useful and elementary notion of set, as
explained in what is called naive set theory, and which is the base
of almost all part of math.
= begin
===
1) SET
Informal definition: a set is a collection of object, called elements,
with the idea that it, the collection or set, can be considered itself
as an object. It is a many seen as a one, if you want. If the set is
not to big, we can describe it exhaustively by listing the elements,
if the set is bigger, we can describe it by some other way. Usually we
use accolades {, followed by the elements, separated by commas, and
then }, in the exhaustive description of a set.
Example/exercise:
1) The set of odd natural numbers which are little than 10. This is a
well defined, and not to big set, so we can describe it exhaustively by
{1, 3, 5, 7, 9}. In this case we say that 7 belongs to {1, 3, 5, 7, 9}.
Exercise 1: does the number 24 belongs to the set {1, 3, 5, 7, 9}?
No
2) the set of even natural number which are little than 13. It is {0,
2, 4, 6, 8, 10, 12}. OK? Some people can have a difficulty which is
not related to the notion of set, for example they can ask themselves
if zero (0) is really an even number. We will come back to this.
3) The set of odd natural numbers which are little than 100. This set
is already too big to describe exhaustively. We will freely describe
such a set by a quasi exhaustion like {1, 3, 5, 7, 9, 11, ... 95, 97,
99}.
Exercise 2: does the number 93 belongs to the set of odd natural
numbers which are little than 100, that is: does 93 belongs to {1, 3,
5, 7, 9, 11, ... 95, 97, 99}?
Yes
4) The set of all natural numbers. This set is hard to define, yet I
hope you agree we can describe it by the infinite quasi exhaustion by
{0, 1, 2, 3, ...}.
Exercise 3: does the number 666 belongs to the set of natural numbers,
that is does 666 belongs to {0, 1, 2, 3, ...}.
Yes
Exercice 4: does the real number square-root(2) belongs to {0, 1, 2,
3, ...}?
No idea what square-root(2) means. When I said I was innumerate I wasn't
kidding! I could of course look
it up or ask my mathematics teacher friends but I just know your explanation
will make theirs seem trite.
5) When a set is too big or cumbersome, mathematician like to give
them a name. They will usually say: let S be the set {14, 345, 78}.
Then we can say that 14 belongs to S, for example.
Exercise 5: does 345 belongs to S?
Clearly, yes
A set is entirely defined by its elements. Put in another way, we will
say that two sets are equal if they have the same elements.
Exercise 6. Let S be the set {0, 1, 45} and let M be the set described
by {45, 0, 1}. Is it true or false that S is equal to M?
True - unless integer position within a given sequence in a set plays a role.
I will guess that it does not
Exercise 7. Let S be the set {666} and M be the set {6, 6, 6}. Is is
true or false that S is equal to M?
False - the commas separate each natural number
Seven exercises are enough. Are you ready to answer them.
Done - apart from the square root question
I hope you
don't find them
Re: The seven step-Mathematical preliminaries
How do you know that there is no biggest number? Have you examined all the natural numbers? How do you prove that there is no biggest number? In my opinion those are excellent questions. I will attempt to answer them. The intended audience of my answer is everyone, so please forgive me if I say something you already know. Firstly, no one has or can examine all the natural numbers. By that I mean no human. Maybe there is an omniscient machine (or a maximally knowledgeable in some paraconsistent way) who can examine all numbers but that is definitely putting the cart before the horse. Secondly, the question boils down to a difference in philosophy: mathematicians would say that it is not necessary to examine all natural numbers. The mathematician would argue that it suffices to examine all essential properties of natural numbers, rather than all natural numbers. There are a variety of equivalent ways to define a natural number but the essential features of natural numbers are that (a) there is an ordering on the set of natural numbers, a well ordering. To say a set is well ordered entails that every =nonempty= subset of it has a least element. (b) the set of natural numbers has a least element (note that it is customary to either say 0 is this least element or say 1 is this least element--in some sense it does not matter what the starting point is) (c) every natural number has a natural number successor. By successor of a natural number, I mean anything for which the well ordering always places the successor as larger than the predecessor. Then the set of natural numbers, N, is the set containing the least element (0 or 1) and every successor of the least element, and only successors of the least element. There is nothing wrong with a proof by contradiction but I think a forward proof might just be more convincing. Consider the following statement: Whenever S is a subset of N, S has a largest element if, and only if, the complement of S has a least element. By complement of S, I mean the set of all elements of N that are not elements of S. Before I give a longer argument, would you agree that statement is true? One can actually be arbitrarily explicit: M is the largest element of S if, and only if, the successor of M is the least element of the compliment of S. If so, then that statement proves that there is no largest element of N: Letting S be N in particular, note that N is a subset of N (albeit not a proper subset). Then the statement reads as the following for this particular choice S: N has a largest element if, and only if, the complement of N has a least element. The compliment of N is the empty set. To elaborate: the compliment of N is the set of all elements of N that are not elements of N. No elements can both be and not be elements of N, so this set is empty. The empty set does not have a least element. In fact, it has no elements at all. Therefore, N does not have a largest element. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
Quentin Anciaux wrote:
2009/6/3 Torgny Tholerus tor...@dsv.su.se:
Bruno Marchal skrev:
On 02 Jun 2009, at 19:43, Torgny Tholerus wrote:
Bruno Marchal skrev:
4) The set of all natural numbers. This set is hard to define, yet I
hope you agree we can describe it by the infinite quasi exhaustion by
{0, 1, 2, 3, ...}.
Let N be the biggest number in the set {0, 1, 2, 3, ...}.
Exercise: does the number N+1 belongs to the set of natural numbers,
that is does N+1 belongs to {0, 1, 2, 3, ...}?
Yes. N+1 belongs to {0, 1, 2, 3, ...}.
This follows from classical logic and the fact that the proposition N
be the biggest number in the set {0, 1, 2, 3, ...} is always false.
And false implies all propositions.
No, you are wrong. The answer is No.
Proof:
Define biggest number as:
a is the biggest number in the set S if and only if for every element e
in S you have e a or e = a.
Now assume that N+1 belongs to the set of natural numbers.
Then you have N+1 N or N+1 = N.
But this is a contradiction. So the assumption must be false. So we
have proved that N+1 does not belongs to the set of natural numbers.
Hi,
No, what you've demonstrated is that there is no biggest number (you
falsified the hypothesis which is there exists a biggest number). You
did a demonstration par l'absurde (in french, don't know how it is
called in english). And you have shown a contradiction, which implies
that your assumption is wrong (there exists a biggest number), not
that this number is not in the set.
Regards,
Quentin
When you arrive at a contradiction it doesn't tell you which assumption
is wrong.
Brent
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Re: The seven step-Mathematical preliminaries
2009/6/3 Brent Meeker meeke...@dslextreme.com:
Quentin Anciaux wrote:
2009/6/3 Torgny Tholerus tor...@dsv.su.se:
Bruno Marchal skrev:
On 02 Jun 2009, at 19:43, Torgny Tholerus wrote:
Bruno Marchal skrev:
4) The set of all natural numbers. This set is hard to define, yet I
hope you agree we can describe it by the infinite quasi exhaustion by
{0, 1, 2, 3, ...}.
Let N be the biggest number in the set {0, 1, 2, 3, ...}.
Exercise: does the number N+1 belongs to the set of natural numbers,
that is does N+1 belongs to {0, 1, 2, 3, ...}?
Yes. N+1 belongs to {0, 1, 2, 3, ...}.
This follows from classical logic and the fact that the proposition N
be the biggest number in the set {0, 1, 2, 3, ...} is always false.
And false implies all propositions.
No, you are wrong. The answer is No.
Proof:
Define biggest number as:
a is the biggest number in the set S if and only if for every element e
in S you have e a or e = a.
Now assume that N+1 belongs to the set of natural numbers.
Then you have N+1 N or N+1 = N.
But this is a contradiction. So the assumption must be false. So we
have proved that N+1 does not belongs to the set of natural numbers.
Hi,
No, what you've demonstrated is that there is no biggest number (you
falsified the hypothesis which is there exists a biggest number). You
did a demonstration par l'absurde (in french, don't know how it is
called in english). And you have shown a contradiction, which implies
that your assumption is wrong (there exists a biggest number), not
that this number is not in the set.
Regards,
Quentin
When you arrive at a contradiction it doesn't tell you which assumption
is wrong.
Brent
Well I agree, but the second assumption depends on the first which is
N exists and well defined. If it was, the second assumption is
trivially false.
Quentin
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Re: The seven step-Mathematical preliminaries 2
Very good answer, Kim,
Just a few comments. and then the sequel.
Exercice 4: does the real number square-root(2) belongs to {0, 1, 2,
3, ...}?
No idea what square-root(2) means. When I said I was innumerate I
wasn't kidding! I could of course look
it up or ask my mathematics teacher friends but I just know your
explanation will make theirs seem trite.
Well thanks.
The square root of 2 is a number x, such that x*x (x times x, x
multiplied by itself) gives 2.
For example, the square root of 4 is 2, because 2*2 is 4. The square
root of 9 is 3, because 3*3 is 9. Her by square root I mean the
positive square root, because we will see (more later that soon) that
numbers can have negative square root, but please forget this.
At this stage, with this definition, you can guess that the square
root of 2 cannot be a natural number. 1*1 = 1, and 2*2 = 4, and it
would be astonishing that x could be bigger than 2. So if there is
number x such that x*x is 2, we can guess that such a x cannot be a
natural number, that is an element of {0, 1, 2, 3 ...}, and the answer
of exercise 4 is no. The square root of two will reappear
recurrently, but more in examples, than in the sequence of notions
which are strictly needed for UDA-7.
A set is entirely defined by its elements. Put in another way, we
will
say that two sets are equal if they have the same elements.
Exercise 6. Let S be the set {0, 1, 45} and let M be the set
described
by {45, 0, 1}. Is it true or false that S is equal to M?
True - unless integer position within a given sequence in a set
plays a role. I will guess that it does not.
You are right.
Exercise 7. Let S be the set {666} and M be the set {6, 6, 6}. Is is
true or false that S is equal to M?
False - the commas separate each natural number
You are right. Also note that there is only one element in the set {6,
6, 6}. It is just a redundant description of the set {6}.
Very excited about doing this. If you can make it all as
approachable as this I am over the moon!
I will try, and it is very kind to play such a candid role. I
appreciate that you have the ability to say I don't know
something. It is very helpful for me to remain approachable, and
eventually it will help everybody.
So let us continue.
=== Intension and extension
Before defining intersection, union and the notion of subset, I would
like to come back on the ways we can define some specific sets.
In the case of finite and little set we have seen that we can define
them by exhaustion. This means we can give an explicit complete
description of all element of the set.
Example. A = {0, 1, 2, 77, 98, 5}
When the set is still finite and too big, or if we are lazy, we can
sometimes define the set by quasi exhaustion. This means we describe
enough elements of the set in a manner which, by requiring some good
will and some imagination, we can estimate having define the set.
Example. B = {3, 6, 9, 12, ... 99}. We can understand in this case
that we meant the set of multiple of the number three, below 100.
A fortiori, when a set in not finite, that is, when the set is
infinite, we have to use either quasi-exhaustion, or we have to use
some sentence or phrase or proposition describing the elements of the
set.
Definition.
I will say that a set is defined IN EXTENSIO, or simply, in extension,
when it is defined in exhaustion or quasi-exhaustion.
I will say that a set is defined IN INTENSIO, or simply in intension,
with a s, when it is defined by a sentence explaining the typical
attribute of the elements.
Example: Let A be the set {2, 4, 6, 8, 10, ... 100}. We can easily
define A in intension: A = the set of numbers which are even and more
little than 100. mathematician will condense this by the following:
A = {x such that x is even and little than 100} = {x ⎮ x is even x
100}. ⎮ is a special character, abbreviating such that, and I
hope it goes through the mail. If not I will use such that, or s.t.,
or things like that.
The expression {x ⎮ x is even} is literally read as: the set of
object x, (or number x if we are in a context where we talk about
number) such that x is even.
Exercise 1: Could you define in intension the following infinite set C
= {101, 103, 105, ...}
C = ?
Exercise 2: I will say that a natural number is a multiple of 4 if it
can be written as 4*y, for some y. For example 0 is a multiple of 4,
(0 = 4*0), but also 28, 400, 404, ... Could you define in extension
the following set D = {x ⎮ x 10x is a multiple of 4}.
A last notational, but important symbol. Sets have elements. For
example the set A = {1, 2, 3} has three elements 1, 2 and 3. For
saying that 3 is an element of A in an a short way, we usually write 3
∈ A. this is read as 3 belongs to A, or 3 is in A. Now 4 does
not belong to A. To write this in a short way, we will write 4 ∉ A,
or we
Re: Consciousness is information?
On Fri, May 22, 2009 at 4:37 PM, Bruno Marchal marc...@ulb.ac.be wrote: Do you believe if we create a computer in this physical universe that it could be made conscious, But a computer is never conscious, nor is a brain. Only a person is conscious, and a computer or a brain can only make it possible for a person to be conscious relatively to another computer. So your question is ambiguous. It is not my brain which is conscious, it is me who is conscious. My brain appears to make it possible for my consciousness to manifest itself relatively to you. Remember that we are supposed to no more count on the physical supervenience thesis. It remains locally correct to attribute a consciousness through a brain or a body to a person we judged succesfully implemented locally in some piece of matter (like when we say yes to a doctor). But the piece of matter is not the subject of the consciousness. It is only the abstract person or program who is the subject of consciousness. To say a brain is conscious consists in doing Searle's'mistake when he confused levels of computations in the Chinese room, as well seen already by Hofstadter and Dennett in Mind's I. Thanks for your response, if I understand you correctly, you are saying that if we run a simulation of a mind, we are not creating consciousness, only adding an additional instantiation to a mind which already has an infinity of indeterminable instantiations. Is that right? Does this imply that it is impossible to create a simulation of a mind that finds it lives in an environment without uncertainty? If so is it because even if the physical laws in one instantiation may be certain, where some of the infinite number of computations that all instantiate that mind may diverge and in particular which one that mind will find itself in is not knowable? The consequence being that all observers everywhere live in QM-like environments? Thanks, I look forward to your reply. Jason or do you count all appearance of matter to be only a description of a computation and not capable of true computation? appearance of matter is a qualia. It does not describe anything but is a subjective experience, which may correspond to something stable and reflecting the existence of a computation (in Platonia) capable to manifest itself relatively to you. Do you believe that the only real computation exists platonically and this is the only source of conscious experience? Computations and their relative implementations exist only in platonia, yes. But even in Platonia, they exist in multiple relative version, all defined eventually through many multiple relations between numbers. If so I find this confusing, as could there not be multiple levels? But they are multiple levels of computations in Platonia or Arithmetic. Even a huge number of them. That is why we have to take into account the first person indeterminacies. For example would a platonic turing machine simulating another turing machine, simulating a mind be consicous? A 3-machine is never conscious. A 3-entity is never conscious. Only a person is. First person can only be associated with the infinities of computations computing them in Platonia. If so, how does that differ from a platonic turing machine simulating a physical reality with matter, simulating a mind? You will have to introduce a magical (assuming comp) selection principle for attaching, in a persistent way, a mind to that physical reality simulation. The mind can only be attached to an infinity of such relative simulations, and this is why if that mind look at itself below its substitution level he will find a trace of those computations. Comp says you have to make the statistic on all the computations. So the Physical has to be a sum on all those computations. That such computations statistically interfere is not so difficult to show. That the comp interference gives the apparent quantum one is not yet discarded. I think you are not taking sufficiently into account the first person (hopefully plural) indeterminacy in front of the universal dovetailer, (or arithmetic) which defined the space of all computations. Does this help a bit? Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
Thank you very much. I realized I made some false statements as well.
It seems likely that reliance on (not P - Q and not Q) - P being a
tautology is the easiest proof of there being no largest natural number.
Brent Meeker wrote:
Brian Tenneson wrote:
How do you know that there is no biggest number? Have you examined all
the natural numbers? How do you prove that there is no biggest number?
In my opinion those are excellent questions. I will attempt to answer
them. The intended audience of my answer is everyone, so please forgive
me if I say something you already know.
Firstly, no one has or can examine all the natural numbers. By that I
mean no human. Maybe there is an omniscient machine (or a maximally
knowledgeable in some paraconsistent way) who can examine all numbers
but that is definitely putting the cart before the horse.
Secondly, the question boils down to a difference in philosophy:
mathematicians would say that it is not necessary to examine all natural
numbers. The mathematician would argue that it suffices to examine all
essential properties of natural numbers, rather than all natural numbers.
There are a variety of equivalent ways to define a natural number but
the essential features of natural numbers are that
(a) there is an ordering on the set of natural numbers, a well
ordering. To say a set is well ordered entails that every =nonempty=
subset of it has a least element.
(b) the set of natural numbers has a least element (note that it is
customary to either say 0 is this least element or say 1 is this least
element--in some sense it does not matter what the starting point is)
(c) every natural number has a natural number successor. By successor
of a natural number, I mean anything for which the well ordering always
places the successor as larger than the predecessor.
Then the set of natural numbers, N, is the set containing the least
element (0 or 1) and every successor of the least element, and only
successors of the least element.
There is nothing wrong with a proof by contradiction but I think a
forward proof might just be more convincing.
Consider the following statement:
Whenever S is a subset of N, S has a largest element if, and only if,
the complement of S has a least element.
Let S={even numbers} the complement of S, ~S={odd numbers} ~S has a
least element, 1. Therefore there is a largest even number.
Brent
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Re: The seven step-Mathematical preliminaries 2
Bruno,
I stopped half-way through because I'm not at all sure of my answers
and would like to have them confirmed or corrected, if necessary, rather than
go on giving wrong answers. marty a.
- Original Message -
From: Bruno Marchal
To: everything-list@googlegroups.com
Sent: Wednesday, June 03, 2009 1:15 PM
Subject: Re: The seven step-Mathematical preliminaries 2
=== Intension and extension
Before defining intersection, union and the notion of subset, I would like
to come back on the ways we can define some specific sets.
In the case of finite and little set we have seen that we can define them
by exhaustion. This means we can give an explicit complete description of all
element of the set.
Example. A = {0, 1, 2, 77, 98, 5}
When the set is still finite and too big, or if we are lazy, we can sometimes
define the set by quasi exhaustion. This means we describe enough elements of
the set in a manner which, by requiring some good will and some imagination, we
can estimate having define the set.
Example. B = {3, 6, 9, 12, ... 99}. We can understand in this case that we
meant the set of multiple of the number three, below 100.
A fortiori, when a set in not finite, that is, when the set is infinite, we
have to use either quasi-exhaustion, or we have to use some sentence or phrase
or proposition describing the elements of the set.
Definition.
I will say that a set is defined IN EXTENSIO, or simply, in extension, when
it is defined in exhaustion or quasi-exhaustion.
I will say that a set is defined IN INTENSIO, or simply in intension, with a
s, when it is defined by a sentence explaining the typical attribute of the
elements.
Example: Let A be the set {2, 4, 6, 8, 10, ... 100}. We can easily define A
in intension: A = the set of numbers which are even and more little than 100.
mathematician will condense this by the following:
A = {x such that x is even and little than 100} = {x ⎮ x is even x 100}.
⎮ is a special character, abbreviating such that, and I hope it goes
through the mail. If not I will use such that, or s.t., or things like that.
The expression {x ⎮ x is even} is literally read as: the set of object x,
(or number x if we are in a context where we talk about number) such that x is
even.
Exercise 1: Could you define in intension the following infinite set C =
{101, 103, 105, ...}
C = ? C={x such that x is odd x 101}
Exercise 2: I will say that a natural number is a multiple of 4 if it can be
written as 4*y, for some y. For example 0 is a multiple of 4, (0 = 4*0), but
also 28, 400, 404, ... Could you define in extension the following set D = {x
⎮ x 10x is a multiple of 4}.D=4*x where x = 0 (but also 1,2,3...10)
A last notational, but important symbol. Sets have elements. For example the
set A = {1, 2, 3} has three elements 1, 2 and 3. For saying that 3 is an
element of A in an a short way, we usually write 3 ∈ A. this is read as 3
belongs to A, or 3 is in A. Now 4 does not belong to A. To write this in a
short way, we will write 4 ∉ A, or we will write ¬ (4 ∈ A) or sometimes just
NOT(4 ∈ A). It is read: 4 does not belong to A, or: it is not the case that 4
belongs to A.
Having those notions and notations at our disposition we can speed up on the
notion of union and intersection.
The intersection of the sets A and B is the (new) set of those elements which
belongs to both A and B. Put in another way:
The intersection of the sets A with the set B is the set of those elements
which belongs to A and which belongs to B.
This new set, obtained from A and B is written A ∩ B, or A inter. B (in case
the special character doesn't go through).
With our notations we can write or define the intersection A ∩ B directly
A ∩ B = {x ⎮ x ∈ A and x ∈ B}.
Example {3, 4, 5, 6, 8} ∩ {5, 6, 7, 9} = {5, 6}
Similarly, we can directly define the union of two sets A and B, written A ∪
B in the following way:
A ∪ B = {x ⎮ x ∈ A or x ∈ B}.Here we use the usual logical or. p or q
is suppose to be true if p is true or q is true (or both are true). It is not
the exclusive or.
Example {3, 4, 5, 6, 8} ∪ {5, 6, 7, 9} = {3, 4, 5, 6, 7, 8}. Question: In
the example above, 5,6 were the intersection because they were the (only) two
numbers BOTH groups had in common. But in this example, 7 is only in the second
group yet it is included in the answer. Please explain.
Exercice 3.
Let N = {0, 1, 2, 3, ...}
Let A = {x ⎮ x 10}
Let B = {x ⎮ x is even}
Describe in extension (that is: exhaustion or quasi-exhaustion) the following
sets:
N ∪ A = {0,1,2,3...} inter {x inter x10}= {0,1,2,3...9}
N ∪ B = {0,1,2,3} inter {x inter x is even}= {0,2,4,6...}
A ∪ B = {x inter x 10} inter {x inter x is even}= {0,2,4,6,8}
B ∪ A = {x inter x is even} inter {x inter x 10}= {0,2,4,6,8}
N
Re: The seven step-Mathematical preliminaries
On Wed, Jun 03, 2009 at 10:11:41AM -0400, Jesse Mazer wrote: The English term for this is proof by contradiction: http://en.wikipedia.org/wiki/Proof_by_contradiction Funnily enough, we were taught to call this by the latin phrase reductio ad absurdum. I think my maths prof came from Cambridge :). Cheers -- Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 hpco...@hpcoders.com.au Australiahttp://www.hpcoders.com.au --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
