Consciousness is information?

[Bruno Marchal]

On 06 Jun 2009, at 23:54, m.a. wrote:

(I'll be here till Tuesday.) Evidently, the symbol you are using for "such that" is being shown on my screen as a small rectangle. In the copy below, I see two rectangles before the A=, two before the B=, two after the A, two after the B. The  UNION symbol (inverted  "U") shows up but is followed by a rectangle in the next two examples and preceded by a rectangle in the last three. In checking a table of logic notaion, I find that the relation "such that" is designated by a reversed capital  "E".   Is this the symbol you are using?     m.a.

Yes, we have a problem. There should be no rectangles at all. We have to switch on english abbreviations. This explains the difficulty you did have with the union ...

You could look on the archive, from here, 

http://www.mail-archive.com/everything-list@.../msg16531.html

the symbols are correct on my computer, but we will think on easier mail symbols. Tell me if you see different symbols in the archive.

Best,

Bruno

----- Original Message -----

From: marchal@...

To: everything-list@...

Sent: Saturday, June 06, 2009 4:36 PM

Subject: Re: The seven step-Mathematical preliminaries 2

 

We do have problem of symbols, with the mail. I don't see any rectangle in the message below!

Take it easy and . We will go very slowly. It will also be the exam periods. There is no rush ...

Have a good holiday

Bruno

 

----- Original Message -----

From: marchal@...

To: everything-list@...

Sent: Wednesday, June 03, 2009 1:15 PM

Subject: Re: The seven step-Mathematical preliminaries 2

∅ ∪ A =

∅ ∪ B =

A ∪ ∅ =

B ∪ ∅ =

N ∩ ∅ =

B ∩ ∅ =

∅ ∩ B =

∅ ∩ ∅ =

∅ ∪ ∅ =

-----------------------

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

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

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[Bruno Marchal]

           I've encountered some difficulty with the examples below. You say that  "in extension" describes  exhaustion or quasi-exhaustion. And you give the example:  "B = {3, 6, 9, 12, ... 99}".

           Then you define "in intension" with exactly the same type of set: "Example: Let A be the set {2, 4, 6, 8, 10, ... 100}".

I give A in extension there, but just to define it in intension after. It is always the same set there. But I show its definition in extension, to show the definition in intension after. You have to read the to sentences.

           Can you see the cause of my confusion?

It is always the same set. I give it in extension, and then in intension.

Incidentally, may I suggest you use "smaller than" rather than  "more little than". Your English is generally too good to include that kind of error.   marty a.

Well sure. Sometimes the correct expression just slip out from my mind. "smaller than " is much better! Thanks for helping,

Bruno

 

 

 

 

----- Original Message -----

From: marchal@...

To: everything-list@...

Sent: Wednesday, June 03, 2009 1:15 PM

Subject: Re: The seven step-Mathematical preliminaries 2

 

=============== Intension and extension ====================

 

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 < 10  &  x 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 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}.

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 =

N ∪ B =

A ∪ B =

B ∪ A =

N ∩ A =

B ∩ A =

N ∩ B =

A ∩ B =

Exercice 4

Is it true that A ∩ B = B ∩ A, whatever A and B are? 

Is it true that A ∪ B = B ∪ A, whatever A and B are?

Now, I could give you exercise so that you would be lead to discoveries, but I prefer to be as simple and approachable as possible, and my goal is not even to give you the taste for doing research, so I will do the discovery by myself here and now. Indeed a natural question occurs. What will happen if we try to find the intersection of two sets which have no elements in common? For example, what is the intersection of A = {x ⎮ x is even} with B = {x ⎮ x is odd} ? At first sight we could say that there is no intersection, given that A and B have no elements in common. But a set is just a bit more than its elements. And if there is no elements in the intersection, it means simply that the set A ∩ B has no elements. So we are very inspired if we let that bizarre set to exist, so we give it a name, and call it the empty set, and we can describe it easily in exhaustion by { }, although many describe it as ∅. So, if A and B have no elements in common, A ∩ B is still well defined and is equal to ∅. having a new toy, we can play with it:

Exercise 5, with A and B the same as in exercise 3.

∅ ∪ A =

∅ ∪ B =

A ∪ ∅ =

B ∪ ∅ =

N ∩ ∅ =

B ∩ ∅ =

∅ ∩ B =

∅ ∩ ∅ =

∅ ∪ ∅ =

-----------------------

SUBSET

We will say that A is a subset of B (A and B being sets) if, whatever object x represents, each time x belongs to A, it belongs to B. Put in another way it means that IF x belongs to A, THEN x belongs to B. It means that all the elements of A are also elements of B. We can write, with 

x ∈ A -> x ∈ B.

             

And this we abbreviate as A ⊆ B, and we read it: A is included in B.

Example:

1) Let us look if the set A = {1, 2} is included in the set B = {1, 2, 3}.  Here A has two elements. To see if A is included in B, we have to look at each element in the set A, and we have to see if they belongs to B. Now A has two elements, 1, and 2, so we have two tasks to accomplish, or two questions to answer:

does 1 belongs also to B. The answer is yes.

does 2 belongs also to B. The answer is yes.

We have thus verify that all elements of A are also elements of B, and thus we can conclude that A is indeed included in B.

2) Let us look if the set A = {1} is included in B = {1, 2, 3}.  Now, A has only one element. So we are lucky, we have only one task to accomplish! Is 1 an element of B? The answer is yes. Thus we have {1} is included in {1, 2, 3}.

3) Let us look if the set A=  { }, the empty set ∅,  is included in B = {1, 2, 3}. Now A has no element. So we are even more lucky, we have no task to accomplish at all. The condition is trivially satisfied. So the empty set is included in {1, 2, 3}. And this shows that the empty set is included in any set. In particular we have that ∅ ⊆ ∅.

Note that all set is a subset of itself. Trivially, all elements of A is an element of A.

Exercise 6 

We will say that a set A is a subset of a set B, if A is included in B.

Could you give all the subsets of the set  {1, 2}.

Could you give all the subsets of the set  {1}

Could you give all the subsets of the set  { }.

The post is long enough, so I spare you the seventh exercise. Also I have to go, I hope there are not to many typo errors and spelling mistakes, and well, I pray for the special symbols going trough. It is possible that they go through for most mailing systems, but not all. Let me know.

Bon courage,

Bruno

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

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

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[m.a.-2]

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On this date, you made the following correction:  "You cannot write D = 4*x ..., "     But you wrote   D= 4*x   in the exercise just above it. I don't get the distinction between your use of the equation and mine.

----- Original Message -----

From: marchal@...

 

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 < 10  &  x is a multiple of 4}.    D=4*x  where x = 0 (but also 1,2,3...10)

You cannot write D = 4*x ..., given that D is a set, and 4*x is a (unknown) number (a multiple of four when x is a natural number).

Read carefully the problem. I gave the set in intension, and the exercise consisted in writing the set in extension. Let us translate in english the definition of the set D = {x ⎮ x < 10  &  x is a multiple of 4}: it means that D is the set of numbers, x, such that x is little than 10, and x is a multiple of four. So D = {0, 4, 8}.

 

SEE BELOW

 

 

 

 

 

Example: the set of multiple of 4 is {0, 4, 8, 12, 16, 20, 24, 28, 32, 36, ...}, all have the shape 4*x, with x = to 0, 1, 2, 3, ...

The set of multiple of 5 is {0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, ...}

Etc.

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.

In the example "above" (that is {3, 4, 5, 6, 8} ∩ {5, 6, 7, 9} = {5, 6}) we were taking the INTERSECTION of the two sets.

But after that, may be too quickly (and I should have made a title perhaps) I was introducing the UNION of the two sets.

If you read carefully the definition in intension, you should see that the intersection of A and B is defined with an "and". The definition of union is defined with a "or". Do you see that? It is just above in the quote.

I hope that your computer can distinguish A ∩ B  (A intersection B) and A ∪ B  (A union B).

In the union of two sets, you put all the elements of the two sets together. In the intersection of two sets, you take only those elements which belongs to the two sets.

It seems you have not seen the difference between "intersection" and "union".  

 

This has indeed been the case. My usual math disabilities have been exacerbated by the confusion of symbols due to E-mail limitations. The profusion of little rectangles replacing the UNION symbol make the formulae difficult to follow.

 

 

 

 

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[Brent Meeker-2]

m.a. wrote:
> *Bruno,*
> *           I've encountered some difficulty with the examples below.
> You say that  "in extension" describes  exhaustion or
> quasi-exhaustion. And you give the example:  "**B = {3, 6, 9, 12, ...
> 99}".*
> *           Then you define "in intension" with exactly the same type
> of set: "Example: Let A be the set {2, 4, 6, 8, 10, ... 100}".*

No, that's not the intensional definition.  This "We can easily define A
in intension:  A = the set of numbers which are even and more little
than 100." is the intensional definition.

Brent

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[m.a.-2]

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Bruno,

               When I tried to copy the symbols from the URL cited below, I found that my email server was not able to reproproduce the intersection or the union symbol. See below:

 

   From: Bruno Marchal
      To: everything-list@...
      

      ∅ ∪ A = I see two rectangles and  "A"
      ∅ ∪ B = I see two rectangles and "B"
      A ∪ ∅ = I see "A" and two rectangles
      B ∪ ∅ = I see "B" and two rectangles
      N ∩ ∅ = I see "N"  Inverted "U"  and a rectangle
      B ∩ ∅ = I see "B"  Inverted "U"  and a rectangle
      ∅ ∩ B = I see a rectangle  an inverted "U"  and "B"
      ∅ ∩ ∅ = I see a rectangle  an inverted "U"  and a rectangle
      ∅ ∪ ∅ = I see three rectangles
         

----- Original Message -----

From: marchal@...

 

You could look on the archive, from here, 

http://www.mail-archive.com/everything-list@.../msg16531.html

 

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[Brent Meeker-2]

Quentin Anciaux wrote:

> 2009/6/6 Torgny Tholerus <torgny@...>:
>  
>> Jesse Mazer skrev:
>>    
>>>      
>>>> Date: Sat, 6 Jun 2009 16:48:21 +0200
>>>> From: torgny@...
>>>> To: everything-list@...
>>>> Subject: Re: The seven step-Mathematical preliminaries
>>>>
>>>> Jesse Mazer skrev:
>>>>        
>>>>> 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.
>>>>>          
>>>> It depends upon how you define "natural number". If you define it by: n
>>>> is a natural number if and only if n belongs to N, the set of all
>>>> natural numbers, then of course BIGGEST+1 is *not* a natural number. In
>>>> that case you have to call BIGGEST+1 something else, maybe "unnatural
>>>> number".
>>>>        
>>> OK, but then you need to define what you mean by "N, the set of all
>>> natural numbers". Specifically you need to say what number is
>>> "BIGGEST". Is it arbitrary? Can I set BIGGEST = 3, for example? Or do
>>> you have some philosophical ideas related to what BIGGEST is, like the
>>> number of particles in the universe or the largest number any human
>>> can conceptualize?
>>>      
>> It is rather the last, the largest number any human can conceptualize.
>> More natural numbers are not needed.
>>    
>
> What is the last number human can invent ? Your theory can't explain
> why addition works... If N is limited, then addition can and will (in
> human lifetime) create "number" which are still finite and not in N.
>  

It is very unlikely that anyone will get to the number 10^10^100 by
addition.  :-)

Would agree that a any given time there is a largest number which has
been conceived by a human being?

> N can be defined solelly as the successor function, you don't need
> anything else. You just have to assert that the function is true
> always.
>
>  
>>> Also, any comment on my point about there being an infinite number of
>>> possible propositions about even a finite set,
>>>      
>> There is not an infinite number of possible proposition.
>>    
>
> Prove it please.
>  

That would seem to turn on the meaning of "possible".  Many (dare I say
"infinitely many") things are logically possible which are not
nomologically possible (although the posters on this list seem to doubt
that).

Brent

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[m.a.-2]

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Okay, so is it true to say that things written in EXTENSION are never in formula style but are translated into formulas when we put them into  INTENSION   form?  You can see that my difficulty with math arises from an inability to master even the simplest definitions.    marty a.

 

 

----- Original Message -----

From: marchal@...

 

           I've encountered some difficulty with the examples below. You say that  "in extension" describes  exhaustion or quasi-exhaustion. And you give the example:  "B = {3, 6, 9, 12, ... 99}".

           Then you define "in intension" with exactly the same type of set: "Example: Let A be the set {2, 4, 6, 8, 10, ... 100}".

I give A in extension there, but just to define it in intension after. It is always the same set there. But I show its definition in extension, to show the definition in intension after. You have to read the to sentences.

           Can you see the cause of my confusion?

It is always the same set. I give it in extension, and then in intension.

 

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

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[Brent Meeker-2]

m.a. wrote:
> *Okay, so is it true to say that things written in EXTENSION are never
> in formula style but are translated into formulas when we put them
> into  INTENSION   form?  You can see that my difficulty with math
> arises from an inability to master even the simplest definitions.    
> marty a.*

It's not that technical.  I could define the set of books on my shelf by
giving a list of titles: "The Comprehensible Cosmos", "Set Theory and
It's Philosophy", "Overshoot", "Quintessence".  That would be a
definition by extension.  Or I could point to them in succession and
say, "That and that and that and that." which would be a definition by
ostension. Or I could just say, "The books on my shelf." which is a
definition by intension.  An intensional definition is a descriptive
phrase with an implicit variable, which in logic you might write as: The
set of things x such that x is a book and x is on my shelf.

Brent

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[russell standish-2]

On Sat, Jun 06, 2009 at 10:22:11AM -0700, Brent Meeker wrote:
> I wonder if anyone has tried work with a theory of finite numbers: where
> BIGGEST+1=BIGGEST or BIGGEST+1=-BIGGEST as in some computers?
>
> Brent
>

The numbers {0,...,p-1} with p prime, and addition and multiplication
given modulo p (ie

a plus b = (a+b) mod p
a times b = (ab) mod p

)

is an interesting mathematical object known as a finite field (or
Galois field) -
http://en.wikipedia.org/wiki/Finite_field

Interesting examples of infinite fields are those quite familiar to
you: rational, real and complex numbers.

It might make sense for Torgny to work with a Galois field for some
large but unnamed prime :)

Cheers

--

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[Bruno Marchal]

Marty,

On 07 Jun 2009, at 02:03, Brent Meeker wrote:

>
> m.a. wrote:
>> *Okay, so is it true to say that things written in EXTENSION are  
>> never
>> in formula style but are translated into formulas when we put them
>> into  INTENSION   form?  You can see that my difficulty with math
>> arises from an inability to master even the simplest definitions.
>> marty a.*
>
> It's not that technical.  I could define the set of books on my  
> shelf by
> giving a list of titles: "The Comprehensible Cosmos", "Set Theory and
> It's Philosophy", "Overshoot", "Quintessence".  That would be a
> definition by extension.  Or I could point to them in succession and
> say, "That and that and that and that." which would be a definition by
> ostension. Or I could just say, "The books on my shelf." which is a
> definition by intension.  An intensional definition is a descriptive
> phrase with an implicit variable, which in logic you might write as:  
> The
> set of things x such that x is a book and x is on my shelf.

This is a good point. A set is just a collection of objects seen as a  
whole.

A definition in extension of a set is just a listing, finite or  
infinite, of its elements.
Like in A = {1, 3, 5}, or B = {2, 4, 6, 8, 10, ...}.

A definition in intension of a set consists in giving the typical  
defining property of the elements of the set.
Like in C= "the set of odd numbers which are smaller than 6". Or D =  
the set of even numbers.

In this case you see that A is the same set as C? And B is the same  
set as D.

Now in mathematics we often use abbreviation. So, for example, instead  
of saying: the set of even numbers, we will write
{x such-that x is even}.

OK?

Bruno




Suppose,





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




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[m.a.-2]

Some parts of this message have been removed. Learn more about Nabble's security policy.

Thank you, Brent,

                             This is quite clear. Hopefully I can apply it as clearly to Bruno's examples.    marty a.

 

 

----- Original Message -----

From: "Brent Meeker" <meekerdb@...>

To: <everything-list@...>

Sent: Saturday, June 06, 2009 8:03 PM

Subject: Re: The seven step-Mathematical preliminaries 2

>

> m.a. wrote:
>> *Okay, so is it true to say that things written in EXTENSION are never
>> in formula style but are translated into formulas when we put them
>> into  INTENSION   form?  You can see that my difficulty with math
>> arises from an inability to master even the simplest definitions.   
>> marty a.*
>
> It's not that technical.  I could define the set of books on my shelf by
> giving a list of titles: "The Comprehensible Cosmos", "Set Theory and
> It's Philosophy", "Overshoot", "Quintessence".  That would be a
> definition by extension.  Or I could point to them in succession and
> say, "That and that and that and that." which would be a definition by
> ostension. Or I could just say, "The books on my shelf." which is a
> definition by intension.  An intensional definition is a descriptive
> phrase with an implicit variable, which in logic you might write as: The
> set of things x such that x is a book and x is on my shelf.
>
> Brent
>
>

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[m.a.-2]

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Bruno,

            Yes, this seems very clear and will be helpful to refer back to if necessary.     m.a.

 

 

 

----- Original Message -----

From: "Bruno Marchal" <marchal@...>

To: <everything-list@...>

Sent: Sunday, June 07, 2009 4:33 AM

Subject: Re: The seven step-Mathematical preliminaries 2

>

> Marty,
>
> On 07 Jun 2009, at 02:03, Brent Meeker wrote:
>
>>
>> m.a. wrote:
>>> *Okay, so is it true to say that things written in EXTENSION are 
>>> never
>>> in formula style but are translated into formulas when we put them
>>> into  INTENSION   form?  You can see that my difficulty with math
>>> arises from an inability to master even the simplest definitions.
>>> marty a.*
>>
>> It's not that technical.  I could define the set of books on my 
>> shelf by
>> giving a list of titles: "The Comprehensible Cosmos", "Set Theory and
>> It's Philosophy", "Overshoot", "Quintessence".  That would be a
>> definition by extension.  Or I could point to them in succession and
>> say, "That and that and that and that." which would be a definition by
>> ostension. Or I could just say, "The books on my shelf." which is a
>> definition by intension.  An intensional definition is a descriptive
>> phrase with an implicit variable, which in logic you might write as: 
>> The
>> set of things x such that x is a book and x is on my shelf.
>
>
> This is a good point. A set is just a collection of objects seen as a 
> whole.
>
> A definition in extension of a set is just a listing, finite or 
> infinite, of its elements.
> Like in A = {1, 3, 5}, or B = {2, 4, 6, 8, 10, ...}.
>
> A definition in intension of a set consists in giving the typical 
> defining property of the elements of the set.
> Like in C= "the set of odd numbers which are smaller than 6". Or D = 
> the set of even numbers.
>
> In this case you see that A is the same set as C? And B is the same 
> set as D.
>
> Now in mathematics we often use abbreviation. So, for example, instead 
> of saying: the set of even numbers, we will write
> {x such-that x is even}.
>
> OK?
>
> Bruno
>
>
>
>
> Suppose,
>
>
>
>
>
> http://iridia.ulb.ac.be/~marchal/

>
>
>
>
>
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[John Mikes]

On Sat, Jun 6, 2009 at 4:20 PM, Jesse Mazer <lasermazer@...> wrote:

> Date: Sat, 6 Jun 2009 21:17:03 +0200

> From: torgny@...
> To: everything-list@...
> Subject: Re: The seven step-Mathematical preliminaries
>
>
> Jesse Mazer skrev:
[[[[[[[>>
>>
>>> Date: Sat, 6 Jun 2009 16:48:21 +0200
>>> From: torgny@...
>>> To: everything-list@...
>>> Subject: Re: The seven step-Mathematical preliminaries
>>>
>>> Jesse Mazer skrev:
>>>>
>>>> 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.
>>>
>>> It depends upon how you define "natural number". If you define it by: n
>>> is a natural number if and only if n belongs to N, the set of all
>>> natural numbers, then of course BIGGEST+1 is *not* a natural number. In
>>> that case you have to call BIGGEST+1 something else, maybe "unnatural
>>> number".
>>
>> OK, but then you need to define what you mean by "N, the set of all
>> natural numbers". Specifically you need to say what number is
>> "BIGGEST". Is it arbitrary? Can I set BIGGEST = 3, for example? Or do
>> you have some philosophical ideas related to what BIGGEST is, like the
>> number of particles in the universe or the largest number any human
>> can conceptualize?
>
> It is rather the last, the largest number any human can conceptualize.
> More natural numbers are not needed.]]]]]]]

Why humans, specifically? What if an alien could conceptualize a larger number? For that matter, since you deny any special role to consciousness, why should it have anything to do with the conceptualizations of beings with brains? A volume of space isn't normally said to "conceptualize" the number of atoms contained in that volume, but why should that number be any less real than the largest number that's been conceptualized by a biological brain?

 

JohnM:

Jesse,

you don't have to go out to 'aliens', just eliminate the format "possible as of 2009". Our un-alien species is well capable of learning (compare to 2000BC) and whatever is restricted today as 'impossible' may be everyday's bread after tomorrow. You are absolutely right - even as of today. 

Especially in your next reply-par below. 

>>
>> Also, any comment on my point about there being an infinite number of
>> possible propositions about even a finite set,
>
> There is not an infinite number of possible proposition. You can only
> create a finite number of proposition with finite length during your
> lifetime. Just like the number of natural numbers are unlimited but
> finite, so are the possible propositions unlimited but finte.

But you said earlier that as long as we admit only a finite collection of numbers, we can prove the "consistency" of mathematics involving only those numbers. Well, how can we "prove" that? If we only show that all the propositions we have generated to date are consistent, how do we know the next proposition we generate won't involve an inconsistency? Presumably you are implicitly suggesting there should be some upper limit on the number of propositions about the numbers as well as on the numbers themselves, but if you define this limit in terms of how many a human could generate in their lifetime, we get back to problems like what if some other being (genetically engineered humans, say) would have a longer lifetime, or what if we built a computer that generated propositions much faster than a human could and checked their consistency automatically, etc.

 
>> or about my question about whether you have any philosophical/logical
>> argument for saying all sets must be finite,
>
> My philosophical argument is about the mening of the word "all". To be
> able to use that word, you must associate it with a value set.

What's a "value set"? And why do you say we "must" associate it in this way? Do you have a philosophical argument for this "must", or is it just an edict that reflects your personal aesthetic preferences?

 Mostly
> that set is "all objects in the universe", and if you stay inside the
> universe, there is no problems.

*I* certainly don't define numbers in terms of any specific mapping between numbers and objects in the universe, it seems like a rather strange notion--shall we have arguments over whether the number 113485 should be associated with this specific shoelace or this specific kangaroo? One of the first thing kids learn about number is that if you count some collection of objects, it doesn't matter what order you count them in, the final number you get will be the same regardless of the order (i.e. it doesn't matter which you point to when you say "1" and which you point to when you say "2", as long as you point to each object exactly once).

Also, am I understanding correctly in thinking you don't believe there can be truths about numbers independent of what humans actually know about them (i.e. there is no truth about the sum of two very large numbers unless some human has actually calculated that sum at one point)? If in fact you don't believe there are truths about numbers independent of human thoughts about them, why do you think there can be truths about the physical universe which humans don't know about? For example, is there a truth about the surface topography of some planet that humans have never and will never see up close or send probes to? In physics most facts about physical systems are quantitative numerical facts, after all, so if you admit truths about the surface topography of a planet in another galaxy there's no reason not to admit truths about the number of atoms in some large volume of space in another galaxy, even if this number is one no human has ever thought about specifically.

 

JohnM:

Not about my questioning 'truth' (who's?) but yur basic question was the 'belief' in numbers... Right on. David Bohm regarded numbers as a human invention and after several questioning on this list and others, I did not receive satisfying response why 'numbers' should be the originating 'truth' of the existence? Only consequences of 'it' and mechanisms.

Would you deny a different mindset in the unlimited? 

 

About the 'physical universe' as truth: I regard it a figment based on the explanations (over millannia) in a steadily enriching epistemic cognitive inventory based on similarly continuously changing explanations of the observations we make in our developing capabilites, 'lately' (?) mostly on a quantified basis (numbers?) building up the edifice of the physical truth. Not vice versa.

>But as soon you go outside universe, 

> you must be carefull with what substitutions you do. If you have "all"
> quantified with all object inside the universe

But I don't, of course. This is an idiosyncratic way of thinking specific to you, and you have not given any philosophical justification for the idea that numbers must be mapped to physical entities. Also, when you say "universe" are you ruling out a priori any cosmological model which says the universe is spatially infinite and contains an infinite number of particles?

 

JohnM:

(could we add: "in an infinite number of universes"? I am not for that, my poor imagination does not cover 'infinite' .)

>
>> as opposed to it just being a sort of aesthetic preference on your
>> part? Do you think there is anything illogical or incoherent about
>> defining a set in terms of a rule that takes any input and decides
>> whether it's a member of the set or not, such that there may be no
>> upper limit on the number of possible inputs that the rule would
>> define as being members? (such as would be the case for the rule 'n is
>> a natural number if n=1 or if n is equal to some other natural number+1')
>
> In the last sentence you have an implicite "all": The full sentence
> would be: For all n in the universe hold that n is a natural number if
> n=1 or if n is equal to some other natural number+1.

I didn't say anything about the universe, I would treat an n as just a possible symbolic input that could be fed into the algorithm that decides whether any given string of symbols fits the definition of a natural number, it doesn't matter if this particular string is ever printed out in the real physical universe.

 

JohnM:

My ignorance asks for a "definition of natural numbers" beyond an illustration "....2,3,4,.." without combining them to countables. The symbols are meaningful only if we assigne meaning to them. You may call 2 as ///// and 4 as // - the world will not collapse. (I asked Bruno and did not get a satisfying (for me) argument.) The axioms are drawn to justify our theories about the world we built in our mind - supported by the fragmental and incomplete continuously developing observations of the day we mustered upon those partially observed phenomena that got to us. (Keeping most of the previous  explanations in store).

A most ingenious edifice humanity erected.

What does not mean that another 'storytelling' is impossible, as was given e.g. by (then young) Paul Churchland by a differently thinking 'tribe' on "caloric" - a liquid.

Thanks for starting an expansion upo the 'possible' hoax.

 

Jesse

 

John M

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[m.a.-2]

Bruno et. al.,
                       Good news! I have discovered that the math symbols copy faithfully here in my Thunderbird email. Henceforth, I will open all list letters here. Please refresh my memory for the following symbols:

1. The   ∅  is called_________________________and means__________________________

2. The    ∪  is called____________________________and means______________________________

3. The   ∩   is called______________________________and means________________________________

----- Original Message -----
      From: Bruno Marchal
      To: everything-list@...
      Sent: Wednesday, June 03, 2009 1:15 PM
      Subject: Re: The seven step-Mathematical preliminaries 2


      ∅ ∪ A =
      ∅ ∪ B =
      A ∪ ∅ =
      B ∪ ∅ =
      N ∩ ∅ =
      B ∩ ∅ =
      ∅ ∩ B =
      ∅ ∩ ∅ =
      ∅ ∪ ∅ =


  

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[Bruno Marchal]

Bravo Thunderbird!

On 07 Jun 2009, at 18:39, m.a. wrote:

Bruno et. al.,
                       Good news! I have discovered that the math symbols copy faithfully here in my Thunderbird email. Henceforth, I will open all list letters here. Please refresh my memory for the following symbols:

1. The   ∅  is called___THE EMPTY SET_____and means______THE SET WITH NO ELEMENTS

The empty set described in extension: { }

The empty set described in intension. Well, let me think. The set of french which are bigger than 42 km tall. 

A cynical definition would be: the set of honest politicians.

A mathematical one: the set of x such that x is different from x.

It is just the set which has no elements. It is empty. 

2. The    ∪  is called________UNION______and means: A ∪ B__= {x such-that x belongs to A  or x belongs to B};

A u B is the set obtained by doing the union of A and B.

3. The   ∩   is called_____INTERSECTION____and means__A ∩ B__=  {x such-that x belongs to A  andr x belongs to B};     A u B is the set obtained by doing the intersection of A and B. It is the set of elements which are in both A and B._

Examples:

{1, 2, 3} ∩ {2, 4, 3} = {2, 3}

{1, 2, 3} u {2, 4, 3} = {1, 2, 3, 4}

{1, 2, 3} ∩ {4, 5, 6} =   ∅

{1, 2, 3} u {4, 5, 6} = {1, 2, 3, 4, 5, 6}

OK?

Bruno

----- Original Message -----
      From: Bruno Marchal
      To: everything-list@...
      Sent: Wednesday, June 03, 2009 1:15 PM
      Subject: Re: The seven step-Mathematical preliminaries 2


      ∅ ∪ A =
      ∅ ∪ B =
      A ∪ ∅ =
      B ∪ ∅ =
      N ∩ ∅ =
      B ∩ ∅ =
      ∅ ∩ B =
      ∅ ∩ ∅ =
      ∅ ∪ ∅ =


  

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

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[Bruno Marchal]

Marty, Kim,

I realize that, now, the message I have just sent does not have the right "symbols". Apparently my computer does not understand the Thunderbird!

From now on I will use capital words for the mathematical symbols. And I will write mathematical expression in bold.

For examples:

{1, 2, 3}  INTERSECTION  {2, 4, 3}   =   {2, 3}

{1, 2, 3}  UNION  {2, 4, 3}   =   {1, 2, 3, 4}

{1, 2, 3}  INTERSECTION  {4, 5, 6}   =   EMPTY

{1, 2, 3}  UNION   {4, 5, 6}   =   {1, 2, 3, 4, 5, 6}

All right? Mathematics will get a FORTRAN look but this is not important, OK? It is just the look. I will do a summary of what we have seen so far.

With those notions you should be able to invent exercises by yourself. Invent simple sets and compute their union, and intersection.

Remenber that the goal consists in building a mathematical shortcut toward a thorugh understanding of step seven. In particular the goal will be to get an idea of a computation is, and what is the difference between a mathemarical computation and a mathematical description of a computation. It helps for the step 8 too.

Marty, have a nice holiday,

Kim, ah ah ... we have two weeks to digest what has been said so far (which is not enormous), OK?

Bruno

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

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[Torgny Tholerus]

Jesse Mazer skrev:

>
>
> > Date: Sat, 6 Jun 2009 21:17:03 +0200
> > From: torgny@...
> > To: everything-list@...
> > Subject: Re: The seven step-Mathematical preliminaries
> >
> > My philosophical argument is about the mening of the word "all". To be
> > able to use that word, you must associate it with a value set.
>
> What's a "value set"? And why do you say we "must" associate it in
> this way? Do you have a philosophical argument for this "must", or is
> it just an edict that reflects your personal aesthetic preferences?
>
> > Mostly that set is "all objects in the universe", and if you stay
> inside the
> > universe, there is no problems.
>
> *I* certainly don't define numbers in terms of any specific mapping
> between numbers and objects in the universe, it seems like a rather
> strange notion--shall we have arguments over whether the number 113485
> should be associated with this specific shoelace or this specific
> kangaroo?

When I talk about "universe" here, I do not mean our physical universe.  
What I mean is something that can be called "everything".  It includes
all objects in our physical universe, as well as all symbols and all
words and all numbers and all sets and all other universes.  It includes
everything you can use the word "all" about.

For you to be able to use the word "all", you must define the "domain"
of that word.  If you do not define the domain, then it will be
impossible for me and all other humans to understand what you are
talking about.

--
Torgny Tholerus

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[Quentin Anciaux-2]

2009/6/9 Torgny Tholerus <torgny@...>:

>
> Jesse Mazer skrev:
>>
>>
>> > Date: Sat, 6 Jun 2009 21:17:03 +0200
>> > From: torgny@...
>> > To: everything-list@...
>> > Subject: Re: The seven step-Mathematical preliminaries
>> >
>> > My philosophical argument is about the mening of the word "all". To be
>> > able to use that word, you must associate it with a value set.
>>
>> What's a "value set"? And why do you say we "must" associate it in
>> this way? Do you have a philosophical argument for this "must", or is
>> it just an edict that reflects your personal aesthetic preferences?
>>
>> > Mostly that set is "all objects in the universe", and if you stay
>> inside the
>> > universe, there is no problems.
>>
>> *I* certainly don't define numbers in terms of any specific mapping
>> between numbers and objects in the universe, it seems like a rather
>> strange notion--shall we have arguments over whether the number 113485
>> should be associated with this specific shoelace or this specific
>> kangaroo?
>
> When I talk about "universe" here, I do not mean our physical universe.
> What I mean is something that can be called "everything".  It includes
> all objects in our physical universe, as well as all symbols and all
> words and all numbers and all sets and all other universes.  It includes
> everything you can use the word "all" about.

It includes all set, but no all set as it N includes all natural
number but not all natural number... excuse-me but this is non-sense.
Either N exists and has an infinite number of member and is
incompatible with an ultrafinitist view or N does not exists because
obviously N cannot be defined in an ultra-finitist way, any set that
contains a finite number of natural number (and still you haven't
defined what it is in an ultrafinitist way) are not the set N.

Also any operation involving two number (addition/multiplication) can
yield as result a number which has the same property as the departing
number (being a natural number) but is not natural number... Also
induction and inference cannot work in such a context.

> For you to be able to use the word "all", you must define the "domain"
> of that word.  If you do not define the domain, then it will be
> impossible for me and all other humans to understand what you are
> talking about.

Well you are the first and only human I know who don't understand
"all" as everybody else does.

Quentin Anciaux

>
> --
> Torgny Tholerus
>
> >
>



--
All those moments will be lost in time, like tears in rain.

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[Brent Meeker-2]

Quentin Anciaux wrote:

> 2009/6/9 Torgny Tholerus <torgny@...>:
>  
>> Jesse Mazer skrev:
>>    
>>>      
>>>> Date: Sat, 6 Jun 2009 21:17:03 +0200
>>>> From: torgny@...
>>>> To: everything-list@...
>>>> Subject: Re: The seven step-Mathematical preliminaries
>>>>
>>>> My philosophical argument is about the mening of the word "all". To be
>>>> able to use that word, you must associate it with a value set.
>>>>        
>>> What's a "value set"? And why do you say we "must" associate it in
>>> this way? Do you have a philosophical argument for this "must", or is
>>> it just an edict that reflects your personal aesthetic preferences?
>>>
>>>      
>>>> Mostly that set is "all objects in the universe", and if you stay
>>>>        
>>> inside the
>>>      
>>>> universe, there is no problems.
>>>>        
>>> *I* certainly don't define numbers in terms of any specific mapping
>>> between numbers and objects in the universe, it seems like a rather
>>> strange notion--shall we have arguments over whether the number 113485
>>> should be associated with this specific shoelace or this specific
>>> kangaroo?
>>>      
>> When I talk about "universe" here, I do not mean our physical universe.
>> What I mean is something that can be called "everything".  It includes
>> all objects in our physical universe, as well as all symbols and all
>> words and all numbers and all sets and all other universes.  It includes
>> everything you can use the word "all" about.
>>    
>
> It includes all set, but no all set as it N includes all natural
> number but not all natural number... excuse-me but this is non-sense.
> Either N exists and has an infinite number of member and is
> incompatible with an ultrafinitist view or N does not exists because
> obviously N cannot be defined in an ultra-finitist way,

That's not obvious to me.  You're assuming that N exists apart from
whatever definition of it is given and that it is the infinite set
described by the Peano axioms or equivalent.  But that's begging the
question of whether a finite set of numbers that we would call "natural
numbers" can be defined.  To avoid begging the question we need some
definition of "natural" that doesn't a priori assume the set is finite
or infinite; something like, "A set of numbers adequate to do all
arithmetic we'll ever need" (unfortunately not very definite).  The
problem is the successor axiom, if we modify it to S{n}=n+1 for n e N
except for the case n=N where S{N}=0 and choose sufficiently large N it
might satisfy the "natural" criteria.

Brent

> any set that
> contains a finite number of natural number (and still you haven't
> defined what it is in an ultrafinitist way) are not the set N.
>
> Also any operation involving two number (addition/multiplication) can
> yield as result a number which has the same property as the departing
> number (being a natural number) but is not natural number... Also
> induction and inference cannot work in such a context.
>
>  
>> For you to be able to use the word "all", you must define the "domain"
>> of that word.  If you do not define the domain, then it will be
>> impossible for me and all other humans to understand what you are
>> talking about.
>>    
>
> Well you are the first and only human I know who don't understand
> "all" as everybody else does.
>
> Quentin Anciaux
>
>  
>> --
>> Torgny Tholerus
>>
>>    
>
>
>
>  

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[Quentin Anciaux-2]

You have to explain why the exception is needed in the first place...

The rule is true until the rule is not true anymore, ok but you have
to explain for what sufficiently large N the successor function would
yield next 0 and why or to add that N and that exception to the
successor function as axiom, if not you can't avoid N+1. But torgny
doesn't evacuate N+1, merely it allows his set to grows undefinitelly
as when he has defined BIGGEST, he still argues BIGGEST+1 makes sense
, is a natural number but not part of the set of natural number, this
is non-sense, assuming your special successor rule BIGGEST+1 simply
does not exists at all.

I can understand this overflow successor function for a finite data
type or a real machine registe but not for N. The successor function
is simple, if you want it to have an exception at biggest you should
justify it.

Regards,
Quentin

2009/6/9 Brent Meeker <meekerdb@...>:

>
> Quentin Anciaux wrote:
>> 2009/6/9 Torgny Tholerus <torgny@...>:
>>
>>> Jesse Mazer skrev:
>>>
>>>>
>>>>> Date: Sat, 6 Jun 2009 21:17:03 +0200
>>>>> From: torgny@...
>>>>> To: everything-list@...
>>>>> Subject: Re: The seven step-Mathematical preliminaries
>>>>>
>>>>> My philosophical argument is about the mening of the word "all". To be
>>>>> able to use that word, you must associate it with a value set.
>>>>>
>>>> What's a "value set"? And why do you say we "must" associate it in
>>>> this way? Do you have a philosophical argument for this "must", or is
>>>> it just an edict that reflects your personal aesthetic preferences?
>>>>
>>>>
>>>>> Mostly that set is "all objects in the universe", and if you stay
>>>>>
>>>> inside the
>>>>
>>>>> universe, there is no problems.
>>>>>
>>>> *I* certainly don't define numbers in terms of any specific mapping
>>>> between numbers and objects in the universe, it seems like a rather
>>>> strange notion--shall we have arguments over whether the number 113485
>>>> should be associated with this specific shoelace or this specific
>>>> kangaroo?
>>>>
>>> When I talk about "universe" here, I do not mean our physical universe.
>>> What I mean is something that can be called "everything".  It includes
>>> all objects in our physical universe, as well as all symbols and all
>>> words and all numbers and all sets and all other universes.  It includes
>>> everything you can use the word "all" about.
>>>
>>
>> It includes all set, but no all set as it N includes all natural
>> number but not all natural number... excuse-me but this is non-sense.
>> Either N exists and has an infinite number of member and is
>> incompatible with an ultrafinitist view or N does not exists because
>> obviously N cannot be defined in an ultra-finitist way,
>
> That's not obvious to me.  You're assuming that N exists apart from
> whatever definition of it is given and that it is the infinite set
> described by the Peano axioms or equivalent.  But that's begging the
> question of whether a finite set of numbers that we would call "natural
> numbers" can be defined.  To avoid begging the question we need some
> definition of "natural" that doesn't a priori assume the set is finite
> or infinite; something like, "A set of numbers adequate to do all
> arithmetic we'll ever need" (unfortunately not very definite).  The
> problem is the successor axiom, if we modify it to S{n}=n+1 for n e N
> except for the case n=N where S{N}=0 and choose sufficiently large N it
> might satisfy the "natural" criteria.
>
> Brent
>
>
>> any set that
>> contains a finite number of natural number (and still you haven't
>> defined what it is in an ultrafinitist way) are not the set N.
>>
>> Also any operation involving two number (addition/multiplication) can
>> yield as result a number which has the same property as the departing
>> number (being a natural number) but is not natural number... Also
>> induction and inference cannot work in such a context.
>>
>>
>>> For you to be able to use the word "all", you must define the "domain"
>>> of that word.  If you do not define the domain, then it will be
>>> impossible for me and all other humans to understand what you are
>>> talking about.
>>>
>>
>> Well you are the first and only human I know who don't understand
>> "all" as everybody else does.
>>
>> Quentin Anciaux
>>
>>
>>> --
>>> Torgny Tholerus
>>>
>>>
>>
>>
>>
>>
>
>
> >
>

--
All those moments will be lost in time, like tears in rain.

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