Consciousness is information?

[Torgny Tholerus]

Kory Heath skrev:

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

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

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

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

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

A is a set, then A belongs to A.

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

--
Torgny Tholerus

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[Quentin Anciaux-2]

2009/6/5 Torgny Tholerus <torgny@...>:

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

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

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

Quentin.

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

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

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[Jesse Mazer]

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

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

Here you're just contradicting yourself. If you say BIGGEST+1 "is then a natural number", that just proves that the set N was not in fact the set "of all natural numbers". The alternative would be to say BIGGEST+1 is *not* a natural number, but then you need to provide a definition of "natural number" that would explain why this is the case.

> The biggest advantage is that everything is finite, and you can then
> really know that the mathematical theory you get is consistent, it does
> not contain any contradictions.

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

Jesse

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[Brian Tenneson]

Torgny Tholerus wrote:

Brian Tenneson skrev:
  

On Thu, Jun 4, 2009 at 8:27 AM, Torgny Tholerus <torgny@... 
torgny@...> wrote:


    Brian Tenneson skrev:
    >
    >
    > Torgny Tholerus wrote:
    >> It is impossible to create a set where the successor of every
    element is
    >> inside the set, there must always be an element where the
    successor of
    >> that element is outside the set.
    >>
    > I disagree.  Can you prove this?
    > Once again, I think the debate ultimately is about whether or not to
    > adopt the axiom of infinity.
    > I think everyone can agree without that axiom, you cannot "build" or
    > "construct" an infinite set.
    > There's nothing right or wrong with adopting any axioms.  What
    results
    > is either interesting or not, relevant or not.

    How do you handle the Russell paradox with the set of all sets
    that does
    not contain itself?  Does that set contain itself or not?

 
If we're talking about ZFC set theory, then the axiom of foundation 
prohibits sets from being elements of themselves.
I think we agree that in ZFC, there is no set of all sets.
    

But there is a set of all sets.  You can construct it by taking all 
sets, and from them doing a new set, the set of all sets.  But note, 
this set will not contain itself, because that set did not exist before.
  

If that set does not contain itself then it is not a set of all sets.

 



    My answer is that that set does not contain itself, because no set can
    contain itself.  So the set of all sets that does not contain
    itself, is
    the same as the set of all sets.  And that set does not contain
    itself.
    This set is a set, but it does not contain itself.  It is exactly the
    same with the natural numbers, *BIGGEST+1 is a natural number, but it
    does not belong to the set of all natural numbers.  *The set of
    all sets
    is a set, but it does not belong to the set of all sets.

How can BIGGEST+1 be a natural number but not belong to the set of all 
natural numbers?
    

One way to represent natural number as sets is:

0 = {}
1 = {0} = {{}}
2 = {0, 1} = 1 union {1} = {{}, {{}}}
3 = {0, 1, 2} = 2 union {2} = ...
. . .
n+1 = {0, 1, 2, ..., n} = n union {n}
. . .

Here you can then define that a is less then b if and only if a belongs 
to b.

With this notation you get the set N of all natural numbers as {0, 1, 2, 
...}.  But the remarkable thing is that N is exactly the same as 
BIGGEST+1.  BIGGEST+1 is a set with the same structure as all the other 
natural numbers, so it is then a natural number.  But BIGGEST+1 is not a 
member of N, the set of all natural numbers.  BIGGEST+1 is bigger than 
all natural numbers, because all natural numbers belongs to BIGGEST+1.
  

Right, so n+1 is a natural number whenever n is. 

 


    >
    >> What the largest number is depends on how you define "natural
    number".
    >> One possible definition is that N contains all explicit numbers
    >> expressed by a human being, or will be expressed by a human
    being in the
    >> future.  Amongst all those explicit numbers there will be one
    that is
    >> the largest.  But this "largest number" is not an explicit number.
    >>
    >>
    > This raises a deeper question which is this: is mathematics
    dependent
    > on humanity or is mathematics independent of humanity?
    > I wonder what would happen to that human being who finally expresses
    > the largest number in the future.  What happens to him when he wakes
    > up the next day and considers adding one to yesterday's number?

    This is no problem.  If he adds one to the explicit number he
    expressed
    yesterday, then this new number is an explicit number, and the number
    expressed yesterday was not the largest number.  Both 17 and 17+1 are
    explicit numbers.

This goes back to my earlier comment that it's hard for me to believe 
that the following statement is false:
every natural number has a natural number successor
We -must- be talking about different things, then, when we use the 
phrase natural number.
I can't say your definition of natural numbers is right and mine is 
wrong, or vice versa.  I do wonder what advantages there are to the 
ultrafinitist approach compared to the math I'm familiar with. 
    

The biggest advantage is that everything is finite, and you can then 
really know that the mathematical theory you get is consistent, it does 
not contain any contradictions.

  

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

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[Bruno Marchal]

Hi Marty,

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

Bruno,

           I don't have dyslexia

Good news.

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

Nor do mine!

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

Like I have to do too.

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

I gave them.

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

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

Please include examples.

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

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

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

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

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

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

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

What is possible is that you understand, but fail ti memorize. This will lead to problems later. So you have to make your own summary and be sure you can easily revise the definition.

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

I think that there is no problem at all. I am just waiting for explicit question from the second round. You can ask any question, and slow me down as much as you want so that we proceed at your own rhythm. 

Don't ask me to slow down in any abstract way. You are the one who have to slow me down by pointing on what you don't understand in a post.

take it easy, and take all your time. Don't try to understand the more advanced replies I give to people who have a bigger baggage.

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

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

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

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

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

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

Examples 

{1, 2, 3} ∩  {3, 4, 5} = {3}

{1, 2, 3} ∪  {3, 4, 5} = {1, 2, 3, 4, 5}

Bruno

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

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[A. Wolf]

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

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

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

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

Anna


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[m.a.-2]

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

Bruno,

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

                                                                                                                                                                                        marty a.

 

 

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

From: marchal@...

To: everything-list@...

Sent: Friday, June 05, 2009 10:03 AM

Subject: Re: The seven step-Mathematical preliminaries 2

Hi Marty,

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

Bruno,

           I don't have dyslexia

Good news.

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

Nor do mine!

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

Like I have to do too.

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

I gave them.

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

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

Please include examples.

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

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

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

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

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

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

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

What is possible is that you understand, but fail ti memorize. This will lead to problems later. So you have to make your own summary and be sure you can easily revise the definition.

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

I think that there is no problem at all. I am just waiting for explicit question from the second round. You can ask any question, and slow me down as much as you want so that we proceed at your own rhythm. 

Don't ask me to slow down in any abstract way. You are the one who have to slow me down by pointing on what you don't understand in a post.

take it easy, and take all your time. Don't try to understand the more advanced replies I give to people who have a bigger baggage.

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

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

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

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

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

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

Examples 

{1, 2, 3} ∩  {3, 4, 5} = {3}

{1, 2, 3} ∪  {3, 4, 5} = {1, 2, 3, 4, 5}

Bruno

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

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[Torgny Tholerus]

Jesse Mazer skrev:

>
>
> > Date: Fri, 5 Jun 2009 08:33:47 +0200
> > From: torgny@...
> > To: everything-list@...
> > Subject: Re: The seven step-Mathematical preliminaries
> >
> >
> > Brian Tenneson skrev:
> >>
> >> How can BIGGEST+1 be a natural number but not belong to the set of all
> >> natural numbers?
> >
> > One way to represent natural number as sets is:
> >
> > 0 = {}
> > 1 = {0} = {{}}
> > 2 = {0, 1} = 1 union {1} = {{}, {{}}}
> > 3 = {0, 1, 2} = 2 union {2} = ...
> > . . .
> > n+1 = {0, 1, 2, ..., n} = n union {n}
> > . . .
> >
> > Here you can then define that a is less then b if and only if a belongs
> > to b.
> >
> > With this notation you get the set N of all natural numbers as {0,
> 1, 2,
> > ...}. But the remarkable thing is that N is exactly the same as
> > BIGGEST+1. BIGGEST+1 is a set with the same structure as all the other
> > natural numbers, so it is then a natural number. But BIGGEST+1 is not a
> > member of N, the set of all natural numbers.
>
> Here you're just contradicting yourself. If you say BIGGEST+1 "is then
> a natural number", that just proves that the set N was not in fact the
> set "of all natural numbers". The alternative would be to say
> BIGGEST+1 is *not* a natural number, but then you need to provide a
> definition of "natural number" that would explain why this is the case.

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".

--
Torgny Tholerus

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[Jesse Mazer]

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> Date: Sat, 6 Jun 2009 16:48:21 +0200
> From: torgny@...
> To: everything-list@...
> Subject: Re: The seven step-Mathematical preliminaries
>
>
> Jesse Mazer skrev:
>>
>>
>>> Date: Fri, 5 Jun 2009 08:33:47 +0200
>>> From: torgny@...
>>> To: everything-list@...
>>> Subject: Re: The seven step-Mathematical preliminaries
>>>
>>>
>>> Brian Tenneson skrev:
>>>>
>>>> How can BIGGEST+1 be a natural number but not belong to the set of all
>>>> natural numbers?
>>>
>>> One way to represent natural number as sets is:
>>>
>>> 0 = {}
>>> 1 = {0} = {{}}
>>> 2 = {0, 1} = 1 union {1} = {{}, {{}}}
>>> 3 = {0, 1, 2} = 2 union {2} = ...
>>> . . .
>>> n+1 = {0, 1, 2, ..., n} = n union {n}
>>> . . .
>>>
>>> Here you can then define that a is less then b if and only if a belongs
>>> to b.
>>>
>>> With this notation you get the set N of all natural numbers as {0,
>> 1, 2,
>>> ...}. But the remarkable thing is that N is exactly the same as
>>> BIGGEST+1. BIGGEST+1 is a set with the same structure as all the other
>>> natural numbers, so it is then a natural number. But BIGGEST+1 is not a
>>> member of N, the set of all natural numbers.
>>
>> Here you're just contradicting yourself. If you say BIGGEST+1 "is then
>> a natural number", that just proves that the set N was not in fact the
>> set "of all natural numbers". The alternative would be to say
>> BIGGEST+1 is *not* a natural number, but then you need to provide a
>> definition of "natural number" that would explain why this is the case.
>
> 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?

Also, any comment on my point about there being an infinite number of possible propositions about even a finite set, or about my question about whether you have any philosophical/logical argument for saying all sets must be finite, 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')

Jesse

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[Brent Meeker-2]

Torgny Tholerus wrote:

> Jesse Mazer skrev:
>  
>>    
>>> Date: Fri, 5 Jun 2009 08:33:47 +0200
>>> From: torgny@...
>>> To: everything-list@...
>>> Subject: Re: The seven step-Mathematical preliminaries
>>>
>>>
>>> Brian Tenneson skrev:
>>>      
>>>> How can BIGGEST+1 be a natural number but not belong to the set of all
>>>> natural numbers?
>>>>        
>>> One way to represent natural number as sets is:
>>>
>>> 0 = {}
>>> 1 = {0} = {{}}
>>> 2 = {0, 1} = 1 union {1} = {{}, {{}}}
>>> 3 = {0, 1, 2} = 2 union {2} = ...
>>> . . .
>>> n+1 = {0, 1, 2, ..., n} = n union {n}
>>> . . .
>>>
>>> Here you can then define that a is less then b if and only if a belongs
>>> to b.
>>>
>>> With this notation you get the set N of all natural numbers as {0,
>>>      
>> 1, 2,
>>    
>>> ...}. But the remarkable thing is that N is exactly the same as
>>> BIGGEST+1. BIGGEST+1 is a set with the same structure as all the other
>>> natural numbers, so it is then a natural number. But BIGGEST+1 is not a
>>> member of N, the set of all natural numbers.
>>>      
>> Here you're just contradicting yourself. If you say BIGGEST+1 "is then
>> a natural number", that just proves that the set N was not in fact the
>> set "of all natural numbers". The alternative would be to say
>> BIGGEST+1 is *not* a natural number, but then you need to provide a
>> definition of "natural number" that would explain why this is the case.
>>    
>
> 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".
>
>  

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

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[A. Wolf]

> 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?

There is a group of faculty who address this problem directly in my
department.  But any general-purpose computer can emulate true, unlimited
natural numbers (which is what people often do, rather than relying on
bounded ints).  The only real limitations that make "computer" not-equal-to
"Turing machine" are memory and the limited patience of humans.  This is one
reason why people spend more time researching P vs. NP than
artificially-imposed limits.

When you add bounds to numbers it requires additional proof obligations,
which makes it more difficult to prove things.  And you can't directly prove
anything about numbers that exist outside the bounds under which you're
working.

Anna


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[m.a.-2]

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

Bruno,

           Before I leave on holiday, I am following your advice to make my own table of symbols. Let me ask first whether the smaller rectangles have a different reference from the larger ones as seen in your example below?

 

----- 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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[Torgny Tholerus]

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.

>
> 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.

> 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.  Mostly
that set is "all objects in the universe", and if you stay inside the
universe, there is no problems.  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, you can not substitute
it with an object outside the universe, because that object was not
included in the original statement.

> 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.  And you may now be
able to understand, that if the number of objects in the universe is
finite, then this sentence will just define a finite set.

--
Torgny Tholerus

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[Jesse Mazer]

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

> 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?

>>
>> 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.

>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?

>
>> 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.

Jesse

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[Bruno Marchal]

Marty,

Bruno,

           Before I leave on holiday, I am following your advice to make my own table of symbols. Let me ask first whether the smaller rectangles have a different reference from the larger ones as seen in your example below?

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/

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[Brent Meeker-2]

Jesse Mazer 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?
>
> >>
> >> 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.

In fact we don't know "truths" about the surface topography of planets,
including the Earth.  Our knowledge is in terms of models, like WGS84,
or topographic maps.  We have confidence in these as approximations in
some degree, but not in the same way as the "truths" we know about the
mathematical models.  For example we can calculate the path distance
between two points on the WGS84 ovoid and we know that value, but it is
only approximately equal to the value we would measure by surveying or
by GPS.  When we use infinities and infinitesimals it greatly simplifies
our mathematical models, but it probably doesn't correspond to anything
in the world.

Brent

>
> >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?
>
> >
> >> 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.
>
> Jesse
>
> >

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[Jesse Mazer]

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If it helps, here's a screenshot of how the symbols are supposed to look:

http://img34.imageshack.us/img34/3345/picture2uzk.png

---

From: marchal@...
To: everything-list@...
Subject: Re: The seven step-Mathematical preliminaries 2
Date: Sat, 6 Jun 2009 22:36:01 +0200

Marty,

Bruno,

           Before I leave on holiday, I am following your advice to make my own table of symbols. Let me ask first whether the smaller rectangles have a different reference from the larger ones as seen in your example below?

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/

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[m.a.-2]

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(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.

----- 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/

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[m.a.-2]

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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}".

           Can you see the cause of my confusion? 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.

 

 

 

 

----- 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/

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[Quentin Anciaux-2]

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.

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.

> You can only
> create a finite number of proposition with finite length during your
> lifetime.

What is a lifetime . What is truth ? Either you ****CAN*** define a
limit or you ***CAN'T***.

> Just like the number of natural numbers are unlimited but
> finite, so are the possible propositions unlimited but finte.

****EVERY*** ***MEMBER*** of the set ***N*** is
****************************FINITE*********************************

>> 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.  Mostly
> that set is "all objects in the universe", and if you stay inside the
> universe, there is no problems.  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, you can not substitute
> it with an object outside the universe, because that object was not
> included in the original statement.
>
>> 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.  And you may now be
> able to understand, that if the number of objects in the universe is
> finite, then this sentence will just define a finite set.
>
> --
> Torgny Tholerus
>
> >
>

I will read the rest (and others) email later unfortunatelly.

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

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