Consciousness is information?

[Torgny Tholerus]

Quentin Anciaux skrev:
> If you are ultrafinitist then by definition the set N does not
> exist... (nor any infinite set countably or not).
>  

All sets are finite.  It it (logically) impossible to construct an
infinite set.

You can construct the set N of all natural numbers.  But that set must
be finite.  What the set N contains depends on how you have defined
"natural number".

> If you pose the assumption of a biggest number for N, you come to a
> contradiction because you use the successor operation which cannot
> admit a biggest number.(because N is well ordered any successor is
> strictly bigger and the successor operation is always valid *by
> definition of the operation*)
>  

You have to define the successor operation.  And to do that you have to
define the definition set for that operation.  So first you have to
define the set N of natural numbers.  And from that you can define the
successor operator.  The value set of the successor operator will be a
new set, that contains one more element than the set N of natural
numbers.  This new element is BIGGEST+1, that is strictly bigger than
all natural numbers.

--
Torgny Tholerus

> So either the set N does not exists in which case it makes no sense to
> talk about the biggest number in N, or the set N does indeed exists
> and it makes no sense to talk about the biggest number in N (while it
> makes sense to talk about a number which is strictly bigger than any
> natural number).
>
> To come back to the proof by contradiction you gave, the assumption
> (2) which is that BIGGEST+1 is in N, is completely defined by the mere
> existence of BIGGEST. If BIGGEST exists and well defined it entails
> that BIGGEST+1 is not in N (but this invalidate the successor
> operation and hence the mere existence of N). If BIGGEST in contrary
> does not exist (as such, means it is not the biggest) then BIGGEST+1
> is in N by definition of N.
>
> Regards,
> Quentin
>
>  

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[Brian Tenneson]

This is a denial of the axiom of infinity.  I think a foundational set theorist might agree that it is impossible to -construct- an infinite set from scratch which is why they use the axiom of infinity.
People are free to deny axioms, of course, though the result will not be like ZFC set theory.  The denial of axiom of foundation is one I've come across; I've never met anyone who denies the axiom of infinity.

For me it is strange that the following statement is false: every natural number has a natural number successor.  To me it seems quite arbitrary for the ultrafinitist's statement: every natural number has a natural number successor UNTIL we reach some natural number which does not have a natural number successor.  I'm left wondering what the largest ultrafinist's number is.

Torgny Tholerus wrote:

Quentin Anciaux skrev:
  

If you are ultrafinitist then by definition the set N does not
exist... (nor any infinite set countably or not).
  
    

All sets are finite.  It it (logically) impossible to construct an 
infinite set.

You can construct the set N of all natural numbers.  But that set must 
be finite.  What the set N contains depends on how you have defined 
"natural number".

  

If you pose the assumption of a biggest number for N, you come to a
contradiction because you use the successor operation which cannot
admit a biggest number.(because N is well ordered any successor is
strictly bigger and the successor operation is always valid *by
definition of the operation*)
  
    

You have to define the successor operation.  And to do that you have to 
define the definition set for that operation.  So first you have to 
define the set N of natural numbers.  And from that you can define the 
successor operator.  The value set of the successor operator will be a 
new set, that contains one more element than the set N of natural 
numbers.  This new element is BIGGEST+1, that is strictly bigger than 
all natural numbers.

  

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[Torgny Tholerus]

Brian Tenneson skrev:

> This is a denial of the axiom of infinity.  I think a foundational set
> theorist might agree that it is impossible to -construct- an infinite
> set from scratch which is why they use the axiom of infinity.
> People are free to deny axioms, of course, though the result will not
> be like ZFC set theory.  The denial of axiom of foundation is one I've
> come across; I've never met anyone who denies the axiom of infinity.
>
> For me it is strange that the following statement is false: every
> natural number has a natural number successor.  To me it seems quite
> arbitrary for the ultrafinitist's statement: every natural number has
> a natural number successor UNTIL we reach some natural number which
> does not have a natural number successor.  I'm left wondering what the
> largest ultrafinist's number is.

It is impossible to lock a box, and quickly throw the key inside the box
before you lock it.
It is impossible to create a set and put the set itself inside the set,
i.e. no set can contain itself.
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.

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.

--
Torgny Tholerus

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[Bruno Marchal]

Hi Jesse,

On 01 May 2009, at 19:36, Jesse Mazer wrote:

I found a paper on the Mandelbrot set and computability, I understand very little but maybe Bruno would be able to follow it:

http://arxiv.org/abs/cs.CC/0604003

The same author has a shorter outline or slides for a presentation on this subject at http://www.cs.swan.ac.uk/cie06/files/d37/PHP_MandelbrotCiE2006Swansea_Jul2006.pdf and at the end he asks the question "If M (Mandelbrot set) not Q-computable, can the Halting Problem be reduced to determining membership of (intersection of M and Q^2), i.e. how powerful a 'hypercomputer' is the Mandelbrot set?" I believe Q^2 here just refers to the set of all possible pairs of rational numbers. Maybe by "reducing" the Halting Problem he means that for any Turing machine + input, there might be some rule that would translate it into a pair of rational numbers such that the computation will halt iff the pair is included in the Mandelbrot set? Whatever he means, it sounds like he's saying it's an open question...

Jesse

> 
> 
> On Thu, Apr 30, 2009 at 10:35 AM, Bruno Marchal <marchal@...> wrote:
>>
>>
>> The mathematical Universal Dovetailer, the splashed universal Turing
>> Machine, the rational Mandelbrot set, or any creative sets in the
>> sense of Emil Post, does all computations. Really all, with Church
>> thesis. This is a theorem in math. The rock? Show me just the 30 first
>> steps of a computation of square-root(2). ...
> 
> Bruno,
> 
> I am interested about your statement regarding the Mandelbrot set
> implementing all computations, could you elaborate on this?

So, indeed the conjecture I made on the Mandelbrot Set concerns the decidability-on-the-rationals of the set M intersected with QXQ. And it is indeed still an open problem. Actually my question is the "creativity" (in the sense of Post) of M, and this would mean that you can reduce the halting problem of any Turing machine into a problem of membership of a rational complex number a+bi (a, b, in Q) to M. There would be one fixed algorithm transforming any computable problem on N into such a membership problem. If the solution is positive, then the Mandelbrot Set would be a compact representation of a Universal Dovetailing. Also, this would entail the existence of interesting relationship between classical computability theory and the theory of Chaos on the reals. The universality in chaos phenomenon (Feigenbaum) would be related to the Turing Universality. Also, each of us would be, in a sense, distributed densely on the boundary of M, and each little Mandelbrot would represent the third person projection view of each of our "first person plenitude". That would be cute, mainly for the pedagogy of the UD, but also, it would made it possible to borrow mathematical tools from chaos theory theory for the pursue of deriving physics from numbers.

Not everything is clear for me in Potgieter paper, probably a result of my incomptence, but it is very interesting. Thanks for the link.

Did I give you the link of the last, impressive M-zoom by phaumann? Look at it with the high quality option + full screen, if you are patient enough. Love it!

http://www.youtube.com/watch?v=x6DD1k4BAUg&feature=channel_page

Bruno

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

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[Brian Tenneson]

Torgny Tholerus wrote:

Brian Tenneson skrev:
  

This is a denial of the axiom of infinity.  I think a foundational set 
theorist might agree that it is impossible to -construct- an infinite 
set from scratch which is why they use the axiom of infinity.
People are free to deny axioms, of course, though the result will not 
be like ZFC set theory.  The denial of axiom of foundation is one I've 
come across; I've never met anyone who denies the axiom of infinity.

For me it is strange that the following statement is false: every 
natural number has a natural number successor.  To me it seems quite 
arbitrary for the ultrafinitist's statement: every natural number has 
a natural number successor UNTIL we reach some natural number which 
does not have a natural number successor.  I'm left wondering what the 
largest ultrafinist's number is.
    

It is impossible to lock a box, and quickly throw the key inside the box 
before you lock it.
  

I disagree.

It is impossible to create a set and put the set itself inside the set, 
i.e. no set can contain itself.
  

No one here is suggesting that you can with regards to natural numbers.

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.

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?

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[Bruno Marchal]

On 03 Jun 2009, at 20:11, Jason Resch wrote:

>
> On Fri, May 22, 2009 at 4:37 PM, Bruno Marchal <marchal@...>  
> wrote:
>
>>> Do you believe if we create a computer in this physical
>>> universe that it could be made conscious,
>>
>> But a computer is never conscious, nor is a brain. Only a person is
>> conscious, and a computer or a brain can only make it possible for a
>> person to be conscious relatively to another computer. So your
>> question is ambiguous.
>> It is not my brain which is conscious, it is me who is conscious. My
>> brain appears to make it possible for my consciousness to manifest
>> itself relatively to you. Remember that we are supposed to no more
>> count on the physical supervenience thesis.
>> It remains locally correct to attribute a consciousness through a
>> brain or a body to a person we judged succesfully implemented locally
>> in some piece of matter (like when we say yes to a doctor).  But the
>> piece of matter is not the subject of the consciousness. It is only
>> the "abstract person" or "program" who is the subject of  
>> consciousness.
>> To say a brain is conscious consists in doing Searle's'mistake when  
>> he
>> confused levels of computations in the Chinese room, as well seen
>> already by Hofstadter and Dennett in Mind's I.
>>
>>
>
> Thanks for your response, if I understand you correctly, you are
> saying that if we run a simulation of a mind, we are not creating
> consciousness, only adding an additional instantiation to a mind which
> already has an infinity of indeterminable instantiations.  Is that
> right?

Yes, you are right. When you implement an emulation of a mind, you are  
just adding such an instanciation relatively to you. Of course you are  
not adding anything in "Platonia".



>
>
> Does this imply that it is impossible to create a simulation of a mind
> that finds it lives in an environment without uncertainty?


That is correct.



>  If so is
> it because even if the physical laws in one instantiation may be
> certain, where some of the infinite number of computations that all
> instantiate that mind may diverge and in particular which one that
> mind will find itself in is not knowable?

Yes. I will come back on this in the seven step thread.


>
>
> The consequence being that all observers everywhere live in QM-like
> environments?

Absolutely. We can consider that we "live" in an infinity of  
computations, but we cannot distinguish them ... until they  
differentiate sufficiently so that they are in principle  
distinguishable (like being in Washington or being in Moscow). This  
entails that below our substitution level
what can be observed depends directly on some average on an infinity  
of computations. The quantum-like aspect of "nature" is, in that  
sense, a consequence of digitalism in the cognitive science. The  
classical, and computational, aspect of physics remains the hard  
things to derive.

Bruno


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




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[Torgny Tholerus]

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?

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.

>
>> 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.

--
Torgny Tholerus

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[Brian Tenneson]

On Thu, Jun 4, 2009 at 8:27 AM, Torgny Tholerus <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.
 

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?
 

>
>> 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. 

--
Torgny Tholerus

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[Jason Resch-2]

On Thu, Jun 4, 2009 at 9:29 AM, Bruno Marchal <marchal@...> wrote:

>
>
> On 03 Jun 2009, at 20:11, Jason Resch wrote:
>
>>
>> On Fri, May 22, 2009 at 4:37 PM, Bruno Marchal <marchal@...>
>> wrote:
>>
>>>> Do you believe if we create a computer in this physical
>>>> universe that it could be made conscious,
>>>
>>> But a computer is never conscious, nor is a brain. Only a person is
>>> conscious, and a computer or a brain can only make it possible for a
>>> person to be conscious relatively to another computer. So your
>>> question is ambiguous.
>>> It is not my brain which is conscious, it is me who is conscious. My
>>> brain appears to make it possible for my consciousness to manifest
>>> itself relatively to you. Remember that we are supposed to no more
>>> count on the physical supervenience thesis.
>>> It remains locally correct to attribute a consciousness through a
>>> brain or a body to a person we judged succesfully implemented locally
>>> in some piece of matter (like when we say yes to a doctor).  But the
>>> piece of matter is not the subject of the consciousness. It is only
>>> the "abstract person" or "program" who is the subject of
>>> consciousness.
>>> To say a brain is conscious consists in doing Searle's'mistake when
>>> he
>>> confused levels of computations in the Chinese room, as well seen
>>> already by Hofstadter and Dennett in Mind's I.
>>>
>>>
>>
>> Thanks for your response, if I understand you correctly, you are
>> saying that if we run a simulation of a mind, we are not creating
>> consciousness, only adding an additional instantiation to a mind which
>> already has an infinity of indeterminable instantiations.  Is that
>> right?
>
>
> Yes, you are right. When you implement an emulation of a mind, you are
> just adding such an instanciation relatively to you. Of course you are
> not adding anything in "Platonia".
>
>

But is the computer emulating the mind not also a platonic object?  If
the computer simulation does not count toward anything then what is
the point of saying yes to the doctor, or to pursue mind uploading
technology as a method to obtain immortality and escape eternal aging
as QM-immortality would predict?

>
>>
>>
>> Does this imply that it is impossible to create a simulation of a mind
>> that finds it lives in an environment without uncertainty?
>
>
> That is correct.
>
>
>
>>  If so is
>> it because even if the physical laws in one instantiation may be
>> certain, where some of the infinite number of computations that all
>> instantiate that mind may diverge and in particular which one that
>> mind will find itself in is not knowable?
>
> Yes. I will come back on this in the seven step thread.
>
>
>>
>>
>> The consequence being that all observers everywhere live in QM-like
>> environments?
>
> Absolutely. We can consider that we "live" in an infinity of
> computations, but we cannot distinguish them ... until they
> differentiate sufficiently so that they are in principle
> distinguishable (like being in Washington or being in Moscow). This
> entails that below our substitution level
> what can be observed depends directly on some average on an infinity
> of computations. The quantum-like aspect of "nature" is, in that
> sense, a consequence of digitalism in the cognitive science. The
> classical, and computational, aspect of physics remains the hard
> things to derive.
>

Interesting, I am curious is there some relationship between ones
substitution level and where one will find the QM uncertainty?  If all
observers live in uncertain environments, and it took us this long to
discover QM behavior, I imagine for some observers it could be much
harder or much easier to find this uncertainty level.

What do you think controls how deep one must look to see the QM
behavior first hand?  I suppose it might also be related to the
complexity of one's observer moment; the more information one takes in
from the environment and has in memory the lower the level the
uncertainty should be.  A God like mind that knew the position of
every particle in the universe in which it lived might not have any
uncertainty, but of course the mind couldn't encode everything about
itself...

Jason

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[Bruno Marchal]

Hi Marty,

On 04 Jun 2009, at 01:11, m.a. wrote:

Bruno,

           I stopped half-way through because I'm not at all sure of my answers and would like to have them confirmed or corrected, if necessary, rather than go on giving wrong answers.   marty a.

No problem.

Exercise 1: Could you define in intension the following infinite set C = {101, 103, 105, ...}

C = ?                          C={x such that x is odd & x <101}

I guess you meant C = {x such that x is odd and x > 101}.  ">" means "bigger than", and "<" means little than. OK.

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}.

Indeed 0 is little than 10, and 0 is a multiple of four (because 0 = 4*0), and

4  is little than 10, and 4 is a multiple of four (because 4 = 4*1)

8 is little than 10, and 8 is a multiple of 4 (because 8 = 4*2)

The next mutiple of 4 is 12. It cannot be in the set because 12 is bigger than 10.

The numbers 1, 2, 3, 5, 7, 9 cannot be in D, because they are not multiple of 4. You cannot write 1 = 4 * (some natural numbers), nor can you write 3 or 5, or 7 or 9 =  4 * x with x a natural number.

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".  I guess you try to go to much quickly, or that the font of your computer are too little, or that you have eyesight problems, or that you have some dyslexia.

Exercice 3. 

Let N = {0, 1, 2, 3, ...}

Let A = {x ⎮ x < 10}

Let B = {x ⎮ x is even}

Describe in extension (that is: exhaustion or quasi-exhaustion) the following sets:

N ∪ A = {0,1,2,3...} inter {x inter x<10}= {0,1,2,3...9}

N ∪ B = {0,1,2,3....} inter {x inter x is even}= {0,2,4,6...}

A ∪ B = {x inter x <10} inter {x inter x is even}= {0,2,4,6,8}

B ∪ A = {x inter x is even} inter {x inter x < 10}= {0,2,4,6,8}

All that would be correct if you were searching the intersection, but "∪" is the UNION symbol. (and "∩" is the INTERSECTION symbol).

also you wrote the "⎮" as "inter", instead of "such that". 

 

N ∩ A = {0,1,2,3...} inter {x inter x<10}= {0,1,2,3...9}

B ∩ A =  {x inter x is even} inter {x inter x < 10}= {0,2,4,6,8}

N ∩ B =  {0,1,2,3....} inter {x inter x is even}= {0,2,4,6...}

A ∩ B =   {x inter x <10} inter {x inter x is even}= {0,2,4,6,8}

All that is correct. Careful you were still using "inter" in place of "such that". Your last line should be

A ∩ B =   {x such that x <10} inter {x such that x is even}= {0,2,4,6,8}

Exercice 4

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

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

Both are correct.

Not bad Marty!  Just read carefully. I thing you have just dismiss the paragraph were I define "UNION". And then, you or your computer seems to have a trouble in distinguishing the symbols "∩" and "∪".  

Example 

{1, 2, 3} ∩  {3, 4, 5} = {3}

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

Tell me if it is OK, now.

And then I let you think on the next exercises. Take the time to read slowly. Have you a problem of dyslexia? Do you see the difference between "<" and ">" ?

If there is a problem with the symbols, I can switch on "english symbol".

Have a nice week-end, don't hesitate to ask questions, clarification of points, or more examples.

Bruno

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

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[Jason Resch-2]

On Thu, Jun 4, 2009 at 7:28 AM, kimjones@...
<kimjones@...> wrote:

>
>
>
>
>
>
>
> On Thu Jun  4  1:15 , Bruno Marchal  sent:
>
>>Very good answer, Kim,
>>Just a few comments. and then the sequel.
>>Exercice 4: does the real number square-root(2) belongs to {0, 1, 2,
>>3, ...}?
>>
>>
>>No idea what square-root(2) means. When I said I was innumerate I wasn't kidding! I
> could of course look
>>it up or ask my mathematics teacher friends but I just know your explanation will make
> theirs seem trite.
>>
>>Well thanks. The square root of 2 is a number x, such that x*x (x times x, x multiplied by
> itself) gives 2.For example, the square root of 4 is 2, because 2*2 is 4. The square root of
> 9 is 3, because 3*3 is 9. Her by "square root" I mean the positive square root, because we
> will see (more later that soon) that numbers can have negative square root, but please
> forget this. At this stage, with this definition, you can guess that the square root of 2
> cannot be a natural number. 1*1 = 1, and 2*2 = 4, and it would be astonishing that x
> could be bigger than 2. So if there is number x such that x*x is 2, we can guess that such
> a x cannot be a natural number, that is an element of {0, 1, 2, 3 ...}, and the answer of
> exercise 4 is "no". The square root of two will reappear recurrently, but more in examples,
> than in the sequence of notions which are strictly needed for UDA-7.
>
>
> OK - I find this quite mind-blowing; probably because I now understand it for the first
> time in my life. So how did it get this rather ridiculous name of "square root"? What's it
> called in French?
>

I don't know what it is called in French, but I can answer the first
part.  I remember the day I first figured out where the term came
from.

When you have a number multiplied by itself, the result is called a
square.  3*3 = 9, so 9 is a square.  Imagine arranging a set of peas,
if you can arrange them in a square (the four cornered kind) with the
same number of rows as columns, then that number is a square.  Some
examples of squares are: 4, 9, 16, 25, 36, 49, 64, 81, see the
pattern?  And the "roots" of those squares are 2, 3, 4, 5, 6, 7, 8,
and 9.  The square root is equal to the number of items in a row, or
column when you arrange them in a square.

This is a completely extraneous fact, but one I consider to be very
interesting: Multiply any 4 consecutive positive whole numbers and the
result will always be 1 less than a square number.  For example,
5*6*7*8 = 1680, which is 1 less than 1681, which is 41*41.  Isn't that
neat?

Jason

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[Brent Meeker-2]

I've never seen an ultrafinitist definition of  the natural numbers.  
The usual definition via Peano's axioms obviously rules out there being
a largest number.  I would suppose that an ultrafinitist definition of
the natural numbers would be something like seen in a computer (which is
necessarily finite). The successor operation would be defined such that
Successor (Biggest) = 0 or -Biggest.

Brent

Quentin Anciaux wrote:

> If you are ultrafinitist then by definition the set N does not
> exist... (nor any infinite set countably or not).
>
> If you pose the assumption of a biggest number for N, you come to a
> contradiction because you use the successor operation which cannot
> admit a biggest number.(because N is well ordered any successor is
> strictly bigger and the successor operation is always valid *by
> definition of the operation*)
>
> So either the set N does not exists in which case it makes no sense to
> talk about the biggest number in N, or the set N does indeed exists
> and it makes no sense to talk about the biggest number in N (while it
> makes sense to talk about a number which is strictly bigger than any
> natural number).
>
> To come back to the proof by contradiction you gave, the assumption
> (2) which is that BIGGEST+1 is in N, is completely defined by the mere
> existence of BIGGEST. If BIGGEST exists and well defined it entails
> that BIGGEST+1 is not in N (but this invalidate the successor
> operation and hence the mere existence of N). If BIGGEST in contrary
> does not exist (as such, means it is not the biggest) then BIGGEST+1
> is in N by definition of N.
>
> Regards,
> Quentin
>
> 2009/6/4 Torgny Tholerus <torgny@...>:
>  
>> Brian Tenneson skrev:
>>    
>>>> How do you know that there is no biggest number?  Have you examined all
>>>> the natural numbers?  How do you prove that there is no biggest number?
>>>>
>>>>
>>>>
>>>>        
>>> In my opinion those are excellent questions.  I will attempt to answer
>>> them.  The intended audience of my answer is everyone, so please forgive
>>> me if I say something you already know.
>>>
>>> Firstly, no one has or can examine all the natural numbers.  By that I
>>> mean no human.  Maybe there is an omniscient machine (or a "maximally
>>> knowledgeable" in some paraconsistent way)  who can examine all numbers
>>> but that is definitely putting the cart before the horse.
>>>
>>> Secondly, the question boils down to a difference in philosophy:
>>> mathematicians would say that it is not necessary to examine all natural
>>> numbers.  The mathematician would argue that it suffices to examine all
>>> essential properties of natural numbers, rather than all natural numbers.
>>>
>>> There are a variety of equivalent ways to define a natural number but
>>> the essential features of natural numbers are that
>>> (a) there is an ordering on the set of natural numbers, a well
>>> ordering.  To say a set is well ordered entails that every =nonempty=
>>> subset of it has a least element.
>>> (b) the set of natural numbers has a least element (note that it is
>>> customary to either say 0 is this least element or say 1 is this least
>>> element--in some sense it does not matter what the starting point is)
>>> (c) every natural number has a natural number successor.  By successor
>>> of a natural number, I mean anything for which the well ordering always
>>> places the successor as larger than the predecessor.
>>>
>>> Then the set of natural numbers, N, is the set containing the least
>>> element (0 or 1) and every successor of the least element, and only
>>> successors of the least element.
>>>
>>> There is nothing wrong with a proof by contradiction but I think a
>>> "forward" proof might just be more convincing.
>>>
>>> Consider the following statement:
>>> Whenever S is a subset of N, S has a largest element if, and only if,
>>> the complement of S has a least element.
>>>
>>> By complement of S, I mean the set of all elements of N that are not
>>> elements of S.
>>>
>>> Before I give a longer argument, would you agree that statement is
>>> true?  One can actually be arbitrarily explicit: M is the largest
>>> element of S if, and only if, the successor of M is the least element of
>>> the compliment of S.
>>>
>>>      
>> I do not agree that statement is true.  Because if you call the Biggest
>> natural number B, then you can describe N as = {1, 2, 3, ..., B}.  If
>> you take the complement of N you will get the empty set.  This set have
>> no least element, but still N has a biggest element.
>>
>> In your statement you are presupposing that N has no biggest element,
>> and from that axiom you can trivially deduce that there is no biggest
>> element.
>>
>> --
>> Torgny Tholerus
>>
>>    
>
>
>
>  

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[Bruno Marchal]

Hi Kim,


On 04 Jun 2009, at 14:28, kimjones@... wrote:

OK - I find this quite mind-blowing; probably because I now understand it for the first

time in my life. So how did it get this rather ridiculous name of "square root"? What's it

called in French?

Racine carrée. Literally "square root".

It comes from the fact that in elementary geometry the surface or area of a square which sides have length x, is given by x*x, also written x^2, which is then called the  "square of x". Taking the square root of a number, consists in doing the inverse of taking the square of a number. It consists in finding the length of a square knowing its area.

Mathematician and especially logician *can* use arbitrary vocabulary. It is the essence of the axiomatic method in "pure mathematics" that what is conveying does not depend on the term which are used. Hilbert said once that he could have use the term "glass of bear" instead of "line" in his work in geometry.

A = {x such that x is even and smaller than 100}  = {x ⎮ x is even & x

special character, abbreviating "such that", and I hope it goes through the mail.

Just an upright line? It comes through as that. I can't seem to get this symbol happening so I will

use "such that"

Yes, "such that" is abbreviated by an upright line. Sometimes also by a half circle followed by a little line, but I don't find it on my palette!

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 objects x, (or number x if we are in a context where we talk

about numbers) such that x is even.

Exercise 1: Could you define in intension the following infinite set C = {101, 103, 105,

...}C = ?

C = {x such that x is odd and x > 101}

Correct.

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 and x is a multiple of 4}?

D = 4*x where x = 0 but also { 1, 2, 3, 4, 8 }

Hmm...
Marty made a similar error. D is a set. May be you wanted to say:

D = {4*x where x = 0 but also { 1, 2, 3, 4, 8 }}. But this does not make much sense. Even if I try to imagine favorably some meaning, I would say that it would mean that D is the set of numbers having the shape 4*x (that is capable of being written as equal to 4*x for some x), and such that x belongs to {0, 1, 2, 3, 4, 8}.
A proper way to describe that set would be

D = {y such that y = 4x and x belongs-to {0, 1, 2, 3, 4, 8}}.

But that would makes D = {0, 4, 8, 12, 32}.

The set D = {x ⎮ x < 10 and x is a multiple of 4} is just, in english, the set of natural numbers which are little than 10 and which are a multiple of 4. The only numbers which are little than 10, and multiple of 4 are the numbers 0, 4, and 8.  D = {0, 4, 8}.

I now realise I am doomed for the next set of exercises because I cannot get to the special

symbols required (yet). As I am adding Internet Phone to my system, I am currently using an

ancient Mac without the correct symbol pallette while somebody spends a few days to flip a single

switch...as soon as I can get back to my regular machine I will complete the rest.

Take it easy. No problem.

In the meantime I am enjoying the N+1 disagreement - how refreshing it is to see that classical

mathematics remains somewhat controversial!

The term is a bit too strong. It is a bit like if I told you that "I am Napoleon", and you conclude that the question of the death of Napoleon is still controversial. I exaggerate a little bit to make my point, but I know only two ultrafinitists *in math*, and I have never understood what they mean by "number", nor did I ever met someone understanding them.

What makes just a little bit more sense (and I guess that's what Torgny really is) is being ultrafinitist *in physics*, and being physicalist. You postulate there is a physical universe, made of a finite number of particles, occupying a finite volume in space-time, etc. Everything is finite, including the "everything".
Then  by using the "unintelligible identity thesis" (and thus reintroducing the mind-body problem), you can prevent the comp white rabbits inflation. Like all form of materialism, this leads to eliminating the person soon or later (by the unsolvability of the mind-body problem by finite means). Ultrafinitist physicalism eliminates also mathematics and all immaterial notions, including all universal machines. Brrr...

The real question is "do *you* think that there is a biggest natural number"? Just tell me at once, because if you really believe that there is a biggest natural number, I have no more clues at all how you could believe in any of computer science nor UDA.

Remember that Thorgny pretends also to be a zombie. It has already eliminate its own consciousness.

Note that after the step seven, you can still use ultrafinitist physicalism to eliminate the inflation of white rabbits *discourses*. After step 8, normally this move stop working unless you eliminate consciousness and persons. I think Thorgny is aware of that, and that is why he defends the idea that he is a zombie. From that point of view he is remarkably coherent with respect to the UD reasoning. But in front of person eliminators I can only say ... Brrr...

Bruno

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

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[Brent Meeker-2]

Torgny Tholerus wrote:

> Brian Tenneson skrev:
>  
>> This is a denial of the axiom of infinity.  I think a foundational set
>> theorist might agree that it is impossible to -construct- an infinite
>> set from scratch which is why they use the axiom of infinity.
>> People are free to deny axioms, of course, though the result will not
>> be like ZFC set theory.  The denial of axiom of foundation is one I've
>> come across; I've never met anyone who denies the axiom of infinity.
>>
>> For me it is strange that the following statement is false: every
>> natural number has a natural number successor.  To me it seems quite
>> arbitrary for the ultrafinitist's statement: every natural number has
>> a natural number successor UNTIL we reach some natural number which
>> does not have a natural number successor.  I'm left wondering what the
>> largest ultrafinist's number is.
>>    
>
> It is impossible to lock a box, and quickly throw the key inside the box
> before you lock it.
> It is impossible to create a set and put the set itself inside the set,
> i.e. no set can contain itself.
> 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.
>  

Depends on how you define "successor".

Brent

> 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.
>
>  


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[Jason Resch-2]

Torngy,

How many numbers do you think exist between 0 and 1?  Certainly not
only the ones we define, for then there would be a different quantity
of numbers between 1 and 2, or 2 and 3.

Jason

On Thu, Jun 4, 2009 at 10:27 AM, Torgny Tholerus <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?
>
> 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.
>
>>
>>> 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.
>
> --
> Torgny Tholerus
>
> >
>

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[Bruno Marchal]

On 04 Jun 2009, at 15:40, Brian Tenneson wrote:

> This is a denial of the axiom of infinity.  I think a foundational  
> set theorist might agree that it is impossible to -construct- an  
> infinite set from scratch which is why they use the axiom of infinity.
> People are free to deny axioms, of course, though the result will  
> not be like ZFC set theory.  The denial of axiom of foundation is  
> one I've come across; I've never met anyone who denies the axiom of  
> infinity.

Among mathematicians nobody denies the axiom of infinity, but many  
philosopher of mathematics are attracted by finitism.
But Torgny is ultrafinitist. That is much rare. he denies the  
existence of natural numbers above some rather putative biggest  
natural number.


>
>
> For me it is strange that the following statement is false: every  
> natural number has a natural number successor.

I thought he would have said this, and accepted that the successor of  
its N is equal to N+1. Nut in a reply he says that N+1 exists but is  
not a natural number, which I think should not be consistent.




> To me it seems quite arbitrary for the ultrafinitist's statement:  
> every natural number has a natural number successor UNTIL we reach  
> some natural number which does not have a natural number successor.  
> I'm left wondering what the largest ultrafinist's number is.

It cannot be a constructive object. It is a number which is so big  
that if you add 1 to it, the "everything" explodes!
I dunno. I still suspect that ultrafinitism in math cannot be  
consistent, unlike the many variate form of finitism. Comp is arguably  
a form of finitism at the ontological level, yet an ultra-infinitism,  
if I can say, at the epistemological level.

Bruno

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




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[Jesse Mazer]

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

> Date: Thu, 4 Jun 2009 15:23:04 +0200
> From: torgny@...
> To: everything-list@...
> Subject: Re: The seven step-Mathematical preliminaries
>
>
> Quentin Anciaux skrev:
>> If you are ultrafinitist then by definition the set N does not
>> exist... (nor any infinite set countably or not).
>>
>
> All sets are finite. It it (logically) impossible to construct an
> infinite set.

What do you mean by "construct"? Do we have to actually write out or otherwise physically embody every element? Why can't we think of a particular "set" as just a type of rule that, given any possible element, tells you whether or not that element is a member or not? In this case there's no reason the rule couldn't be such that there are an infinite number of possible inputs that the rule would identify as valid members.

>
> You can construct the set N of all natural numbers. But that set must
> be finite. What the set N contains depends on how you have defined
> "natural number".

How do *you* define "natural number", if not according to the usual recursive rule that 1 is a natural number and that if N is a natural number, N+1 is also a natural number? Hopefully you agree that there can be no finite upper limit on possible inputs you could give this rule that the rule would identify as valid natural numbers? I think your claim would be that simply describing the rule is not a valid way of "constructing" the set of natural numbers. If so, why *isn't* it valid? *You* may prefer to adopt the rule that we should only be allowed to call something a "set" if we can actually write out every member, but do you have any argument as to why it's "invalid" for the rest of us to define sets simply as general rules that decide whether a given input is a member or not? This seems more like an aesthetic preference on your part rather than something you have a compelling philosophical argument for (or at least if you have such an argument you haven't provided it).

Jesse

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[Kory Heath-3]

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?

-- Kory


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[m.a.-2]

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

Bruno,

           I don't have dyslexia but my keyboard doesn't contain either the UNION symbol or the INTERSECTION symbol (unless I want to go into an INSERT pull down menu every time I use those symbols). 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 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. Please include examples. I tend to be somewhat careless when dealing with very fine distinctions and may type the wrong symbol while intending to type the correct one. Also, I must admit that the lessons are going too fast for me and are moving ahead before I've mastered the previous material. 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.

 

 

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

From: marchal@...

To: everything-list@...

Sent: Thursday, June 04, 2009 12:04 PM

Subject: Re: The seven step-Mathematical preliminaries 2

Hi Marty,

On 04 Jun 2009, at 01:11, m.a. wrote:

Bruno,

           I stopped half-way through because I'm not at all sure of my answers and would like to have them confirmed or corrected, if necessary, rather than go on giving wrong answers.   marty a.

No problem.

Exercise 1: Could you define in intension the following infinite set C = {101, 103, 105, ...}

C = ?                          C={x such that x is odd & x <101}

I guess you meant C = {x such that x is odd and x > 101}.  ">" means "bigger than", and "<" means little than. OK.

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}.

Indeed 0 is little than 10, and 0 is a multiple of four (because 0 = 4*0), and

4  is little than 10, and 4 is a multiple of four (because 4 = 4*1)

8 is little than 10, and 8 is a multiple of 4 (because 8 = 4*2)

The next mutiple of 4 is 12. It cannot be in the set because 12 is bigger than 10.

The numbers 1, 2, 3, 5, 7, 9 cannot be in D, because they are not multiple of 4. You cannot write 1 = 4 * (some natural numbers), nor can you write 3 or 5, or 7 or 9 =  4 * x with x a natural number.

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".  I guess you try to go to much quickly, or that the font of your computer are too little, or that you have eyesight problems, or that you have some dyslexia.

Exercice 3. 

Let N = {0, 1, 2, 3, ...}

Let A = {x ⎮ x < 10}

Let B = {x ⎮ x is even}

Describe in extension (that is: exhaustion or quasi-exhaustion) the following sets:

N ∪ A = {0,1,2,3...} inter {x inter x<10}= {0,1,2,3...9}

N ∪ B = {0,1,2,3....} inter {x inter x is even}= {0,2,4,6...}

A ∪ B = {x inter x <10} inter {x inter x is even}= {0,2,4,6,8}

B ∪ A = {x inter x is even} inter {x inter x < 10}= {0,2,4,6,8}

All that would be correct if you were searching the intersection, but "∪" is the UNION symbol. (and "∩" is the INTERSECTION symbol).

also you wrote the "⎮" as "inter", instead of "such that". 

 

N ∩ A = {0,1,2,3...} inter {x inter x<10}= {0,1,2,3...9}

B ∩ A =  {x inter x is even} inter {x inter x < 10}= {0,2,4,6,8}

N ∩ B =  {0,1,2,3....} inter {x inter x is even}= {0,2,4,6...}

A ∩ B =   {x inter x <10} inter {x inter x is even}= {0,2,4,6,8}

All that is correct. Careful you were still using "inter" in place of "such that". Your last line should be

A ∩ B =   {x such that x <10} inter {x such that x is even}= {0,2,4,6,8}

Exercice 4

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

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

Both are correct.

Not bad Marty!  Just read carefully. I thing you have just dismiss the paragraph were I define "UNION". And then, you or your computer seems to have a trouble in distinguishing the symbols "∩" and "∪".  

Example 

{1, 2, 3} ∩  {3, 4, 5} = {3}

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

Tell me if it is OK, now.

And then I let you think on the next exercises. Take the time to read slowly. Have you a problem of dyslexia? Do you see the difference between "<" and ">" ?

If there is a problem with the symbols, I can switch on "english symbol".

Have a nice week-end, don't hesitate to ask questions, clarification of points, or more examples.

Bruno

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

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[Torgny Tholerus]

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.  BIGGEST+1 is bigger than
all natural numbers, because all natural numbers belongs to BIGGEST+1.

>  
>
>
>     >
>     >> 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.

--
Torgny Tholerus

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