subject:"The Axiom Of Choice and ComputationalismT"

Re: The Axiom Of Choice and ComputationalismT

2015-09-28 Thread John Clark

On Mon, Sep 28, 2015 Bruno Marchal  wrote:

t
> > 
> he same set of axiom, ZF, can have a model verifying AC and a model
> verifying ~AC,
>

Yes, we've known that since 1963 and therefore AC is independent of ZF. 

> >>
>> emulated people have access to arithmetic just like non emulated
>> people, so regardless of if they are emulated or not why do the employees
>> at INTEL bother to use silicon?
>
>
> >
> To share the computations that they are living. To manifest themselves
> relatively to each other.
>  
> Even the people in arithmetic needs hardware to do that.
>

Yes I know they need hardware, and I have a explanation why, your
explanation makes no sense. 

> > 
> But us, from outside the arithmetical reality we can see that their
> hardware are given by the internal FPI limits.
>

I don't see the relevance
Fixed
 P
oint
I
teration
 has to all this.

> > 
> hardware is not something which exist, it is only an appearance.
>

Even if that is true Is it important, what does "
only an appearance
" even mean? 
How would things be the slightest bit different if matter did not

just appear to exist but did exist?  

> > 
> Arithmetic let physics emerging from the (arithmetical) computations,
>  
> and the physics let human to implement computations in it.
>

I don't see why you're so certain that isn't backward. There is ZERO
evidence that arithmetical computations independent of physics even
exists, but there is plenty of evidence that physical matter exists. So why
wouldn't it be more likely that arithmetic is just something that helps
humans understand how matter behaves?

> > 
> And that relation is not transitive,
>

But why? Even a high level computer language can access a machine language
subroutine when needed, so when a human wants to make a computation why
can't he do the same and bypass to tedious physical matter stage and make
calculations without matter?

> 
> If we are machine, physics is a branch of machine theology
>

I have no opinion on that because I don't know what the word "
theology
" means in Brunospeak. 

> > 
> I don't want introduce too much technical jargon here.
>

That's not technical jargon that's homemade jargon, aka baby-talk.

>
>>> >>
>>> OK, but then we are not Turing emulable, and you need to explain me
>>> what magical thing, or actual infinite, you are using for that primitive
>>> matter to select the computations, or just abandon comp, and revised the
>>> contract asking for them to keep intact the actual infinities in the
>>> primitive matter of your brain (good luck explaining them what you mean).
>>
>>
>> >> 
>> I can not parse that sentence, it does not compute.
>
>
> > 
> I said that if primitive matter is assumed in the theory, then you cannot
> be a machine,
>

I can't see why. Matter can be primitive and still be generic, in fact if
it's truly fundamental, a brute fact, it would more or less have to be.

> > 
> and you should say no to the doctor to remain consistent.
>

I can't see why
. I am just the way atoms behave when they're organized in a johnkclarkian
way and all hydrogen atoms are identical so replacing one with another
would make no difference.

> 
> you would not criticize a theoy like QM fro not being able to make a pizza,
>
but you do that critics for RA.
>

To make a pizza a chef is required. A theory can't make anything, without
matter nothing can make anything into anything.  Mathematics is eternal
and static; if you want change (such as thought) you need matter.

> > 
> You are confusing the theory and what is supposed to be explained by the
> theory.
>

You are confusing
 the explanation of an act with the act itself.

  John K Clark

-- 
You received this message because you are subscribed to the Google Groups 
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email 
to everything-list+unsubscr...@googlegroups.com.
To post to this group, send email to everything-list@googlegroups.com.
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.

Re: The Axiom Of Choice and ComputationalismT

2015-09-27 Thread John Clark

On Sun, Sep 27, 2015 at 12:04 PM, Bruno Marchal  wrote:

> 
> The constructible set of Gödel can be use to show that ZF and ZFC proves
> the same arithmetical theorems.

That is incorrect, ZF can not prove that the Banach-Tarski construction
works, ZFC can. What Godel discovered in 1938 was that the theorems that ZF
can prove will produce  no contradictions if you assume that the Axiom of
Choice is TRUE; what Paul Cohen discovered in 1963 was that the theorems
that ZF can prove will produce  no contradictions if you assume that the
Axiom of Choice is FALSE, thus in 1963 we knew that ZF has nothing to with
choice and that's why it's called a axiom.

>> 
>> If a axiom has been verified, that is to say if it can be derived from
>> other axioms,
>
>
> > 
> I meant verified in a model, not prove in a theory.
>

In this case ZF is the model. 

> > 
> If a proposition is verified in a model, its negation can still be
> verified in another model.
>

Yes,
 a set of axioms other than ZF could
derive
the Axiom Of Choice and yet another set
of axioms 
could
derive the negation of 
the Axiom Of Choice
.

> 
> But of course richer theory can prove more theorem

ZF is intuitively true and it is powerful, although not powerful enough to
derive the Axiom of Choice; finding another set of axioms that are equally
intuitive but even more powerful is not easy. 

>> >>
>> Yes perfectly true, you need physical hardware. But my question is* WHY*?
>> The only answer can be that physical hardware has something that
>> "arithmetical truth" does not.
>
>
> > 
> Then you artificial brain is not Turing emulable, and computationalism is
> false.
>

The above statement does not compute. In other word the above statement is
bullshit.

> >> 
>> We may not be certain what that something is but the fact that computer
>> hardware companies have non zero manufacturing costs is proof that one
>> has something the other does not.
>
>
> > 
> No, because that relative cost exost also in arithmetic reelatively to the
> people emulated in arithmetic.
>

That is another statement that does not compute; emulated people have
access to arithmetic just like non emulated people, so regardless of if
they are emulated or not why do the employees at INTEL bother to use
silicon?  

>>>   
>>  I need to implement the universal machine in that hardware.
>
>
> 
>> >
>> > 
>> Y
>> es exactly you need to implement it, but to implement it mathematics
>> needs help, it needs physics!
>>
>
> > 
> We agree on this,
>

Good.

> but that is not a proof that hardware exist,
>

If "

W
e agree on this
" and if computations exist then physical hardware exists.

>
>> >>
>> 
>> Numbers ==> computations ==> dreams ===> physical reality ===> physical
>> computation ===> hardware company
>>
>
> 
> >> 
> OK, but the hardware company certainly has access to numbers so why
> doesn't INTEL just make calculations directly and forget about all that
> unnecessary and expensive messing around with silicon?
>
> > 
> Because if we want to share computations, we need to implement them
>

And you agreed above that physics is needed to do that, physics can do
something that arithmetic can not.

> > 
> in the first person plural reality that we share to begin with. but that
> reality is itself emerging from infinitely many computations in arithmetic
>

Then I was right, matter can do something arithmetic can't, a finite
amount of 

matter can embody a infinite amount of mathematics but a finite amount of
mathematics can not.

>> 
>> arithmetical truth
>>  is certainly lacking something that physics has.
>>
>
> > 
> That is a theorem in machine's theology.
>

No, that is a machine in theology's theorem. Hey... if words no longer
have any meaning I can arrange them in any sequence I want just like you
do.

> >
>> >>
>> 
>> like someone can emulate Einstein's brain
>>
>
> 
> >> 
> Then that emulation is Einstein.
>
>
> > 
> Better: that emulation makes it possible for Einstein to manifest itself.
>

You can call it "manifest" if you like or "implement" or "emulate" or
"simulate" but the fact remains that if you want anything to change
anything in any way you're going to need physics, there is no evidence
 that mathematics by itself can do a damn thing.

> >
>> >> 
>> 
>> making a course in GR without any understanding of GR.
>>
>
> 
> >> 
> Then Einstein didn't "understand" GR
>
> > 
> No, you confuse the level.
>

Like hell I do!

> 
The guy who manipulate the pages of the book can talk with Einstein, but it
does not become Einstein by emulating it!

I really REALLY hope I'm misunderstanding you and you're not refereeing to

S
earle
 and his imbecilic Chinese room. 

> > 
> ZF can prove that PA is consistent.
>

But can not prove that PA is complete and that's a good thing because if
it could then ZF would be inconsistent because there are true

Re: The Axiom Of Choice and ComputationalismT

2015-09-26 Thread Bruno Marchal

On 25 Sep 2015, at 19:16, John Clark wrote:

On Fri, Sep 25, 2015 at 11:36 AM, Bruno Marchal   
wrote:

 >>Paul Cohen not Godel proved that arithmetical reality is  
independent of the Axiom of Choice

> I don't think so. The independence of arithmetic from AC in  
ZF follows from Gödel's proof that V=L -> AC. A model of ZF where  
all sets are "constructible" (V = L) verifies the choice axiom, and  
that proves the consistency of AC.

The consistency of AC does not enter into it, the independence  
of AC from ZF does. Godel proved that if you assume that AC is true  
ZF will produce no contradictions, 25 years later Paul Cohen  
proved that if you assume AC is false ZF will STILL not produce any  
contradictions, and so AC must be independent of ZF and can not be  
derived from ZF.

Yes, but this has nothing to do with what I am saying. The fact that  
the arithmetical truth is independent of the choice axiom can be seen  
as a corollary of Gödel's proof that AC is consistent with ZF, in fact  
that V = L (all sets are constructible) is consistent with ZF.

 >> If physics is ZFC

> ZFC is a set theory.

I know.

> Physics is a theory about a possible physical reality

I know. So if physical reality is ZFC ( a big "if" I admit  
but it could be) then physical reality has something that  
arithmetic derived from just ZF does not have.

"physical reality is ZFC" means nothing to me.

And with computationalism, neither the physical reality, nor the  
psychological reality have to be described by arithmetic.

Anyway, I do not assume that there is a physical reality. I derived it  
from some coherence condition on the number-dream relations, which are  
provably emulated by a tiny part of arithmetic.

> Performing all computations is already done in the tiny  
sigma_1 part of the arithmetical truth,

And yet despite repeated requests you are unable or unwilling to  
explain why you can't start the Tiny Sigma_1 Computer  
Hardware Corporation and become the richest man on the planet.

Not at all. I think you don't read the answer.

The answer, I repeat again, is that to build an hardware corporation I  
need hardware and I need to implement the universal machine in that  
hardware.

But to have a computation running, I need only arithmetic.

Then I explain the existence of the illusion of hardware by the  
coherence of the computations existing in arithmetic.

Numbers ==> computations ==> dreams ===> physical reality ===>  
physical computation ===> hardware company

You can't explain it but I can, you can't do it because a physical  
silicone microprocessor chip has something that Robinson  
arithmetic and "the tiny sigma_1 part of the arithmetical  
truth" lacks.

Yes, that is right. But that things which is lacking is an illusion,  
entirely explained by the dreaming machine living in RA. Now such  
machines will believe in much more than what RA can prove. RA just  
emulate machines much more powerful than RA, like someone can emulate  
Einstein's brain making a course in GR without any understanding of GR.

I'm not sure exactly what it's lacking, maybe it's the Axiom of  
Choice and maybe it's something else,

It is the primitive matter which is lacking.

but it sure as hell is lacking SOMETHING because nobody has been  
able to start a computer hardware company with zero manufacturing  
costs.

But the hardware and the primitive matter are explained in RA, so we  
don't need to assume it, and with step 8 we know that even if we  
assume it, we cannot use it. A good thing because an explanation of  
matter which assumes matter is just circular.

Bruno

  John K Clark

--
You received this message because you are subscribed to the Google  
Groups "Everything List" group.
To unsubscribe from this group and stop receiving emails from it,  
send an email to everything-list+unsubscr...@googlegroups.com.

To post to this group, send email to everything-list@googlegroups.com.
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.

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

--
You received this message because you are subscribed to the Google Groups 
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email 
to everything-list+unsubscr...@googlegroups.com.
To post to this group, send email to everything-list@googlegroups.com.
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.

Re: The Axiom Of Choice and ComputationalismT

2015-09-26 Thread John Clark

On Sat, Sep 26, 2015  Bruno Marchal  wrote:

>
>>> >>
>>> Paul Cohen not Godel proved that arithmetical reality is independent of
>>> the
>>>  
>>> Axiom of Choice
>>
>>
>>
> > 
>> I don't think so. The independence of arithmetic from AC in ZF follows
>> from Gödel's proof that V=L -> AC. A model of ZF where all sets are
>> "constructible" (V = L) verifies the choice axiom
>>
> If a axiom has been verified, that is to say if it can be derived from
other axioms, then it no longer needs to be a axiom and is just the result
of more fundamental axioms.   
Paul Cohen
 proved that
AC can not be derived from ZF.

>>
>> Godel proved that if you assume that AC is true ZF will produce no
>> contradictions, 25 years later Paul Cohen
>>  proved that if you assume AC is false ZF will STILL not produce
>> any contradictions, and so AC must be independent of ZF and can not be
>> derived from ZF.
>>
>
> > 
> Yes, but this has nothing to do with what I am saying. The fact that the
> arithmetical truth is independent of the choice axiom
>

Then a lot of stuff that mathematicians think is true is not true, or at
least can't be proven to be part of "
arithmetical truth
"
 because the Axiom of Choice is needed to prove them. 

> > 
> can be seen as a corollary of Gödel's proof that AC is consistent with ZF,
>

Godel proved in 1938 that AC was consistent with ZF but for all Godel
knew The Axiom of Choice could be derived from Zermelo-Fraenkel; and that
is in fact what Godel believed at the time and what most mathematicians
thought,

even Paul Cohen thought so and was as surprised as anyone when he found in
1963 that the negation of AC was consistent with ZF too and thus
independent of ZF.

> >
>>> >>
>>> Physics is a theory about a possible physical reality
>>
>>
> 
>> >> 
>> I know. So if physical reality
>>  is ZFC ( a big "if" I admit but it could be) then physical reality has
>> something that arithmetic derived from just ZF does not have.
>>
> >> 
> "physical reality is ZFC" means nothing to me.
>

Physical reality is 
Zermelo-Fraenkel
 plus the Axiom of Choice,  
"
arithmetical truth
"
 is just
Zermelo-Fraenkel
. I'm not saying it's true, I'm just saying that's what it means; it might
be wrong but it's not gibberish.

> 
> Anyway, I do not assume that there is a physical reality.
>

Hmm. Margaret Fuller once said "I accept the universe" to which Thomas
Carlyle replied "Gad,
you
'd better".

Unlike you at leas Fuller accepted the universe,
 I wonder what Carlyle would say to you.

> >> 
>> And yet despite repeated requests you are unable or unwilling to explain
>> why you can't start the
>>  
>> Tiny
>>  
>> Sigma_1 Computer Hardware Corporation and become the richest man on the
>> planet.
>
>
> > 
> N
> 
> ot at all. I think you don't read the answer. The answer, I repeat again,
> is that to build an hardware corporation I need hardware
>

Yes perfectly true, you need physical hardware. But my question is* WHY*?
The only answer can be that physical hardware has something that
"arithmetical truth" does not. We may not be certain what that something is
but the fact that computer hardware companies have non zero manufacturing
costs is proof that one has something the other does not.

> > 
> and I need to implement the universal machine in that hardware.
>

Yes exactly you need to implement it, but to implement it mathematics
needs help, it needs physics!

> > 
> to have a computation running, I need only arithmetic.
>

But it is a fact that to have a successful company that provides answers
to arithmetical problems arithmetic is *NOT* all you need. 

> 
> Numbers ==> computations ==> dreams ===> physical reality ===> physical
> computation ===> hardware company
>

OK, but the hardware company certainly has access to numbers so why
doesn't INTEL just make calculations directly and forget about all that
unnecessary and expensive messing around with silicon?

> >> 
>> You can't explain it but I can, you can't do it because a physical
>> silicone microprocessor chip has something that Robinson arithmetic
>>  and
>>  "the tiny sigma_1 part of the arithmetical truth"
>>  
>> lacks.
>
>

> 
> Yes, that is right. But that things which is lacking is an illusion,
>

An Illusion is a perfectly respectable subjective phenomenon, and so is
consciousness; so you're saying that
subjective phenomenon
 is
 the thing that that matter that obeys the laws of physics can create that
arithmetical truth can not create.  Well maybe,
arithmetical truth
 is certainly lacking something that physics has.

> > 
> like someone can emulate Einstein's brain
>

Then that emulation is Einstein.

> > 
> making a course in GR without any understanding of GR.
>

Then Einstein didn't "understand" GR and like "God" and more recently
"theology" the word has lost all meaning. This destruction of words you're
engages in is getting

Re: The Axiom Of Choice and ComputationalismT

2015-09-25 Thread Bruno Marchal

On 24 Sep 2015, at 20:49, John Clark wrote:

On Thu, Sep 24, 2015 at  Bruno Marchal  wrote:

> You can define prime number in arithmetic,

Who cares? I'm not interested in arithmetic or in anything else  
defining prime numbers, I'm interested in CALCULATING prime numbers.

> That the arithmetical reality is independent of the axiom of  
choice has been proved by Gödel

Paul Cohen not Godel proved that arithmetical reality is independent  
of the Axiom of Choice,

I don't think so. The independence of arithmetic from AC in ZF follows  
from Gödel's proof that V=L -> AC. A model of ZF where all sets are  
"constructible" (V = L) verifies the choice axiom, and that proves the  
consistency of AC. It is not related to the proof of the consistency  
of the negation of the axiom of choice made by Cohen, using models of  
ZF in which V≠L.

Godel just proved it was consistent with it, Cohen proved its  
negation was consistent with it too.  And if arithmetical reality is  
independent of the axiom of choice then something that was dependent  
on BOTH arithmetical reality AND the Axiom of Choice would be  
different from just arithmetical reality, maybe something like  
physical reality.

Or set theoretical reality, or analysis, or arithmetic + "arithmetic  
is consistent". yes, the arithmetical truth is inexhaustible, ZF knows  
much more than PA, and ZF + kappa knows much more than ZF, etc.

> ZF and ZFC see exactly the same arithmetical reality,

In ZFC the Banach-Tarski construction is part of reality, in ZF  
it is not.

Yes, it like Euclid's parallel axioms.

  If physics is ZFC

ZFC is a set theory. Physics is a theory about a possible physical  
reality (primitive or not). You can't equate them.

then Banach-Tarski is physical reality even if it's not  
arithmetical reality and physics can do stuff that arithmetic can't;  
but we already knew that, arithmetical reality isn't sufficient to  
perform calculations.

Performing all computations is already done in the tiny sigma_1 part  
of the arithmetical truth, which is so big that no axiomatizable  
theory at all can proves its propositions.

>> Also if the the Axiom Of Choice is true then the Banach- 
Tarski construction (sometimes called paradox) can be done. If you  
cut up a solid sphere and then put all the pieces back together in a  
way specified by Banach and Tarski you can create TWO solid spheres  
of a size equal to the original single sphere. This can’t happen in  
the real physical world so does this fact work against my idea that  
Physics is arithmetic plus the Axiom Of Choice? Maybe not because  
maybe it does happen in the real physical world. We know from  
astronomical observation that space is expanding, new space is being  
created, and maybe Banach-Tarski is how physics does it.

> That seems quite speculative

It is, but not as speculative as the idea that the human  
biological brain needs dark matter to operate.

I used that as a counter-example.

Bruno

 John K Clark

--
You received this message because you are subscribed to the Google  
Groups "Everything List" group.
To unsubscribe from this group and stop receiving emails from it,  
send an email to everything-list+unsubscr...@googlegroups.com.

To post to this group, send email to everything-list@googlegroups.com.
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.

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

--
You received this message because you are subscribed to the Google Groups 
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email 
to everything-list+unsubscr...@googlegroups.com.
To post to this group, send email to everything-list@googlegroups.com.
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.

Re: The Axiom Of Choice and ComputationalismT

2015-09-25 Thread John Clark

On Fri, Sep 25, 2015 at 11:36 AM, Bruno Marchal  wrote:

>> >>
>> Paul Cohen not Godel proved that arithmetical reality is independent of
>> the
>>  
>> Axiom of Choice
>
>
>
> 
> I don't think so. The independence of arithmetic from AC in ZF follows
> from Gödel's proof that V=L -> AC. A model of ZF where all sets are
> "constructible" (V = L) verifies the choice axiom, and that proves the
> consistency of AC.
>

The consistency of AC does not enter into it, the independence of AC from
ZF does. Godel proved that if you assume that AC is true ZF will produce no
contradictions, 25 years later
Paul Cohen
 proved that if you assume AC is false ZF will STILL not produce
any contradictions, and so AC must be independent of ZF and can not be
derived from ZF.

>> >>
>> If physics is ZFC
>
>
> > 
> ZFC is a set theory.
>

I know.

> > 
> Physics is a theory about a possible physical reality
>

I know. So if 
physical reality
 is ZFC ( a big "if" I admit but it could be) then physical reality has
something that arithmetic derived from just ZF does not have.

> 
> Performing all computations is already done in the tiny sigma_1 part of
> the arithmetical truth,
>

And yet despite repeated requests you are unable or unwilling to explain
why you can't start the

Tiny

Sigma_1 Computer Hardware Corporation and become the richest man on the
planet. You can't explain it but I can, you can't do it because a physical
silicone microprocessor chip has something that Robinson arithmetic
 and
 "the tiny sigma_1 part of the arithmetical truth"

lacks. I'm not sure exactly what it's lacking, maybe it's the Axiom of
Choice and maybe it's something else, but it sure as hell is lacking
SOMETHING because nobody has been able to start a computer hardware company
with zero manufacturing costs.

  John K Clark

-- 
You received this message because you are subscribed to the Google Groups 
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email 
to everything-list+unsubscr...@googlegroups.com.
To post to this group, send email to everything-list@googlegroups.com.
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.

Re: The Axiom Of Choice and ComputationalismT

2015-09-24 Thread Bruno Marchal



On 23 Sep 2015, at 23:59, John Clark wrote:

It seems to me the debate I’v  been having with Bruno, the one  
about Arithmetic being able to perform calculations all by itself  
without the help of matter that obeys the laws of physics, comes  
down to the Axiom Of Choice. I would humbly propose that maybe just  
maybe mathematics is everything EXCEPT for the Axiom of Choice and  
physics is mathematics PLUS the Axiom of Choice If this is true  
then for something to be really real and not just sorta real physics  
must be able to calculate (choose) it.



The Axiom of Choice says that if you have an infinite number of bins  
with two or more different types of things in them then you can  
always create a new bin containing exactly one item from each bin.  
Bertrand Russell gave this example: “To choose one sock from each  
of infinitely many pairs of socks requires the Axiom of Choice, but  
for shoes the Axiom is not needed.” With shoes you could have a  
finite number of rules (just one in this case)  that would work,  
always pick the left shoe from each bin, but no corresponding finite  
number of rules exists for socks so you’d have to invoke the Axiom  
of Choice. This may have some relevance to the following question:  
If it exceeds the computational power of the entire universe to  
calculate (choose) does the 423rd prime number greater than  
10^100^100 really exist or only sorta exist?


To create a bin containing all the integers the Axiom of Choice is  
not needed, the 8 Zermelo-Fraenkel Axioms are enough; thus you could  
create a bin containing all the integers and only the integers  
{1,2,3,4...} , you can also create bins with {2,3,4,5...} and  
another with {3,4,5,6...} etc. A finite number of rules (just 8) can  
create such bins (sets) . But what about a bin that contains all the  
prime numbers and only the prime numbers?




You don't need the axiom of choice to, prove the existence of the set  
of prime numbers (and only prime numbers).



Without the Axiom of Choice there is no rule of finite length that  
would allow you to choose one and only one prime number from all the  
bins I listed above and use them to come up with a new bin  
containing all the prime numbers and nothing but the prime numbers.


You can define prime number in arithmetic, and PA can prove their  
existence. Everything is even computable. All recursive and  
recursively enumerable set are representable already in the theory of  
finite sets, or the theory of integers, etc.






Godel proved in 1938 that if you assume the Axiom of Choice is true  
then it will cause no contradictions in Zermelo-Fraenkel or in  
arithmetic, and Paul Cohen proved in 1963 that if you assume the the  
Axiom of Choice is false it will cause no contradictions in Zermelo- 
Fraenkel or in arithmetic. In other words the Axiom of Choice is  
independent of arithmetic and independent of the Zermelo-Fraenkel  
Axioms.



No. Independent of ZF.

That the arithmetical reality is independent of the axiom of choice  
has been proved by Gödel using his notion of constructible set. It  
shows that ZF and ZFC see exactly the same arithmetical reality, and  
so the axiom of choice has no role for proving new arithmetical  
theorem. This does not mean that the axiom of choice does not simplify  
the search of such truth, but in principle the use of the axiom of  
choice can be eliminated. That is a different result than the  
independence of AC from ZF (by Gödel and Cohen).







The Axiom of Choice has always been far more controversial than the  
8 Zermelo-Fraenkel Axioms, and mathematicians are reluctant to use  
it in their proofs unless they have to, in fact it’s almost as  
controversial as Euclid’s Fifth Postulate. As I’ve stated it the  
Axiom seems intuitively true, almost bland;




It is not an effective axioms. It is a highly non computational axiom,  
but then the axiom of infinity also. ZF is doin high level theology  
all the times, and has very strong belief.





but the trouble is that you can state the same thing in a different  
way that is absolutely equivalent but when stated that way it seems  
intuitively false. For example, the Axiom of Choice can also be  
stated as "every set can be well ordered” and that seems false;  
“well ordered” means it has a least element, it’s easy to see  
that the set of positive integers is well ordered but how would you  
well order the real numbers? Mathematicians think it’s ugly for the  
Axiom Of Choice to produce a set as if by magic with no instructions  
on how to actually build it.


OK. And Solovay proved that all set of real numbers is measurable, as  
a consequence of the axiom of choice.
We need the choice axiom also to prove the completeness of infinitary  
logics (but I avoid them usually).


Another equivalent version of the Choice axiom is: all linear spaces  
have a base.



Also if the the Axiom Of Choice is true then the Banach-Tarski  
construction (sometimes

Re: The Axiom Of Choice and ComputationalismT

2015-09-24 Thread John Clark

On Thu, Sep 24, 2015 at  Bruno Marchal  wrote:

> 
> You can define prime number in arithmetic,
>

Who cares? I'm not interested in arithmetic or in anything else defining
prime numbers, I'm interested in *CALCULATING *prime numbers.

> 
> That the arithmetical reality is independent of the axiom of choice has
> been proved by Gödel
>

Paul Cohen not Godel proved that arithmetical reality is independent of the
Axiom of Choice, Godel just proved it was consistent with it, Cohen proved
its negation was consistent with it too.  And if arithmetical reality is
independent of the axiom of choice then something that was dependent on
BOTH arithmetical reality AND the Axiom of Choice would be different from
just arithmetical reality, maybe something like physical reality.

> > 
> ZF and ZFC see exactly the same arithmetical reality,
>

In ZFC the Banach-Tarski construction is part of reality, in ZF it is
not.  If physics is ZFC then Banach-Tarski is physical reality even if
it's not arithmetical reality and physics can do stuff that arithmetic
can't; but we already knew that, arithmetical reality isn't sufficient to
perform calculations.

>> 
> Also if the the Axiom Of Choice is true then the Banach-Tarski
> construction (sometimes called paradox) can be done. If you cut up a solid
> sphere and then put all the pieces back together in a way specified by
> Banach and Tarski you can create TWO solid spheres of a size equal to the
> original single sphere. This can’t happen in the real physical world so
> does this fact work against my idea that Physics is arithmetic plus
> the Axiom Of Choice? Maybe not because maybe it does happen in the real
> physical world. We know from astronomical observation that space is
> expanding, new space is being created, and maybe Banach-Tarski is how
> physics does it.

> 
> That seems quite speculative
>

It is, but not as 
speculative
 as the idea that the human biological brain needs dark matter to operate.

 John K Clark

-- 
You received this message because you are subscribed to the Google Groups 
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email 
to everything-list+unsubscr...@googlegroups.com.
To post to this group, send email to everything-list@googlegroups.com.
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.

The Axiom Of Choice and ComputationalismT

2015-09-23 Thread John Clark

It seems to me the debate I’v  been having with Bruno, the one about
Arithmetic being able to perform calculations all by itself without the
help of matter that obeys the laws of physics, comes down to the Axiom Of
Choice. I would humbly propose that maybe just maybe mathematics is
everything EXCEPT for the Axiom of Choice and physics is mathematics PLUS the
Axiom of Choice If this is true then for something to be really real and
not just sorta real physics must be able to calculate (choose) it.

The Axiom of Choice says that if you have an infinite number of bins with
two or more different types of things in them then you can always create a
new bin containing exactly one item from each bin. Bertrand Russell gave
this example: “To choose one sock from each of infinitely many pairs
of socks requires the Axiom of Choice, but for shoes the Axiom is not
needed.” With shoes you could have a finite number of rules (just one in
this case)  that would work, always pick the left shoe from each bin, but
no corresponding finite number of rules exists for socks so you’d have to
invoke the Axiom of Choice. This may have some relevance to the following
question: If it exceeds the computational power of the entire universe to
calculate (choose) does the 423rd prime number greater than 10^100^100
really exist or only sorta exist?

To create a bin containing all the integers the Axiom of Choice is not
needed, the 8 Zermelo-Fraenkel Axioms are enough; thus you could create a
bin containing all the integers and only the integers {1,2,3,4...} , you
can also create bins with {2,3,4,5...} and another with {3,4,5,6...} etc. A
finite number of rules (just 8) can create such bins (sets) . But what
about a bin that contains all the prime numbers and only the prime numbers?
Without the Axiom of Choice there is no rule of finite length that would
allow you to choose one and only one prime number from all the bins I
listed above and use them to come up with a new bin containing all the
prime numbers and nothing but the prime numbers.

Godel proved in 1938 that if you assume the Axiom of Choice is true then it
will cause no contradictions in Zermelo-Fraenkel or in arithmetic, and Paul
Cohen proved in 1963 that if you assume the the Axiom of Choice is false it
will cause no contradictions in Zermelo-Fraenkel or in arithmetic. In other
words the Axiom of Choice is independent of arithmetic and independent of
the Zermelo-Fraenkel Axioms.

The Axiom of Choice has always been far more controversial than the 8
Zermelo-Fraenkel Axioms, and mathematicians are reluctant to use it in
their proofs unless they have to, in fact it’s almost as controversial as
Euclid’s Fifth Postulate. As I’ve stated it the Axiom seems intuitively
true, almost bland; but the trouble is that you can state the same thing in
a different way that is absolutely equivalent but when stated that way it
seems intuitively false. For example, the Axiom of Choice can also be
stated as "every set can be well ordered” and that seems false; “well
ordered” means it has a least element, it’s easy to see that the set of
positive integers is well ordered but how would you well order the real
numbers? Mathematicians think it’s ugly for the Axiom Of Choice to produce
a set as if by magic with no instructions on how to actually build it.

Also if the the Axiom Of Choice is true then the Banach-Tarski construction
(sometimes called paradox) can be done. If you cut up a solid sphere and
then put all the pieces back together in a way specified by Banach and
Tarski you can create TWO solid spheres of a size equal to the original
single sphere. This can’t happen in the real physical world so does this
fact work against my idea that Physics is arithmetic plus the Axiom Of
Choice? Maybe not because maybe it does happen in the real physical world.
We know from astronomical observation that space is expanding, new space is
being created, and maybe Banach-Tarski is how physics does it.

  John K Clark

-- 
You received this message because you are subscribed to the Google Groups 
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email 
to everything-list+unsubscr...@googlegroups.com.
To post to this group, send email to everything-list@googlegroups.com.
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.

Re: The Axiom Of Choice and ComputationalismT

Re: The Axiom Of Choice and ComputationalismT

Re: The Axiom Of Choice and ComputationalismT

Re: The Axiom Of Choice and ComputationalismT

Re: The Axiom Of Choice and ComputationalismT

Re: The Axiom Of Choice and ComputationalismT

Re: The Axiom Of Choice and ComputationalismT

Re: The Axiom Of Choice and ComputationalismT

The Axiom Of Choice and ComputationalismT

9 matches

Site Navigation

Mail list logo

Footer information