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From: Bruno Marchal <marchal.domain.name.hidden>

Date: Thu, 4 Jun 2009 20:31:30 +0200

On 04 Jun 2009, at 19:28, Brent Meeker wrote:

*>
*

*> Bruno Marchal wrote:
*

*>> Hi Ronald,
*

*>>
*

*>>
*

*>> On 02 Jun 2009, at 16:45, ronaldheld wrote:
*

*>>
*

*>>
*

*>>> Bruno:
*

*>>> Since I program in Fortran, I am uncertain how to interpret things.
*

*>>>
*

*>>
*

*>> I was alluding to old, and less old, disputes again programmers,
*

*>> about
*

*>> which programming language to prefer.
*

*>> It is a version of Church Thesis that all algorithm can be written in
*

*>> FORTRAN. But this does not mean that it is relevant to define an
*

*>> algorithm by a fortran program. I thought this was obvious, and I was
*

*>> using that "known" confusion to point on a similar confusion in Set
*

*>> Theory, like Langan can be said to perform.
*

*>>
*

*>> In Set Theorist, we still find often the error consisting in defining
*

*>> a mathematical object by a set. I have done that error in my youth.
*

*>> What you can do, indeed, is to *represent* (almost all) mathematical
*

*>> objects by sets. Langan seems to make that mistake.
*

*>>
*

*>> The point is just that we have to distinguish a mathematical object
*

*>> and the representation of that object in some mathematical theory.
*

*>>
*

*>
*

*> Just so I'm sure I understand you; do you mean that, for example, the
*

*> natural numbers exist in a way that is independent of Peano's axioms
*

Not just the existence of the natural numbers, all the true relations

are independent of the Peano Axioms, and of me, ZF, ZFC and you.

*> and
*

*> the theorems that can be proven from them.
*

A formal theory is just a machine which put a tiny light on those truth.

*> In other words you could add
*

*> to Peano's axioms something like Goldbach's conjecture and you would
*

*> still have the same mathematical object?
*

The whole point of logic is to consider the "Peano's axioms" as a

mathematical object itself, which is studied mathematically in the

usual informal (yet rigorous and typically mathematica) way.

PA, and PA+GOLDBACH are different mathematical objects. They are

different theories, or different machines.

Now if GOLDBACH is provable by PA, then PA and PA+GOLDBACH shed the

same light on the same arithmetical truth. In that case I will

identify PA and PA+GOLDBACH, in many contexts, because most of the

time I identify a theory with its set of theorems. Like I identify a

person with its set of (possible) beliefs.

If GOLDBACH is true, but not provable by PA, then PA and PA+GOLDBACH

still talk on the same reality, but PA+GOLDBACH will shed more light

on it, by proving more theorems on the numbers and numbers relations

than PA. I do no more identify them, and they have different set of

theorems.

If GOLDBACH is false. Well GOLBACH is PI_1, that is, its negation is

SIGMA_1, that is, it has the shape "it exist a number such that it

verify this decidable property". Indeed the negation of Goldbach

conjecture is "it exists a number bigger than 02 which is not the sum

of two primes". This, if true, is verifiable already by the much

weaker RA (Robinson arithmetic). So, if GOLDBACH is false PA +

GOLDBACH is inconsistent. That is a mathematical object quite

different from PA!

Here, you would have taken the twin primes conjecture, and things

would have been different, and more complex.

Note that a theory of set like ZF shed even much more large light on

arithmetical truth, (and is still incomplete on arithmetic, by

Gödel ...).

Incidentally it can be shown that ZF and ZFC, although they shed

different light on the mathematical truth in general, does shed

exactly the same light on arithmetical truth. They prove the same

arithmetical theorems. On the numbers, the axiom of choice add

nothing. This is quite unlike the ladder of infinity axioms.

I would say it is and will be particularly important to distinguish

chatting beings like RA, PA, ZF, ZFC, etc... and what those beings are

talking about.

Bruno

*>
*

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

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Received on Thu Jun 04 2009 - 20:31:30 PDT

Date: Thu, 4 Jun 2009 20:31:30 +0200

On 04 Jun 2009, at 19:28, Brent Meeker wrote:

Not just the existence of the natural numbers, all the true relations

are independent of the Peano Axioms, and of me, ZF, ZFC and you.

A formal theory is just a machine which put a tiny light on those truth.

The whole point of logic is to consider the "Peano's axioms" as a

mathematical object itself, which is studied mathematically in the

usual informal (yet rigorous and typically mathematica) way.

PA, and PA+GOLDBACH are different mathematical objects. They are

different theories, or different machines.

Now if GOLDBACH is provable by PA, then PA and PA+GOLDBACH shed the

same light on the same arithmetical truth. In that case I will

identify PA and PA+GOLDBACH, in many contexts, because most of the

time I identify a theory with its set of theorems. Like I identify a

person with its set of (possible) beliefs.

If GOLDBACH is true, but not provable by PA, then PA and PA+GOLDBACH

still talk on the same reality, but PA+GOLDBACH will shed more light

on it, by proving more theorems on the numbers and numbers relations

than PA. I do no more identify them, and they have different set of

theorems.

If GOLDBACH is false. Well GOLBACH is PI_1, that is, its negation is

SIGMA_1, that is, it has the shape "it exist a number such that it

verify this decidable property". Indeed the negation of Goldbach

conjecture is "it exists a number bigger than 02 which is not the sum

of two primes". This, if true, is verifiable already by the much

weaker RA (Robinson arithmetic). So, if GOLDBACH is false PA +

GOLDBACH is inconsistent. That is a mathematical object quite

different from PA!

Here, you would have taken the twin primes conjecture, and things

would have been different, and more complex.

Note that a theory of set like ZF shed even much more large light on

arithmetical truth, (and is still incomplete on arithmetic, by

Gödel ...).

Incidentally it can be shown that ZF and ZFC, although they shed

different light on the mathematical truth in general, does shed

exactly the same light on arithmetical truth. They prove the same

arithmetical theorems. On the numbers, the axiom of choice add

nothing. This is quite unlike the ladder of infinity axioms.

I would say it is and will be particularly important to distinguish

chatting beings like RA, PA, ZF, ZFC, etc... and what those beings are

talking about.

Bruno

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

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Received on Thu Jun 04 2009 - 20:31:30 PDT

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