Bruno Marchal wrote:
> ...
>> Bruno Marchal wrote:
>
> 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!
So what then is the status of the natural numbers? Are there many
different objects in Platonia which we loosely refer to as "the natural
numbers" or is there only one such object and the Goldbach conjecture is
either true of false of this object?
>
> Here, you would have taken the twin primes conjecture, and things
> would have been different, and more complex.
Because, even if it is false, it cannot be proven false by exhibiting an
example?
>
> 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
Do you mean PA talks about the natural numbers but PA+theorems is a
different mathematical object than N?
Brent
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Received on Thu Jun 04 2009 - 12:23:29 PDT