On 04 Jun 2009, at 19:28, Brent Meeker wrote:
> Bruno Marchal wrote:
>> Hi Ronald,
>> On 02 Jun 2009, at 16:45, ronaldheld wrote:
>>> Since I program in Fortran, I am uncertain how to interpret things.
>> I was alluding to old, and less old, disputes again programmers,
>> 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.
> 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
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 2 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
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
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