On 04 Jun 2009, at 21:23, Brent Meeker wrote:
>
> 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?
Nobody can answer this question in your place.
But if you believe that the principle of excluded middle can be
applied to closed arithmetical sentences, like 99,999% of the
mathematician, then you have to believe that the Goldbach conjecture
is either true or false.
Even intuitionist will admit that Goldabch conjecture is true or
false, given its Sigma_1 character. This means that, about the (true-
or-false) nature of GOLDBACH is doubtable only for an ultrafinitist.
BTW, Goldbach conjecture asserts that all female (even) numbers can be
written as a sum of two primes, except the number two. (I forget the
word "even" in my enunciation above!).
>
>>
>> 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?
Yes. And this entails that both PA+TPC and PA + (~TPC) could be
consistent, yet one of those theory has to be unsound, or if you
prefer has to enunciate false arithmetical statements (yet consistent
with PA).
"Sound" is relative to the usual understanding of the natural numbers
which is presupposed in any work in mathematical logic or computer
science, like it is presupposed in any part of any physical theory.
That usual meaning is taught in primary school without any trouble.
In model theory, this notion of soundness can be made more precise,
through the notion of standard model of PA for example, but this
presupposes, in the meta-theory, an understanding of that usual notion
of numbers.
Nobody doubts the consistency and soundness of the theories like RA
and PA. (Even Torgny, who fakes that he doubts them for a
philosophical purpose unrelated to our discussion, like he fakes to be
a faking zombie, etc. This is clear from older post by Torgny).
>
>
>>
>> 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?
I am not sure I understand what you mean. PA is an (immaterial)
machine, or a program if you want. I guess that, by PA+theorems, you
mean the set of theorems of PA. In some context we can identify PA and
PA+theorems, because the context makes things unambiguous. But
strictly speaking those are different mathematical object: PA is
finite (well, as I defined it usually), But PA+theorems is infinite.
Both talk about N, and both are different of N. Indeed PA is a finite
(or infinite in the usual first order presentation) set of axioms and
rules, PA+theorems is an infinite set of formula, and N is an infinite
set of numbers. That is very different. Of course both PA and PA
+theorems (your wording) talk really about the structure (N, +, x),
that is the set of numbers N together with its additive and
multiplicative structure, as studied in school.
It is important to distinguish a theory or a machine (usually a finite
object), with the set of statements proved by that theory or machine
(usually an infinite set).
And it is important to distinguish both of them with the semantical
content of those statements produce by that theory or machine. In
metamathematics (or mathematical logic) that "semantical content" will
itself be represented by a mathematical object (a model) in some other
theory (usually set theory, or category theory, or model theory).
With respect to the current thread on the seven step, this is of
course sort of advanced remarks. But mathematical logic is not an easy
subject. Many things which are not distinguished in the usual practice
of mathematics or physics are distinguished by logicians.
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Fri Jun 05 2009 - 15:18:27 PDT