Re: What is the 1st order formal system known as PA?

On Fri, 18 Nov 2005 04:49:09 GMT, Nam Nguyen <namducnguyen@xxxxxxx>
wrote:

>A 1st order formal system T is, of course, defined by:
>
>i) 1st order language L(T),
>ii) logical axioms of L(T), plus some non-logical axioms,
>iii) some rules of inferences.
>
>[based on Shoenfield's "Mathematical Logic".]
>
>So what would make a system T unique from another, say, T'?
>Obviously either L(T) is different from L'(T'), or non-logical
>axioms of T are different than those of T', or both. Suppose though
>T and T' are of the same language L [i.e. L(T) = L(T')], then T and
>T' could be different iff at there is at least one non-logical
>axiom, say, A of T differs from one axiom A' of T' [assuming both
>T and T' have the same number of axioms].
>
>But when could 2 formulae A, A' of the same language become 2
>different axioms? There seem to be only 2 cases:
>
>(a) A and A' are syntactically different.
>(b) A and A' are syntactically _identical_, but the n-ary relation(s)
> stipulated in this one formula are different, from one being's
> point of view to another's.
>
>It's the case (b) that, imho, PA has an issue that has long been
>neglected. The issue is that from the theory, or FOL framework,
>level [and not necessarily from the model level], given any successor
>function S of PA, which is 1-1, one always could infer another
>_different_ successor function S' that one could use to define the
>binaries + and *. Therefore, when 2 [human?] beings refer to PA
>theory, they might in fact be talking about _2 different PA theories_.
>Or, shall we say 2 different PA-ish theories! [Note that all PA-ish
>theories are of the same language L(PA)]
>
>Two consequences seem to stem from such multi-PA-ish-theories
>observation:
>
>(1) what we normally consider as a (PA) theorem now should be a
> _common_ theorem: a theorem in all the PA-ish theories.
>(2) Given a particular formula F of L(PA), it's quite possible that
> one could not - within the framework of FOL - determine a
> particular PA-ish theory in which F is a decidable, or undecidable.
>(3) It's suspected that: if GC (Goldbach Conjecture) is false then
> it's provable in all PA-ish theories; otherwise one could not use
> FOL framework to determine which of the PA-ish theories GC is
> (un)decidable.
>
>Just an opinion though. And I'd like to thank in advance for any
>constructive correction, or comments.

The parts of this that make sense, and are not simple
misunderstandings on your part, have not been neglected,
they've very well known. To be more specific:

You evidently think that the function symbols in FOL
actually refer to specific n-ary relations. They do not.

They do not _until_ we start talking about an _interpretation_
of the language (a "model"). If we're talking about
interpretations of first-order PA then yes, "S" can have
many different interpretations. Nobody has every said
otherwise - it's awesomely obvious that "S" can have
many different interpretations.

>---Nam


************************

David C. Ullrich
.

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