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

David C. Ullrich wrote:

> it's awesomely obvious that "S" can have
> many different interpretations.

I understand that point very well. But I am unclear about a few things
regarding PA. In some instances I'm not sure what people mean when they
refer to 'first order PA'. Is the predicate 'is a number' always a
predicate of the language and do the axioms always include the common
clauses 'if n is a number'?

If there is a 1-place predicate 'is a number' in the language, and the
axioms include such clauses as 'if n is a number', then am I correct
that the predicate symbol 'is a number' can be assigned by a structure
to a proper subset of the domain? In other words, it might be that only
a proper subset of the domain of interpretation is such that 0 and the
successor operation provide for a Peano system. Right? So, for example,
as I mentioned in another post, there can be models of the theory in
which S is not 1-1 on the entire domain. Is that correct?

Another point I am unclear on is addition and multiplication. Can we
define these in Peano arithmetic? Don't we have to add axioms for them?
To use recursion to define them, we have to step out into a broader
theory such as set theory, right? Or we could take the addition
operation symbol and the multiplication operation symbol as primitive
and then add the usual four axioms for them (two axioms for each
symbol). Do I have that right?

So axiomatizations that include axioms for addition, multiplication, or
ordering are not PA, but rather they are axiomatizations of different
theories (ones that are incomplete in the sense that not all sentences
true in the standard model are provable from the axioms axioms). Am I
correct or am I missing something?

Also, in set theory we prove that any two Peano systems are isomorphic.
And, if I'm not mistaken, we prove that any two completely ordered
fields are isomorphic. Yet, Lowenheim-Skolem tells us that for the
axioms of Peano arithmetic there are non-isomorphic models, and for the
axioms for a complete ordered field, there are non-isomorphic models.
But I'm having a hard time picturing how to reconcile these facts. In
set theory they're isomorphic, but in the meta-theory for first-order
logic, they're not isomorphic. But the meta-theory for first-order
logic is, we can suppose, set theory. So I am having a hard time
picturing why things come out differently. What am I missing?

Those are a lot of questions, I know, so I appreciate whatever help
with any of them that anyone would lend.

Thanks,

MoeBlee

.

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