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

Rupert wrote:

> Yes, but on this occasion you don't need to talk about that theory.

> Yes, but I don't see why we need to refer to a metatheory for set
> theory on this occasion.

In the object theory, I'm looking at isomorphisms among structures
(such as algebraic structures, Peano systems, fields, etc.). In the
meta-theory (in which is conducted model theory), I'm looking at
isomorphisms among models.

Then we can take a definition of a certain kind of (algebraic or other)
structure, and "recast" it as axioms for a theory in a different
language from set theory but with all its predicates and operation
symbols definable in set theory. Then we can look at the models for
that theory to see how they fare for isomorphism and homomorphism. And
vice versa: For axiomatizations of certain theories in certain
languages, we can define their predicates and operation symbols in set
theory and "recast" the axioms as a definition of a certain kind of
structure. Then we can look to see whether all structures so defined
are isomorphic.

This is complicated by the fact that definitions that talk about
subsets have to be "recast" as axiom schemata, and conversely, axiom
schemata have to be "recast" as definitions that talk about subsets.

I'm just wondering what kind of generalizations we can make about this.
For example, if all the algebraic structures of a certain kind are
isomorphic, then what else is required to ensure that the theory of the
axioms (from "recasting" definitions into axioms) is categorical, or
categorical within cardinalities?

I'm confident that the answers I'm looking for will come to me once
I've made more progress in my reading. But for now, does what I've
described make sense?

Thanks,

MoeBlee

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