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


MoeBlee wrote:
> 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.

Well, sometimes you will be able to do that, sometimes you won't.

> 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.

Yes, that's true.

> 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,

Well, as we've seen, that's not a fully satisfactory way of doing it,
the models of your resulting theory won't be exactly the same thing as
the structures you want to talk about.

> and conversely, axiom
> schemata have to be "recast" as definitions that talk about subsets.
>

Again, you can do this if you want to, but it will change the class of
structures you are looking at.

> 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?
>

It depends on what way of recasting definitions you have in mind. As
we've seen, the ways of recasting definitions you tried definitely
don't work.

> 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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