[FOM] Question about Set Theory as a formal basis for mathematics

[Harvey Friedman]

On 2/26/06 11:03 AM, "Andrea Proli" <aprol at tin.it> wrote:

> Harvey, by "stable" I meant "something that is not going to collapse on
> itself", ... on the one hand, Set Theory (say ZF, but I suppose it could
> be any first-order Set Theory) leans on Model Theory

I was trying to tell you that this is not true.

>because, as a  
> first-order theory, its semantics ("what is it about") is given by the
> family of all possible first-order structures ("states of affairs") in
> which their formulas are valid - i.e. the admissible arrangements of the
> world it describes;

I can now see how you might be led to say this IF the system of set theory
is based on one or more axiom schemes - like ZFC is. You have to know what
you mean by a first order formula.

However, one can instead use formulations of set theory that have only
finitely many axioms, of which there are plenty. Then you can merely read
each axiom one by one, without having to rely on predicate calculus. There
are plenty of good finitely axiomatized systems that do about the same job
as ZFC does for foundational purposes. They are of two kinds:

1. Theory of classes, like NBG. It is finitely axiomatizable. One can
actually write down the finitely many axioms, although it is messy. I know
how to modify this somewhat and make it prettier.
2. Substantial fragments of ZFC. For example, bounded Zermelo set theory.
This is also messy, but can be cleaned up somewhat.

>on the other hand, Model Theory leans on Set Theory
> because such descriptions, i.e. first-order structures (interpretations),
> are made up from a domain of discourse, which is a "set" of individuals,
> from a "set" of relations, and from a "set" of functions, relations and
> functions being in turn "set"s.

This point I addressed in my earlier reply.

I want to merely emphasize that the set theoretic foundations of mathematics
in no way shape or form rests on model theory.

Model theory comes in indirectly, when you want to make a claim that the
"logical part" of ZFC and other set theories is "best possible" or
"complete". However, set theoretic foundations does not rely on our ability
to obtain this marvelous completeness result (due to Godel).

> Aatu, thank you too for your explanation. What I ambiguously called the
> "semantics of sets" (I apologize for this ambiguity, Harvey) should
> actually have been "what variables in set theory are allowed to range
> over". Aatu, you say:

The variables in set theory range over all sets. The notion of sets is taken
as primitive. 

Perhaps you are not comfortable with the cumulative hierarchy of sets.

There are very good set theories for foundations, weaker than ZFC, where the
variables can be construed as ranging over a restricted notion of set that
many people are more comfortable with. Specifically, the sets that come up
when one iterates the power set operation starting with omega, any finite
number of times. The relevant set theory is that of ZC = Zermelo set theory
with the axiom of choice. An alternative is to think of omega as the set of
all natural numbers, where natural numbers are NOT viewed as sets, but
rather as objects left undefined (urelements).

Harvey Friedman