Re: Minimal set theory for model theory?

On Apr 24, 7:13 am, Bill Taylor <w.tay...@xxxxxxxxxxxxxxxxxxxxx>
wrote:
This query is rather vague. I'm hoping some expert might make
the question appropriately precise, and then answer it.

The query is prompted by the recent thread on Skolem's paradox,
but can be considered entirely independently of that.
It is a query that informally comes up here from time to time.

The query is not one that bothers me personally - I'm quite happy with
the situation; but I find myself at a slight loss when trying to
answer it,
when posed by someone else. So perhaps this is a pedagogical query,
really.
- - -

It concerns the matter of treating set theory formally and thoroughly,
in particular using model theory. The thing being, that model theory
itself relies quite a bit on set theortetical concepts,
although rather simpler ones than full set theory itself.

But learners are often bothered by the fact that they see us trying to
"define" or "explain" what sets are, formally, by using the details of
FOL
plus its model theory, which is itself reliant on a form of set
theory.
So they detect a circularity here, at least an informal one, and it is
a complaint that needs addressing.

Now I and others have long noted, that the amount of set theory
required
to do model theory is *very* limited compared with full set theory
itself.
So my query is - if one treats FOL, proof theory & model theory
*formally*,
(as one must in mathematical logic), then WHAT is the minimum amount
of simple set theory one can get away with? It presumably includes
simple ideas of finite unions and intersections, finite cross products
(to deal with multiple arities) and basic ideas of membership.
It presumably doesn't have to deal with power sets, cardinality
beyond the finite/infinite distinction, regularity, well-ordering,
replacement, or even very much separation.

It would be nice to know if this basic amount of set theory has itself
been made the topic of a formal (mathematical) theory. Has it?
In particular, what is the MINIMAL amount of formal set theory
required to support the basic concepts of a formal approach to FOL,
proof theory, and model theory.
- - -

I hope the question is clear enough to admit of some precise answer!

------------------------------------------------------------------------
Bill Taylor W.Tay...@xxxxxxxxxxxxxxxxxxxxx
------------------------------------------------------------------------
The math is done right, but is the right math done?
------------------------------------------------------------------------

A previous post hasn't appeared.

Well, take into account Orayen's paradox. It is caused by the fact
that we use FOL to formalize set theory and set theory to interpret
FOL formulae. The paradox shows that there is no standard model for
first order set theory and derives this from Cantor's paradox: set
theory is supposed to speak about any set, so that the universe of
discourse of the corresponding interpretation should be the set of all
sets, which does not exist. This we usually prove from Powerset and
Cantor's theorem.

So, contrary to what it might seem, model theory for FOL could require
quite a bit of set theory.

Regards

.

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