# [FOM] Rigorous Foundations of Model Theory?

Rupert McCallum Rupert.McCallum at acu.edu.au
Thu Oct 8 23:32:14 EDT 2009

Stephen Simpson's "Subsystems of Second Order Arithmetic" examines the proof of some model-theoretic results in RCA_0 and WKL_0.

-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of Monroe Eskew
Sent: Thursday, 8 October 2009 12:16 PM
To: Foundations of Mathematics
Subject: Re: [FOM] Rigorous Foundations of Model Theory?

If you have clear definitions for the basic notions of model theory,
that should suffice for a rigorous foundation.  The notions can be
defined within ZF(C) set theory, but other set theories can probably
make sense out of model theory too.

A standard treatment starts with a language L as a collection of
constant symbols, relation and function symbols of various finite
arity.  One also has fixed symbols for conjunction, negation,
quantifier, parentheses, variables.  One defines recursively what
counts as a term and a formula.  An L-model M is an ordered pair
(A,f), where A is the universe of the model, and f is a function
assigning (a) to each constant symbol, an element of A, (b) to each
n-ary relation symbol, a subset of A^n, (c) to each n-ary function
symbol, a function from A^n to A.  One then defines by induction on
complexity of formulas what it is for a model M to satisfy a formula.
To answer your question, if M assigns to the n-ary relation symbol R a
subset R' of A^n, then M satisfies R(a_1,...,a_n) iff (a_1,...,a_n) is
a member of R'.

In some set theories like Morse-Kelley, you can have proper-class
models.  There might even be a reasonable way to interpret model
theory within axiomatic category theory, for all I know.