[FOM] explicit construction; choice and model theory

[Stephen G Simpson]

Harvey Friedman in his posting of Tue, 1 Jul 2003 18:12:02 -0400
suggested that Borel model theory may be a dead subject in need of
revival.  I want to mention that one of my favorite model theorists,
James Schmerl, recently obtained some nice results in this subject.

  Theorem (Schmerl).  Every consistent countable theory has an
  $\aleph_0$-saturated model which is Borel.  In fact, the full
  satisfaction relation of the model is Borel.

  Special Case of the Theorem.  There is a Borel model of true
  arithmetic whose standard system (a.k.a., Scott system) is the
  entire powerset of the natural numbers.

  [ If M is a model of arithmetic, the standard system of M is defined
  to be the set of intersections of definable sets in M with the set
  of standard integers of M, identified with the natural numbers. ]

These results bear on the recent discussion of explicit constructions
in model theory.

The usual construction of saturated models (e.g., the construction
presented in the recent model theory textbook by David Marker) uses a
heavy dose of set-theoretic machinery, namely transfinite recursion
along an uncountable initial segment of the ordinal numbers, with the
Axiom of Choice at each transfinite stage.  This means that we cannot
expect the models so obtained to be explicitly definable.

By way of contrast, Schmerl's construction of $\aleph_0$-saturated
models is quite different.  It goes through in ZF set theory without
the Axiom of Choice, and it produces models which are explicitly
definable, indeed Borel.

Another nice methodological aspect of Schmerl's construction is that
it uses some interesting combinatorics, namely polarized partition
relations.

-- Stephen G. Simpson
Professor of Mathematics
Pennsylvania State University

P.S.

Here are some admittedly disingenuously naive questions, just to get
the ball rolling.

Why do some model theorists apparently think that model theory is
somehow independent of or free of set theory?  After all, even the
most basic definitions in model theory are overtly set-theoretical in
nature.  For example, a *model* is defined to be a *set* D, the domain
of the model, together with some relations on D, etc.  David Marker's
recent model theory textbook even has an appendix presenting the
set-theoretical background which is essential to understanding the
rest of the book.

Does anyone know of an alternative way to set up model theory using
some approach other than the familiar one, which is overtly
set-theoretical?

Do model theorists have some special reason for wanting to declare
their independence from set theory?  (I thought of this question
recently on July 4, the American holiday known as Independence Day.)