[FOM] The use of replacement in model theory
John Baldwin
jbaldwin at uic.edu
Thu Feb 4 09:51:32 EST 2010
On Sat, 30 Jan 2010, Harvey Friedman wrote:
>
> On Jan 28, 2010, at 5:23 PM, John Baldwin wrote:
>
Harvey quoted a couple of paragraphs from my post about a possible
essential use of
replacement in model theory that ended with a reference to
> An introductory account of this topic appears in the paper by Kim and
> Pillay: From stability to simplicity, Bulletin of Symbolic Logic,
> 4, (1998), 17-36.
Harvey continues:
>
> The definition of forking starts on page 20 of [Kim,Pillay]. However,
> the second sentence says "and we work in a saturated model C of T of
> cardinality kappa for some large kappa. It is sometimes convenient to
> assume that kappa is strongly inaccessible". So I read this as an
> indication that the definition is not "about countable sets of
> formulas".
My response attempting to explain that the definition is indeed about
countable sets of formulas was posted as:
http://cs.nyu.edu/pipermail/fom/2010-February/014370.html
I now address the general issue of `monster models'.
A pdf file of the rest of this note (with bibliography) is at:
http://www.math.uic.edu/~jbaldwin/pub/monster.pdf
Here is the text for those who want a quick look:
Contemporary model theorists often begin papers by assuming `we are
working in a saturated model of cardinality kappa for sufficiently large
kappa (a monster model). In every case I know such a declaration is not
intended to convey a reliance on the existence of large cardinals.
Rather, in Marker's phrase, it is declaration of laziness, `If the stakes
were high enough I could write down a ZFC proof'. As we note below, in
standard cases the author isn't being very lazy; but formalizing a
metatheorem expressing this intuition remains interesting.
The easiest way to find such a model is to choose kappa strongly
inaccessible, thereby extending ZFC. I know of no first order example
where this is necessary. In contrast there are uses of extensions of ZFC
in infinitary model theory but they are explicitly addressed and do not
arise through the monster model convention. In many cases the necessity of
the extension is an open problem.
The fundamental unit of study is a particular first order theory. The need
is for a monster model of the theory $T$. If $M$ is a $\kappa$ saturated
model of $T$, then every model $N$ of $T$ with cardinality at most
$\kappa$ is elementarily embedded in $M$ and every type over a set of size
$<\kappa$ is realized in $M$. So every configuration of size less than
$\kappa$ that could occur in any model of $T$ occurs in $M$.
Many use of this convention are to the study of $\omega$-stable countable
models (saturated models exist in every cardinal) or stable countable
theories (there is a saturated model in $\lambda$ if $\lambda^{\omega} =
\lambda$. So there is no difficulty finding a monster. As model theory
advanced to the detailed study of unstable theories, the choice of a
monster model became more delicate.
In fact, the requirement that the monster model be saturated in its own
cardinality is excessive.
A more refined version of the `monster model hypothesis' asserts: Any
first order model theoretic properties of sets of size less than kappa can
be proved in a $\kappa$-saturated strongly $\kappa$-homogenous model M
(any two isomorphic submodels of card less than $\kappa$ are conjugate by
an automorphism of M). Such a model exists(provably in ZFC) in some
$\kappa'$ not too much bigger than $\kappa$. See Hodges (big
models)\cite{Hodgesbook}
or or my new monograph on categoricity \cite{Baldwincatmon} for the
refined version. (Hodges's considition is ostensbibly stronger and
slightly more complicated to state; but existence is also provable in
ZFC.) Buechler \cite{Buechlerbook}, Shelah \cite{Shelahbook}
Marker\cite{Markerbook} expound harmless nature of the fully saturated
version. Ziegler \cite{Zieglerbasic} adopts a class approach that could
be formulated in G\"odel Bernays set theory.
Replacing for all $\kappa$ there exists $\kappa'$ by `there is one
monster' is just a convenient shorthand for saying we can repeat the same
proof for any given set of initial data.
There is of course a flaw in my description. What does `any model
theoretic property' mean?
It would be valuable to formalize this notion but it has seemed
unproblematic.
Recently, however, there has been a concrete example of a property where
finding the monster model is difficult.
Arising from problems is studying groups without the independence
property, Newelski (in a preprint)\cite{Newelskihanf} asked, what is the
Hanf number for the property:
Let $(T,T_1,p)$ be a
triple of two countable first order theories in vocabularies $\tau \subset
\tau_1$ and $p$ be a $\tau_1$-type over the empty set.
Specifically, Newelski asks, "What is the least cardinal kappa such that
if there is a model $N$ (of cardinality $\kappa$) of $T_1$ omitting p but
such that the reduct of $N$ to $\tau$ is saturated, then there are
arbitrarily large such models?"
(Newelski saw computing this Hanf number (depending on the cardinality of
$\tau_1$) as an issue of computing the cardinality of the `monster
model').
Baldwin and Shelah show the Hanf number for this property is the same as
the L\"owenheim number for second order logic\cite{BaldwinShelahnewhanf}.
That is, `as big as you want it be'.
http://www.math.uic.edu/~jbaldwin/pub/shnew8
This makes the meta-model theoretic problem more interesting. The
formulation and proof of a general metatheorem is analogous to but seems
much more tractable than the `universes issue' in number theory and
geometry.
John Baldwin
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