[FOM] The use of replacement in model theory

[John Baldwin]

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