[FOM] The use of replacement in model theory
John Baldwin
jbaldwin at uic.edu
Mon Feb 1 11:51:39 EST 2010
I reply below to Harvey's comment.
On Sat, 30 Jan 2010, Harvey Friedman wrote:
>
> On Jan 28, 2010, at 5:23 PM, John Baldwin wrote:
>
>> Byunghan Kim proved that for a simple first order theory non-forking
>> is
>> equivalent to
>> non-dividing. The notions of simple, non-forking, and non-dividing
>> are all
>> statements about countable sets of formulas. Nevertheless, the
>> argument
>> for the result employs Morley's technique for omitting types; that
>> is it
>> uses the Erdos-Rado theorem on all cardinals less than $
>> \beth_{\omega_1}$.
>>
>> Thus, a priori, this is a use of the replacement axiom for a result
>> whose
>> statement does not require replacement. (I use this more technical
>> example rather than the original Hanf number computation, precisely
>> for
>> this reason).
>>
>> Does any one know whether this use is essential?
>>
>> 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.
>
> 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".
I think this is an ill-founded conclusion. The conventions that Kim and
Pillay adopt are just that. I will address the `monster model' issue
again in a separate note. It is not germane to the current question.
For a statement and proof sketch of the Kim result which is directly
combinatorial see my paper : Forking and multiplicity in First Order
Theories
It is from 2001 so is about 30th among the papers available at
http://www.math.uic.edu/~jbaldwin/model.html
I did skip over some model theoretic folklore. The results are stated
for types over sets of arbitrary cardinality. But it is easy to see by
compactness and Lowenheim Skolem that p in S(A) forks over B iff the
restriction of p to a countable subset of A forks over a countable
subset of B (indeed finite sets). Conditions such as
stability and simplicitity for a theory T are statements about countable
sets of formulas without parameters. This is explicit in most model theory
books. (The definitive account for simple theories is Wagner's, Simple
Theories).
In particular consider the characterizations of simplicity: Fact 2.3 and
Proposition 2.11 a) of Kim Pillay. These
are
certainly theorems of ZFC and should use very little of replacement. My
question could be posed as: Show that `less replacement' is used for
these results than for the Kim result on forking iff dividing.
>
> Is there a theorem in countable mathematics in this area of model
> theory whose only known proof uses more than ZC? If so, I would
> appreciate a self contained indication in an FOM sposting of what the
> statement is in purely countable terms.
I think any statement of the result is too long for a fom posting. I have
tried to direct the reader to accessible statements.
>
> In the case of Hnaf numbers, which is NOT such an example, readers may
> wish to look at
>
> H. Friedman, On Existence Proofs of Hanf Numbers, J. of Symbolic
> Logic, Vol. 39, No. 2, (1974), pp. 318-324.
>
> Harvey Friedman
>
>
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