Minimal elementary extensions of models of set theory and arithmetic*. Ali Enayat. Department of Mathematics and Statistics, The American University, ...
Arch. Math. Logic (1990) 30:181-192
Archive for
Mathematical
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9 Springer-Verlag 1990
Minimal elementary extensions of models of set theory and arithmetic* Ali Enayat Department of Mathematics and Statistics, The American University, Washington, DC20016, USA Received August 31, 1989/in revised form August 8, 1990
Abstract. We discuss minimal elementary extensions of models of set theory and contrast the behavior of models of set theory and arithmetic as regarding such extensions. Our main result, proved using a Boolean ultrapower argument, is: Theorem. Every model of ZFC has a conservative elementary extension which possesses a cofinal minimal elementary extension. An application of Boolean ultrapowers to models of full arithmetic is also presented.
1 Introduction In this paper we continue the comparative study of the behavior of models of set theory and those of Peano arithmetic, initiated in [E-3] and continued in I-E-4]. We do not, however, assume familiarity with the cited papers. Our context here is that of minimal elementary extensions. Recall that an elementary extension ~3 of a model 9.I is said to be minimal if there is no structure ~ such that 9I ~(~ < ~. Phillips and Gaifman (independently) showed that the McDowell-Specker method [MS] for building end extensions of models of PA can be refined so as to incorporate minimality and conservativity
Theorem 1.1 ([Ph-2], [G]). Every model of PA has a minimal conservative elementary end extension. is a conservative extension of 9~ iff the intersection of any parametrically definable subset of ~ with 91 is parametrically definable in 91. This notion, which also plays a key role at the "other end" of Model theory known as stability theory, has a central role in the model theory of Peano Arithmetic. It is not hard to see that * The results of this paper were presented at the spring meeting of the Association for Symbolic Logic held at Pennsylvania State University during April 7 and 8, 1990
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conservative elementary extensions of models of PA are end extensions. However, the situation is different for models of set theory, as witnessed by the following theorem: Theorem 1.2 ([E-5]). Suppose ~3 is a conservative elementary extension of a model 92( of ZF. (a) ~B is not an end extension. (b) I f 91 satisfies Dependent Choice and ~ fixes ~o~ then 91 is C O F I N A L in ~. Note that Theorem 1.2 above is the corrected version of a stronger, yet false, theorem of the author (Theorem 2.2 of [E-3]), which claimed that all conservative elementary extensions of models of ZF are cofinal. Here we wish to put the emphasis on the notion of minimality of elementary extensions. It is known that countable models of Z F C have minimal cofinal elementary extensions (see the proof of Theorem 2.12 of [E-2]) and that if the model also has a definable well-ordering, i.e., it satisfies "3x(V = HOD(x))", then one is guaranteed an end extension which is a minimal elementary extension as well (see [Kn, Theorem 2.3]). In light of Kunen's result [Ku] concerning the consistency of Z F C +"there is no selective ultrafilter", the countability assumption turns out to be crucial, by the following theorem, first announced in FE-1]: Theorem 1.3. Let ~cdenote the first strongly inaccessible cardinal. The model (R~, ~) has a minimal elementary extension iff there exists a selective ultrafilter on ~o. Proof. Recall that an ultrafilter q / o n co is selective iff every function f : c o ~ is constant or one-to-one on a member of og. It is routine to show that the selectivity of q/implies the minimality of the ultrapower (R~, &/~//over (R~, e). To prove the other direction, assume that 9Jr is a minimal elementary extension of (R~, 5). By a result of Keisler and Silver [KS] 93/is not an end extension, therefore implying that 99l enlarges some ordinal ~. Let ~o be the first such, and note that ~0 must be a cardinal. Next, choose a new member c of C~oand consider the collection q/defined as
{x
9 pc X}.
q/is easily seen to be a ~0-complete ultrafilter over ~o, therefore implying that ~o is measurable. But then ~o has to be the only measurable cardinal below ~, i.e., ~0 ~--~-(D.
Let us now see why q/is a selective ultrafilter. Suppose not and choose g such that g: co--co and g is neither one-to-one, nor onto, on any member of q/. N o w look at a new ultrafilter ~ defined by
To reach our contradiction we claim that: (i)
9Jr ~ (R~, e)~/q/,
(ii)
(R~, 5)-~ (R~, ~)~/~-~ (R~, ~)~/q/.
and (i) is easy to check using the minimality of ~ by noting that the map [-f]~u ~ f(c)
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is an elementary embedding. The crucial idea behind (ii) is that the map
j:(R~, ~)~/~ ~(R~, e)~/~ defined by j ( [ f ] ~ ) = [ f o g]~ i s a well-defined elementary embedding which does not include the J//-equivalence class of the identity function in its range. This can be verified by a simple calculation, hence allowing us to proclaim the end of the proof. [] The following theorem acts as a prelude to our main result (Theorem 2.4) and is proved by means of some very general model theoretic arguments. We are presenting it here to clarify the effect of the conservativity clause in Theorem 2.4, whose proof seems to make an essential use of a Boolean ultrapower argument. Theorem 1.4. Every model 9/ of Z F C has an elementary extension 9.I* which has a
minimal cofinal elementary extension. Furthermore, if 9/.Ihas a definable global wellordering then 9/* also has a minimal elementary end extension. Proof. Let 9/* be a resplendent elementary extension of 9/.1.To see that 9/* has a minimal elementary extension we first wish to embed 9/* in a model of Z F C in which it is a countable model. To this end, consider the following theory T in the language of set theory e, a constant symbol for each member of 9/*, and a new constant symbol E: T = Z F C + { E ~ q): q) is a sentence of Th(9/*, a),~.} + " ~ is countable". Using the Lowenheim-Skolem theorem it is straightforward to see that Tis finitely satisfiable and hence has a model 93l (note that we also need the assumption that ZF is consistent to show the consistency of T, but if ZF is inconsistent then the theorem is trivially true!). This shows that 9/* can be elementary extended to a model E which is a countable model in some universe of set theory. But usual coding tricks allow one to write a Z~-sentence a such that for all models ~3, "~3 is a countable model in some model of set theory" iff ~ ~ a. Since 9/* is resplendent, 9/* also satisfies a and can without loss of generality be identified with ~. Now since 9/* is countable in g)t, we have g)l~"9/* has a cofinal minimal elementary extension". This is because Z F C proves "every countable model of Z F C has a cofinal minimal elementary extension". To finish the proof, we will show that 9/* has a cofinal minimal elementary extension in the real world by noting that: if ~ is a cofinal minimal elementary extension of 9/* then there is a proper initial segment I of 9/*, and an ultrafilter r on N(I)~*= the collection of subsets of I coded in 9/*, such that
(l)
~3 is isomorphic to the ultrapower (9/,)i/og
and
(2)
Every function from I to 9/* which is coded in 9/* is constant or one to one on a member of ~//.
Here I is the initial segment of 9/.1"consisting of elements which do not get enlarged in the passage from 9/* to ~3, and the ultrapower (9/,)i/q/ consists of the
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equivalence classes of functions from I in to 9.:[* which are coded in 9.:[6*. The definition of (9.:[.)i is absolute for all models of set theory containing I and otherwise does not depend on the model of set theory in which 9.:[* is viewed from. Moreover, condition (2) above concerning q/which is initially granted to us by the model 991is clearly an absolute one also. Therefore we can conclude that ~ is also a minimal elementary extension of 9.:[* in the real world. If 9.:[ happens to have a definable global well-ordering then 9.:[* will have a minimal elementary end extension ~3 from the point of view of 93/. Although ~ is clearly an elementary end extension of 96" in the real world, it is not at all clear that ~3 retains its minimality in the passage from 931to the real world. To overcome this obstacle we first claim that 9.:[* has a blunt elementary end extension ~, i.e., an elementary extension ~B which has an ordinal 7 such that 7 =min(Ord03)-9.:[). Note that the countability of a model of set theory is not a sufficient condition for the existence of a blunt elementary end extension of that model. For instance, it is well-known that Cohen's minimal model does not have a blunt elementary end extension. However, resplendent models of Z F C do have blunt elementary end extensions, since if 9.:[* is resplendent then in particular it is recursively saturated and therefore the following type S(~), which by the reflection theorem is finitely consistent, is realized in 9.:[*: S(~) = "~ is an ordinal"+ {"(R~, e)standard, Y has to be co~-complete, hence implying that the cardinality of ~ is at least as large as the first uncountable measurable cardinal). With this in mind, we define the standard part of ~2~, denoted s t ( ~ ) , to be the model obtained from (2~ by restricting the signature to the signature of the standard model ~. Note the important fact that:
st (O~) = Q . / g . To see that (2~/~ has a nonconservative elementary end extension we first note that ~ , in the eyes of 91, is an co-standard model of PA of countable signature. This in turn implies that 9l "believes" that there is a nonconservative elementary end extension 9~=(A, ...) of (2~. This is a consequence of the following theorem of ZFC, which easily follows from the ordinary compactness theorem of first order logic: (*)
Given any countable family ~4__c~(~o) there is a model ~i which is a nonconservative elementary end extension of
(co, +,., X ) x ~ (proof: choose any Y=
