End Extension Research Articles

Decidability and invariant classes for degree structures

We present a decision procedure for the ∀ ∃ \forall \exists -theory of D [ 0 , 0 ′ ] \mathcal {D}[{\mathbf {0}},\,{\mathbf {0}}’ ] , the Turing degrees below 0 ′ {\mathbf {0}}’ . The two main ingredients are a new extension of embeddings result and a strengthening of the initial segments results below 0 ′ {\mathbf {0}}’ of [Le1]. First, given any finite subuppersemilattice U U of D [ 0 , 0 ′ ] \mathcal {D}[{\mathbf {0}},\,{\mathbf {0}}’ ] with top element 0 ′ {\mathbf {0}}’ and an isomorphism type V V of a poset extending U U consistently with its structure as an usl such that V V and U U have the same top element and V V is an end extension of U − { 0 ′ } U - \{ {\mathbf {0}}’ \} , we construct an extension of U U inside D [ 0 , 0 ′ ] \mathcal {D}[{\mathbf {0}},\,{\mathbf {0}}’ ] isomorphic to V V . Second, we obtain an initial segment W W of D [ 0 , 0 ′ ] \mathcal {D}[{\mathbf {0}},\,{\mathbf {0}}’ ] which is isomorphic to U − { 0 ′ } U - \{ {\mathbf {0}}’ \} such that W ∪ { 0 ′ } W \cup \{ {\mathbf {0}}’ \} is a subusl of D \mathcal {D} . The decision procedure follows easily from these results. As a corollary to the ∀ ∃ \forall \exists -decision procedure for D \mathcal {D} , we show that no degree a > 0 {\mathbf {a}} > {\mathbf {0}} is definable by any ∃ ∀ \exists \forall -formula of degree theory. As a start on restricting the formulas which could possibly define the various jump classes we classify the generalized jump classes which are invariant for any ∀ \forall or ∃ \exists -formula. The analysis again uses the decision procedure for the ∀ ∃ \forall \exists -theory of D \mathcal {D} . A similar analysis is carried out for the high/low hierarchy using the decision procedure for the ∀ ∃ \forall \exists -theory of D [ 0 , 0 ′ ] \mathcal {D}[{\mathbf {0}},\,{\mathbf {0}}’ ] . (A jump class C \mathcal {C} is σ \sigma -invariant if σ ( a ) \sigma ({\mathbf {a}}) holds for every a {\mathbf {a}} in C \mathcal {C} .)

Conservative extensions of models of set theory and generalizations

An attempt to answer the following question gave rise to the results of the present paper. Let be an arbitrary model of set theory. Does there exist an elementary extension of satisfying the two requirements: (1) contains an ordinal exceeding all the ordinals of ; (2) does not enlarge any (hyper) integer of ? Note that a trivial application of the ordinary compactness theorem produces a model satisfying condition (1); and an internal ultrapower modulo an internal ultrafilter produces a model satisfying condition (2) (but not (1), because of the axiom of replacement). Also, such a satisfying both conditions (1) and (2) exists if the external cofinality of the ordinals of is countable, since by [KM], would then have an elementary end extension.Using a class of models constructed by M. Rubin using in [RS], and already employed in [E1], we prove that our question in general has a negative answer (see Theorem 2.3). This result generalizes the results of M. Kaufmann and the author (appearing respectively in [Ka] and [E1]) concerning models of set theory with no elementary end extensions.In the course of the proof it was necessary to establish that all conservative extensions (see Definition 2.1) of models of ZF must be cofinal. This is in direct contrast with the case of Peano arithmetic where all conservative extensions are end extensional (as observed by Phillips in [Ph1]). This led the author to introduce two useful weakenings of the notion of a conservative end extension which, as shown by the “completeness” theorems in §3, can exist.

Weakly compact cardinals in models of set theory

The central notion of this paper is that of a κ-elementary end extension of a model of set theory. A model is said to be a κ-elementary end extension of a model of set theory if > and κ, which is a cardinal of , is end extended in the passage from to , i.e., enlarges κ without enlarging any of its members (see §0 for more detail). This notion was implicitly introduced by Scott in [Sco] and further studied by Keisler and Morley in [KM], Hutchinson in [H] and recently by the author in [E]. It is not hard to see that if has a κ-elementary end extension then κ must be regular in . Keisler and Morley [KM] noticed that this has a converse if is countable, i.e., if κ is a regular cardinal of a countable model then has a κ-elementary end extension. Later Hutchinson [H] refined this result by constructing κ-elementary end extensions 1 and 2 of an arbitrary countable model in which κ is a regular uncountable cardinal, such that 1 adds a least new element to κ while 2 adds no least new ordinal to κ. It is a folklore fact of model theory that the Keisler-Morley result gives soft and short proofs of countable compactness and abstract completeness (i.e. recursive enumera-bility of validities) of the logic L(Q), studied extensively in Keisler's [K2]; and Hutchinson's refinement does the same for stationary logic L(aa), studied by Barwise et al. in [BKM]. The proof of Keisler-Morley and that of Hutchinson make essential use of the countability of since they both rely on the Henkin-Orey omitting types theorem. As pointed out in [E, Theorem 2.12], one can prove these theorems using “generic” ultrapowers just utilizing the assumption of countability of the -power set of κ. The following result, appearing as Theorem 2.14 in [E], links the notion of κ-elementary end extension to that of measurability of κ. The proof using (b) is due to Matti Rubin.

Blunt and topless end extensions of models of set theory

AbstractLet be a well-founded model of ZFC whose class of ordinals has uncountable cofinality, such that has a Σn end extension for each n ∈ ω. It is shown in Theorem 1.1 that there is such a model which has no elementary end extension. In the process some interesting facts about topless end extensions (those with no least new ordinal) are uncovered, for example Theorem 2.1: If is a well-founded model of ZFC, such that has uncountable cofinality and has a topless Σ3 end extension, then has a topless elementary end extension and also a well-founded elementary end extension, and contains ordinals which are (in ) highly hyperinaccessible. In §3 related results are proved for κ-like models (κ any regular cardinal) which need not be well founded. As an application a soft proof is given of a theorem of Schmerl on the model-theoretic relation κ → λ. (The author has been informed that Silver had earlier, independently, found a similar unpublished proof of that theorem.) Also, a simpler proof is given of (a generalization of) a characterization by Keisler and Silver of the class of well-founded models which have a Σn end extension for each n ∈ ω. The case κ = ω1 is investigated more deeply in §4, where the problem solved by Theorem 1.1 is considered for non-well-founded models. In Theorems 4.1 and 4.4, ω1-like models of ZFC are constructed which have a Σn end extension for all n ∈ ω but have no elementary end extension. ω1-like models of ZFC which have no Σ3 end extension are produced in Theorem 4.2. The proof uses a notion of satisfaction class, which is also applied in the proof of Theorem 4.6: No model of ZFC has a definable end extension which satisfies ZFC. Finally, Theorem 5.1 generalizes results of Keisler and Morley, and Hutchinson, by asserting that every model of ZFC of countable cofinality has a topless elementary end extension. This contrasts with the rest of the paper, which shows that for well-founded models of uncountable cofinality and for κ-like models with κ regular, topless end extensions are much rarer than blunt end extensions.

Definable Ultrafilters and end Extension of Constructible Sets

Mathematical Logic QuarterlyVolume 28, Issue 27-32 p. 395-412 Article Definable Ultrafilters and end Extension of Constructible Sets Evangelos Kranakis, Evangelos Kranakis West Lafayette, Indiana (U.S.A.)Search for more papers by this author Evangelos Kranakis, Evangelos Kranakis West Lafayette, Indiana (U.S.A.)Search for more papers by this author First published: 1982 https://doi.org/10.1002/malq.19820282703Citations: 3AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume28, Issue27-321982Pages 395-412 RelatedInformation

End extensions and numbers of countable models

AbstractWe prove that every model of T = Th(ω, <,…) (T countable) has an end extension; and that every countable theory with an infinite order and Skolem functions has nonisomorphic countable models; and that if every model of T has an end extension, then every ∣T∣-universal model of T has an end extension definable with parameters.

Models Without Indiscernibles

For T any completion of Peano Arithmetic and for n any positive integer, there is a model of T of size 2, with no (n + 1)-length sequence of indiscernibles. Hence the Hanf number for omitting over T, H(T), is at least 2, (Now, using an upper bound previously obtained by Julia Knight H (true arithmetic) is exactly 2b,.) If TX true arithmetic, then H(T) = 2,. If 8 -A (p)'-, then any completion of Peano Arithmetic has a model of size 8 with no set of indiscernibles of size p. There are similar results for theories strongly resembling Peano Arithmetic, e.g., ZF + V = L. ?0. Introduction. The main accomplishment of the research presented here is the lifting of the Specker-MacDowell-Gaifman technology for over arithmetic to a technique for building models from carefully constructed n-types. In Specker and MacDowell [1959] it is shown that every model of Peano Arithmetic, (PA), has a proper elementary extension. Gaifman [1965], [1968] shows how to construct end extension 1-types which give rise to such extensions, investigates the iteration of such extension techniques, and gives a construction of independent extension types which is the direct forebear of the generic types we use here. It is implicit in these papers (and, in fact, is a special case of the M(I) in Gaifman [1976]) that a model of PA generated by 21-many mutually generic will have no two distinct elements with the same 1-type.3 Here we give a direct exposition of such a model, in a way that clearly suggests how to construct a model of PA with no (n + 1)-length sequence of indiscernibles. Carrying through this suggestion involves far more complicated combinatorics than in the case of 2i. What is needed is a certain generalization of finitary Ramsey's Theorem proved independently by Neset'ril and Rodl [1976] and by the present authors. Our original exposition also involved some complicated machinery specially designed for our particular results. We owe a large debt to Haim Gaifman who pointed out some structural properties of our models which are of interest in their own right and which can be used directly to obtain our results. The approach of the present paper is that suggested to us by Gaifman, and we will clearly indicate the specific results which are due to him. Received April 19, 1976; revised September 12, 1977. University of Wisconsin-Milwaukee. This author's research was supported, in part, by an American Mathematical Society Postdoctoral Research Fellowship and by NSF Grant Number

A model of peano arithmetic with no elementary end extension

AbstractWe construct a model of Peano arithmetic in an uncountable language which has no elementary end extension. This answers a question of Gaifman and contrasts with the well-known theorem of MacDowell and Specker which states that every model of Peano arithmetic in a countable language has an elementary end extension. The construction employs forcing in a nonstandard model.

End extensions, conservative extensions, and the Rudin-Frolík ordering

The ordering of ultrafilters on the natural numbers defined by “E-prod N is an end extension of D-prod N,” the ordering defined by “E-prod N is a conservative extension of D-prod N,” and the Rudin-Frolik ordering are proved to be distinct if the continuum hypothesis holds. These three orderings are also characterized in terms of (not necessarily internal) ultrafilters in the Boolean algebra of internal sets of natural numbers in a nonstandard universe.

On models of arithmetic—Answers to two problems raised by H. Gaifman

In a recent paper [3] H. Gaifman investigated some model theoretic consequences of Matijasevič's theorem [5], and posed some further problems which naturally arise. We provide here partial answers to two of these problems, the results having been previously announced in the postscript of [3].Firstly, it is shown in [3] that if M1 and M2 are models of the Peano axioms P and M1 ⊆ M2, then M1 is closed under the recursive functions of M2. The converse of this statement is false. Moreover, Gaifman asks: Is every initial segment of a model M of P which is closed under the recursive functions of M (or the ∑n-definable functions) also a model of P? We show that this is false and our method gives, en route, another proof of a theorem of Rabin [7] stating the P is not implied by any consistent set of ∑n sentences for any n.Secondly, we partially answer a question posed on p. 129 of [3] by proving (some-what more than) every countable nonstandard model of P has an end extension in which a diophantine equation, not solvable in the original model, has a solution. We can, in fact, take the new model to be isomorphic to the original one. This generalises (apart from the countability restriction) a theorem of Rabin [6].