Pointwise definable models of set theory

2016

A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V = HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are continuum many pointwise definable models of ZFC. If there is a transitive model of ZFC, then there are continuum many pointwise definable transitive models of ZFC. What is more, every countable model of ZFC has a class forcing extension that is pointwise definable. Indeed, for the main contribution of this article, every countable model of Gödel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters. 2000 Mathematics Subject Classification. 03E55. Key words and phrases. set theory, forcing. 1 See [Ani] for an instance of the argument at MathOverflow, which surely serves a brisk cup of math tea online. We leave aside the remark of Horatio, eight-year-old son of the first author, who announced, "Sure, papa, I can describe any number. Let me show you: tell me any number, and I'll tell you a description of it!"

Proceedings of The American Mathematical Society, 1988

Every countable model M of PA or ZFC, by a theorem of S. Simpson, has a "class" X which has the curious property: Every element of the expanded structure (M, X) is definable. Here we prove: THEOREM A. Every completion T of PA has a countable model M (indeed there are 2" many such M 's for each T) which is not pointwise definable and yet becomes pointwise definable upon adjoining any undefinable class X to M. THEOREM B. Let M 1= ZF + "V = HOD" be a well-founded model of any cardinality. There exists an undefinable class X such that the definable points of M and (M, X) coincide. THEOREM C. Let M t= PA or ZF +"V = HOD". There exists an undefinable class X such that the definable points of M and (M, X) coincide if one of the conditions below is satisfied. (A) The definable elements o/M are cofinal in M. (B) M is recursively saturated and cf (M) = uj.

Archive for Mathematical Logic, 2005

A DO model (here also referred to a Paris model) is a model of set theory all of whose ordinals are first order definable in . Jeffrey Paris (1973) initiated the study of DO models and showed that (1) every consistent extension T of ZF has a DO model, and (2) for complete extensions T, T has a unique DO model up to isomorphism iff T proves V=OD. Here we provide a comprehensive treatment of Paris models. Our results include the following: 1. If T is a consistent completion of ZF+V≠OD, then T has continuum-many countable nonisomorphic Paris models. 2. Every countable model of ZFC has a Paris generic extension. 3. If there is an uncountable well-founded model of ZFC, then for every infinite cardinal κ there is a Paris model of ZF of cardinality κ which has a nontrivial automorphism. 4. For a model ZF, is a prime model ⇒ is a Paris model and satisfies AC⇒ is a minimal model. Moreover, Neither implication reverses assuming Con(ZF).

We show that the analogues of the Hamkins embedding theorems [Ham13], proved for the countable models of set theory, do not hold when extended to the uncountable realm of ω 1 -like models of set theory. Specifically, under the ♦ hypothesis and suitable consistency assumptions, we show that there is a family of 2 ω 1 many ω 1 -like models of ZFC, all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive ω 1 -like model of ZFC that does not embed into its own constructible universe; and there can be an ω 1 -like model of PA whose structure of hereditarily finite sets is not universal for the ω 1 -like models of set theory.

Notre Dame Journal of Formal Logic, 1983

数理解析研究所講究録, 2014

Mathematics

In this paper we prove that for any m≥1 there exists a generic extension of L, the constructible universe, in which it is true that the set of all constructible reals (here subsets of ω) is equal to the set D1m of all reals definable by a parameter free type-theoretic formula with types bounded by m, and hence the Tarski ‘definability of definable’ sentence D1m∈D2m (even in the form D1m∈D21) holds for this particular m. This solves an old problem of Alfred Tarski (1948). Our methods, based on the almost-disjoint forcing of Jensen and Solovay, are significant modifications and further development of the methods presented in our two previous papers in this Journal.

Proceedings of the American Mathematical Society, 1980

We determine when a model M \mathfrak {M} of ZF can be expanded to a model ⟨ M , X ⟩ \langle \mathfrak {M},\mathfrak {X}\rangle of a weak extension of Gödel Bernays: GB + {\text {GB}} + the Δ 1 1 \Delta _1^1 comprehension axiom. For nonstandard M \mathfrak {M} , the ordinal of the standard part of M \mathfrak {M} must equal the inductive closure ordinal of M \mathfrak {M} , and M \mathfrak {M} must satisfy the axioms of ZF with replacement and separation for formulas involving predicates for all hyperelementary relations on M \mathfrak {M} . We also consider expansions to models of GB + Σ 1 1 {\text {GB}} + \Sigma _1^1 choice, observe that the results actually apply to more general theories of well-founded relations, and observe relationships to expansibility to models of other second order theories.

2021

We investigate systems of transitive models of ZFC which are elementarily embeddable into each other and the influence of definability properties on such systems. One of Ken Kunen’s best-known and most striking results is that there is no elementary embedding of the universe of sets into itself other than the identity. Kunen’s result is best understood in a theory that includes proper classes as genuine objects, such as von Neumann-Gödel-Bernays set theory (NBG), which we take in this article as our background theory. The argument proceeds by assuming there is a nontrivial j : V → V and using Replacement and Comprehension for formulas involving the parameter j, arriving at a contradiction. It does not exclude the possibility of the existence of an inner model M of Zermelo-Fraenkel set theory (ZFC) and a nontrivial elementary j : M → M , where M is not the class of sets of an NBG-universe that includes the class j. Indeed, Kunen’s early work on 0 showed an equivalence between the non...

Mathematical Notes, 2017

We make use of a finite support product of Jensen forcing to define a model in which there is a countable non-empty Π 1 2 set X of reals containing no ordinal-definable real. 1 * Revised version. The revision includes an updated proof of Lemma 4.5 (the densitypreservation lemma for the product).