Ordinal Definability Research Articles

Strongly compact cardinals and ordinal definability

This paper explores several topics related to Woodin’s HOD conjecture. We improve the large cardinal hypothesis of Woodin’s HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. We prove that the HOD hypothesis holds if and only if every regular cardinal above the first strongly compact cardinal carries an ordinal definable [Formula: see text]-Jónsson algebra. We show that if the HOD hypothesis holds and HOD satisfies the Ultrapower Axiom, then every supercompact cardinal is supercompact in HOD.

A Generic Model in Which the Russell-Nontypical Sets Satisfy ZFC Strictly between HOD and the Universe

The notion of ordinal definability and the related notions of ordinal definable sets (class OD) and hereditarily ordinal definable sets (class HOD) belong to the key concepts of modern set theory. Recent studies have discovered more general types of sets, still based on the notion of ordinal definability, but in a more blurry way. In particular, Tzouvaras has recently introduced the notion of sets nontypical in the Russell sense, so that a set x is nontypical if it belongs to a countable ordinal definable set. Tzouvaras demonstrated that the class HNT of all hereditarily nontypical sets satisfies all axioms of ZF and satisfies HOD⊆HNT. In view of this, Tzouvaras proposed a problem—to find out whether the class HNT can be separated from HOD by the strict inclusion HOD⫋HNT, and whether it can also be separated from the universe V of all sets by the strict inclusion HNT⫋V, in suitable set theoretic models. Solving this problem, a generic extension L[a,x] of the Gödel-constructible universe L, by two reals a,x, is presented in this paper, in which the relation L=HOD⫋L[a]=HNT⫋L[a,x]=V is fulfilled, so that HNT is a model of ZFC strictly between HOD and the universe. Our result proves that the class HNT is really a new rich class of sets, which does not necessarily coincide with either the well-known class HOD or the whole universe V. This opens new possibilities in the ongoing study of the consistency and independence problems in modern set theory.

Ordinal definability and combinatorics of equivalence relations

Assume [Formula: see text]. Let [Formula: see text] be a [Formula: see text] equivalence relation coded in [Formula: see text]. [Formula: see text] has an ordinal definable equivalence class without any ordinal definable elements if and only if [Formula: see text] is unpinned. [Formula: see text] proves [Formula: see text]-class section uniformization when [Formula: see text] is a [Formula: see text] equivalence relation on [Formula: see text] which is pinned in every transitive model of [Formula: see text] containing the real which codes [Formula: see text]: Suppose [Formula: see text] is a relation on [Formula: see text] such that each section [Formula: see text] is an [Formula: see text]-class, then there is a function [Formula: see text] such that for all [Formula: see text], [Formula: see text]. [Formula: see text] proves that [Formula: see text] is Jónsson whenever [Formula: see text] is an ordinal: For every function [Formula: see text], there is an [Formula: see text] with [Formula: see text] in bijection with [Formula: see text] and [Formula: see text].

John R. Steel and W. Hugh Woodin, HOD as a core model, Ordinal Definability and Recursion Theory: The Cabal Seminar, vol. III (A. S. Kechris, B. Löwe, and J. R. Steel, editors), Lecture Notes in Logic 43, Association for Symbolic Logic and Cambridge University Press, 2016, pp. 257–343.

John R. Steel and W. Hugh Woodin, <i>HOD as a core model</i>, Ordinal Definability and Recursion Theory: The Cabal Seminar, <b><i>vol. III</i></b> (A. S. Kechris, B. Löwe, and J. R. Steel, editors), Lecture Notes in Logic 43, Association for Symbolic Logic and Cambridge University Press, 2016, pp. 257–343.

Kenneth McAloon. On the sequence of models HODn. Fundamenta mathematicae, vol. 82 (1974), pp. 85–93. - Thomas J. Jech. Forcing with trees and ordinal definability. Annals of mathematical logic, vol. 7 no. 4 (1975), pp. 387–409. - Włodzimierz Zadrożny. Iterating ordinal definability. Annals of pure and applied logic, vol. 24 (1983), pp. 263–310.

Kenneth McAloon. On the sequence of models HOD_n. Fundamenta mathematicae, vol. 82 (1974), pp. 85–93. - Thomas J. Jech. Forcing with trees and ordinal definability. Annals of mathematical logic, vol. 7 no. 4 (1975), pp. 387–409. - Włodzimierz Zadrożny. Iterating ordinal definability. Annals of pure and applied logic, vol. 24 (1983), pp. 263–310.

Ordinal definability in Jensen's model

In [J] R. Jensen proved that any model of ZFC can be generically extended to a model of ZFC + (∃a ⊆ ω)(V = L[a]). This extension was made via a class P of forcing conditions. One can ask what can be said about the class HOD in such extensions. In particular one can ask whether V = HOD holds in Jensen's model. A full answer to this question is given by the following assertion:Theorem. Let N = L[a] be given by forcing with Jensen's conditions (the classP).Then:1. N ⊨ V ≠ HOD,2. N ⊨ (∀n &lt; ω)(HODn + 1 ⊈ HODω).3. N ⊨ (∀α ≥ ω)(HODα = HODω), or equivalently HODω + 1 = HODω.The result is established by investigating homogeneity properties of certain complete Boolean algebras that are associated with the partial orderings of forcing conditions for coding using almost disjoint sets.We explain shortly the first of our two technical results: It is rather well known that if we have a subset then by performing threefold almost disjoint coding we come to a set a0 ⊆ ω such that ; cf. [JS]. This is achieved by iterating over almost disjoint forcing . That means we first code by a subset , of [ω1ω2) using the almost disjoint forcing conditions . Then we code by aω, contained in [ω, ω1), using forcing with over .

Iterating ordinal definability

The paper presents a uniform way of obtaining by forcing descending sequences of the iterations of HOD and describes their structure. Among other things the following results are proved: Theorem. Any model M of ZFC can be obtained by the transfinite iteration of HOD in some generic extension N of M, i.e. M=(HOD On) N. Theorem (ZFC). For any ordinal α there is a complete Boolean algebra B with a decreasing sequence of derivatives of length α. Moreover some other results about ordinal definability are reproved using the introduced method. This method is a refinement of the classical method of coding introduced by McAloon in [11].

Cardinal collapsing and ordinal definability

We shall describe Boolean extensions of models of set theory with the axiom of choice in which cardinals are collapsed by mappings definable from parameters in the ground model. In particular, starting from the constructible universe, we get Boolean extensions in which constructible cardinals are collapsed by ordinal definable sets.Let be a transitive model of set theory with the axiom of choice. Definability of sets in the generic extensions of is closely related to the automorphisms of the corresponding Boolean algebra. In particular, if G is an -generic ultrafilter on a rigid complete Boolean algebra C, then every set in [G] is definable from parameters in . Hence if B is a complete Boolean algebra containing a set of forcing conditions to collapse some cardinals in , it suffices to construct a rigid complete Boolean algebra C, in which B is completely embedded. If G is as above, then [G] satisfies “every set is -definable” and the inner model [G ∩ B] contains the collapsing mapping determined by B. To complete the result, it is necessary to give some conditions under which every cardinal from [G ∩ B] remains a cardinal in [G].The absolutness is granted for every cardinal at least as large as the saturation of C. To keep the upper cardinals absolute, it often suffices to construct C with the same saturation as B. It was shown in [6] that this is always possible, namely, that every Boolean algebra can be completely embedded in a rigid complete Boolean algebra with the same saturation.

Forcing with trees and ordinal definability

Forcing with trees and ordinal definability

Ordinal definability in the rank hierarchy

Myhill and Scott [ 15 ] have characterized the ordinal definable sets as those definable without parameters in some level R(a) of the rank hierarchy. In the same paper they showed how the class HOD of hereditarily ordinal definable sets could be used to simplify GSdel's proof of the consistency of the axiom of choice with Zermelo-Fraenkel set theory. Subsequently the class HOD has been applied by--L6vy [ 10] to obtain refinement of Cohen's independence proof for the axiom of choice, and more recently by Solovay [22] in establishing the consistency of ZF+ Every set of reals is Lebesgue measurable relative to ZF+ There is an inaccessible cardinal. In addition, the concept of ordinal definability was apparently explored by both Post and GSdel many years prior to its exposition in [ 15 ], and was later rediscovered independently by Vop~nka. Despite this pattern of repeated discovery and application to important consistency questions, the ordinal definable sets have never been systematically investigated. Indeed, the only published work in this area, aside from [ 15 ], appears to be that of McAloon [ 12 ] devoted to consistency questions concerning the inclusions L c_ HOD ~ V. Yet in view of the significance of Jensen's recent investigations [7] of the fine structure of L, it might be hoped that knowledge of the fine structure of the ordinal definable sets could be similarly productive. With this in mind, we here study the relation a is definable without parameters in (R(3'), ~) between ordinals ,~, 3' in models of set theory.

Consistency results about ordinal definability

Consistency results about ordinal definability