[bibtex key=”GroszekHamkins2019:The-implicitly-constructible-universe”]
Abstract. We answer several questions posed by Hamkins and Leahy concerning the implicitly constructible universe
[bibtex key=”GroszekHamkins2019:The-implicitly-constructible-universe”]
Abstract. We answer several questions posed by Hamkins and Leahy concerning the implicitly constructible universe
[bibtex key=FuchsGitmanHamkins2018:EhrenfeuchtsLemmaInSetTheory]
This is joint work with Gunter Fuchs and Victoria Gitman.
Abstract. Ehrenfeucht’s lemma asserts that whenever one element of a model of Peano arithmetic is definable from another, then they satisfy different types. We consider here the analogue of Ehrenfeucht’s lemma for models of set theory. The original argument applies directly to the ordinal-definable elements of any model of set theory, and in particular, Ehrenfeucht’s lemma holds fully for models of set theory satisfying
[bibtex key=HamkinsLeahy2016:AlgebraicityAndImplicitDefinabilityInSetTheory]
We aim in this article to analyze the effect of replacing several natural uses of definability in set theory by the weaker model-theoretic notion of algebraicity and its companion concept of implicit definability. In place of the class HOD of hereditarily ordinal definable sets, for example, we consider the class HOA of hereditarily ordinal-algebraic sets. In place of the pointwise definable models of set theory, we examine its (pointwise) algebraic models. And in place of Gödel’s constructible universe L, obtained by iterating the definable power set operation, we introduce the implicitly constructible universe Imp, obtained by iterating the algebraic or implicitly definable power set operation. In each case we investigate how the change from definability to algebraicity affects the nature of the resulting concept. We are especially intrigued by Imp, for it is a new canonical inner model of ZF whose subtler properties are just now coming to light. Open questions about Imp abound.
Before proceeding further, let us review the basic definability definitions. In the model theory of first-order logic, an element
In the second-order context, a subset or class
The main theorems of the paper are:
Theorem. The class of hereditarily ordinal algebraic sets is the same as the class of hereditarily ordinal definable sets:
Theorem. Every pointwise algebraic model of ZF is a pointwise definable model of ZFC+V=HOD.
In the latter part of the paper, we introduce what we view as the natural algebraic analogue of the constructible universe, namely, the implicitly constructible universe, denoted Imp, and built as follows:
Theorem. Imp is an inner model of ZF with
Theorem. It is relatively consistent with ZFC that
Theorem. In any set-forcing extension
Open questions about Imp abound. Can
This article arose from a question posed on MathOverflow by my co-author Cole Leahy and our subsequent engagement with it.
This is a talk May 10, 2013 for the CUNY Set Theory Seminar.
Abstract. An element a is definable in a model M if it is the unique object in M satisfying some first-order property. It is algebraic, in contrast, if it is amongst at most finitely many objects satisfying some first-order property φ, that is, if { b | M satisfies φ[b] } is a finite set containing a. In this talk, I aim to consider the situation that arises when one replaces the use of definability in several parts of set theory with the weaker concept of algebraicity. For example, in place of the class HOD of all hereditarily ordinal-definable sets, I should like to consider the class HOA of all hereditarily ordinal algebraic sets. How do these two classes relate? In place of the study of pointwise definable models of set theory, I should like to consider the pointwise algebraic models of set theory. Are these the same? In place of the constructible universe L, I should like to consider the inner model arising from iterating the algebraic (or implicit) power set operation rather than the definable power set operation. The result is a highly interesting new inner model of ZFC, denoted Imp, whose properties are only now coming to light. Is Imp the same as L? Is it absolute? I shall answer all these questions at the talk, but many others remain open.
This is joint work with Cole Leahy (MIT).