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2016 Algebraicity and Implicit Definability in Set Theory
Joel David Hamkins, Cole Leahy
Notre Dame J. Formal Logic 57(3): 431-439 (2016). DOI: 10.1215/00294527-3542326


We analyze the effect of replacing several natural uses of definability in set theory by the weaker model-theoretic notion of algebraicity. We find, for example, that the class of hereditarily ordinal algebraic sets is the same as the class of hereditarily ordinal definable sets; that is, HOA=HOD. Moreover, we show that every (pointwise) algebraic model of ZF is actually pointwise definable. Finally, we consider the implicitly constructible universe Imp—an algebraic analogue of the constructible universe—which is obtained by iteratively adding not only the sets that are definable over what has been built so far, but also those that are algebraic (or, equivalently, implicitly definable) over the existing structure. While we know that Imp can differ from L, the subtler properties of this new inner model are just now coming to light. Many questions remain open.


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Joel David Hamkins. Cole Leahy. "Algebraicity and Implicit Definability in Set Theory." Notre Dame J. Formal Logic 57 (3) 431 - 439, 2016.


Received: 27 February 2012; Accepted: 30 December 2013; Published: 2016
First available in Project Euclid: 20 April 2016

zbMATH: 06621300
MathSciNet: MR3521491
Digital Object Identifier: 10.1215/00294527-3542326

Primary: 03E47
Secondary: 03C55

Rights: Copyright © 2016 University of Notre Dame


Vol.57 • No. 3 • 2016
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