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Substandard models of finite set theory

Laurence Kirby

Mathematical Logic Quarterly 56 (6):631-642 (2010)

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Every countable model of set theory embeds into its own constructible universe.Joel David Hamkins - 2013 - Journal of Mathematical Logic 13 (2):1350006.details The main theorem of this article is that every countable model of set theory 〈M, ∈M〉, including every well-founded model, is isomorphic to a submodel of its own constructible universe 〈LM, ∈M〉 by means of an embedding j : M → LM. It follows from the proof that the countable models of set theory are linearly pre-ordered by embeddability: if 〈M, ∈M〉 and 〈N, ∈N〉 are countable models of set theory, then either M is isomorphic to a submodel of N (...) Download Export citation Bookmark 8 citations
Ordinal operations on graph representations of sets.Laurence Kirby - 2013 - Mathematical Logic Quarterly 59 (1-2):19-26.details Any set x is uniquely specified by the graph of the membership relation on the set obtained by adjoining x to the transitive closure of x. Thus any operation on sets can be looked at as an operation on these graphs. We look at the operations of ordinal arithmetic of sets in this light. This turns out to be simplest for a modified ordinal arithmetic based on the Zermelo ordinals, instead of the usual von Neumann ordinals. In this arithmetic, addition (...) Download Export citation Bookmark 3 citations

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