Interpreting extended set theory in classical set theory
We exhibit an interpretation of the Extended Set Theory proposed by Dave Childs in classical Zermelo-Fraenkel set theory with the axiom of choice and an axiom asserting the existence of arbitrarily large inaccessible cardinals. In particular, if the existence of arbitrarily large inaccessible cardinals is consistent with ZFC, then Childs's Extended Set Theory is also consistent.
Zero-one laws: Thesauri and parametric conditions, joint with Yuri Gurevich (Bull. European Assoc. Theoret. Comp. Sci. 91 (2007) 125-144)
The 0-1 law for first-order properties of finite structures and its proof via extension axioms were first obtained in the context of arbitrary finite structures for a fixed finite vocabulary. But it was soon observed that the result and the proof continue to work for structures subject to certain restrictions. Examples include undirected graphs, tournaments, and pure simplicial complexes. We discuss two ways of formalizing these extensions, Oberschelp's (1982) parametric conditions and our (2003) thesauri. We show that, if we restrict thesauri by requiring their probability distributions to be uniform, then they and parametric conditions are equivalent. Nevertheless, some situations admit more natural descriptions in terms of thesauri, and the thesaurus point of view suggests some possible extensions of the theory.
Some Questions Arising from Hindman's Theorem
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We present, with some background material, four open questions connected with Neil Hindman's Finite Unions Theorem. Two questions are in set theory and the other two are in computability theory.
This paper is a contribution to the ceremony awarding the Japan Association for Mathematical Sciences International Prize for 2003 to Neil Hindman.
Resource Consciousness in Classical Logic (in Games, Logic, and Constructive Sets, Proceedings of LLC9, the 9th conference on Logic, Language, and Computation, held at CSLI (ed. G. Mints and R. Muskens) (2003) 61-74)
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Using Herbrand's Theorem, we define simple Herbrand validity, a sort of resource consciousness that makes sense in classical predicate logic. We characterize the propositional formulas all of whose first-order instances are simply Herbrand valid. The characterization turns out to coincide with a known characterization of game semantical validity for multiplicative formulas.
A Faithful Modal Interpretation of Propositional Ontology (Math. Japonica 40 (1994) 217--223)
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The propositional ontology introduced by Ishimoto is essentially the first-order universal fragment of Lesniewski's ontology. We give a faithful interpretation of this system in the modal logic K.