Abstract
Positive solutions to the decision problem for a class of quantified formulae of the first order set theoretic language based on ϕ, ε, =, involving particular occurrences of restricted universal quantifiers and for the unquantified formulae of ϕ, ε, =, {...}, η, where {...} is the tuple operator and η is a general choice operator, are obtained. To that end a method is developed which also provides strong reflection principles over the hereditarily finite sets. As far as finite satisfiability is concerned such results apply also to the unquantified extention of ϕ, ε, =, {...}, η, obtained by adding the operators of binary union, intersection and difference and the relation of inclusion, provided no nested term involving η is allowed.
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References
Ackermann, W. (1937). ‘Die Wiederspruchsfreiheit der allgemeinen mengenkhere’, Mathematische Annalen 114.
AXL Newsletter no. 4.
BrebanM., FerroA., OmodeoE., and SchwartzJ. T. (1981). ‘Decision Procedures for elementary sublanguages of set theory II. Formulas involving restricted quantifiers together with ordinal, integer, map and domain notions’, Comm. Pure and Appl. Math. XXXIV/177, 177–195.
Ferro, A. (1988). ‘Decision procedures for elementary sublanguages of set theory XII. Multilevel syllogistic extended with singleton and choice operators’, this volume.
Omodeo, E. (1984). ‘Decidability and proof procedures for set theory with a choice operator’, Ph.D. Thesis, New York University.
Parlamento F. and Policriti A. (1988a). ‘Decision Procedures for elementary sublanguages of set theory IX. Unsolvability of the decision problem for a restricted subclass of the Δ0-formulas of set theory’, Comm. Pure Appl. Math. XLI/2, 221–251.
ParlamentoF. and PolicritiA. (1988b). ‘The logically simplest form of the infinity axiom’, Proc. AMS 103/1, 274–276.
Policriti, A. (1988). ‘The NP-completeness of MLS’, Research Report, Department of Mathematics and Computer Science, University of Udine.
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Parlamento, F., Policriti, A. Decision procedures for elementary sublanguages of set theory: XIII. Model graphs, reflection and decidability. J Autom Reasoning 7, 271–284 (1991). https://doi.org/10.1007/BF00243810
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DOI: https://doi.org/10.1007/BF00243810