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1 results

  • Truth In V for ∃*∀∀-Sentences is Decidable

  • D. Bellé, F. Parlamento
  • Journal:
    The Journal of Symbolic Logic / Volume 71 / Issue 4 / December 2006
    Published online by Cambridge University Press:
    12 March 2014, pp. 1200-1222
    Print publication:
    December 2006

  • Let V be the cumulative set theoretic hierarchy, generated from the empty set by taking powers at successor stages and unions at limit stages and. following [2], let the primitive language of set theory be the first order language which contains binary symbols for equality and membership only. Despite the existence of ∀∀-formulae in the primitive language, with two free variables, which are satisfiable in ∀ but not by finite sets ([5]). and therefore of ∃∃∀∀ sentences of the same language, which are undecidable in ZFC without the Axiom of Infinity, truth in V for ∃*∀∀-sentences of the primitive language, is decidable ([1]). Completeness of ZF with respect to such sentences follows.