Search results for "Zermelo–Fraenkel set theory"

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Towards Axiomatic Basis of Inductive Inference

2001

The language for the formulation of the interesting statements is, of course, most important. We use first order predicate logic. Our main achievement in this paper is an axiom system which we believe to be more powerful than any other natural general purpose discovery axiom system. We prove soundness of this axiom system in this paper. Additionally we prove that if we remove some of the requirements used in our axiom system, the system becomes not sound. We characterize the complexity of the quantifier prefix which guaranties provability of a true formula via our system. We prove also that if a true formula contains only monadic predicates, our axiom system is capable to prove this formula…

SoundnessDiscrete mathematicsPredicate logicSMorse–Kelley set theoryComputer scienceNon-well-founded set theoryZermelo–Fraenkel set theoryConstructive set theoryInductive reasoningAxiom schemaUrelementScott's trickMonad (functional programming)First-order logicAxiom of extensionalityMathematics::LogicTheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGESTheoryofComputation_LOGICSANDMEANINGSOFPROGRAMSCalculusAxiom of projective determinacyAxiom of choiceKripke–Platek set theoryAction axiomAxiom
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Finiteness in a Minimalist Foundation

2008

We analyze the concepts of finite set and finite subset from the perspective of a minimalist foundational theory which has recently been introduced by Maria Emilia Maietti and the second author. The main feature of that theory and, as a consequence, of our approach is compatibility with other foundational theories such as Zermelo-Fraenkel set theory, Martin-Lof's intuitionistic Type Theory, topos theory, Aczel's CZF, Coquand's Calculus of Constructions. This compatibility forces our arguments to be constructive in a strong sense: no use is made of powerful principles such as the axiom of choice, the power-set axiom, the law of the excluded middle.

General set theoryMorse–Kelley set theoryNon-well-founded set theoryZermelo–Fraenkel set theoryConstructive set theoryminimalist foundation; finite sets; finite subsets; type theory; constructive mathematicsconstructive mathematicsfinite subsetsUrelementMathematics::LogicType theorytype theoryComputer Science::Logic in Computer ScienceAxiom of choicefinite setsminimalist foundationMathematical economicsMathematics
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