A theory of sets with the negation of the axiom of infinity

Baratella, Stefano; Ferro, Ruggero

doi:10.1002/malq.19930390138

Cited by 11 publications

(14 citation statements)

References 5 publications

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“…It is not hard to see that with this interpretation, the induction scheme is provable within EST (see [BF,Sec. 1]), thereby allowing EST to interpret PA. (h) For a model M EST, and x ∈ M, we say that x is N -finite if there is a bijection in M between x and some element of N. Let: (i) τ (n, x) is the term expressing "the n-th approximation to the transitive closure of {x} (where n is a natural number)".…”

Section: Preliminariesmentioning

confidence: 89%

“…It is easy to see that EST proves Fin N → Fin D → ¬ Infinity. By a theorem of Vopȇnka , Power set and the well-ordering theorem (and therefore the axiom of choice) are provable within EST + Fin N (see [BF,Theorem 5] for an exposition). (c) Kunen [BF,Sec.…”

Section: Preliminariesmentioning

confidence: 99%

“…By a theorem of Vopȇnka , Power set and the well-ordering theorem (and therefore the axiom of choice) are provable within EST + Fin N (see [BF,Theorem 5] for an exposition). (c) Kunen [BF,Sec. 7] has shown the consistency of the theory EST + ¬Infinity + ¬Fin N using the Fraenkel-Mostowski permutations method.…”

Section: Preliminariesmentioning

confidence: 99%

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ω-models of finite set theory

Enayat¹,

Schmerl²,

Visser³

2011

Set Theory, Arithmetic, and Foundations of Mathematics

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Abstract. Finite set theory, here denoted ZF fin , is the theory obtained by replacing the axiom of infinity by its negation in the usual axiomatization of ZF (Zermelo-Fraenkel set theory). An ω-model of ZF fin is a model in which every set has at most finitely many elements (as viewed externally). Mancini and Zambella (2001) employed the Bernays-Rieger method of permutations to construct a recursive ω-model of ZF fin that is nonstandard (i.e., not isomorphic to the hereditarily finite sets V ω ). In this paper we initiate the metamathematical investigation of ω-models of ZF fin . In particular, we present a perspicuous method for constructing recursive nonstandard ω-model of ZF fin without the use of permutations. We then use this method to establish the following central theorem. Theorem A. For every simple graph (A, F ), where F is a set of unordered pairs of A, there is an ω-model M of ZF fin whose universe contains A and which satisfies the following two conditions:(1) There is parameter-free formula ϕ(x, y) such that for all elements a andTheorem A enables us to build a variety of ω-models with special features, in particular:

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Section: Preliminariesmentioning

confidence: 89%

Section: Preliminariesmentioning

confidence: 99%

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ω-models of finite set theory

Enayat¹,

Schmerl²,

Visser³

2011

Set Theory, Arithmetic, and Foundations of Mathematics

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“…The definition of ω above is as in [4, pp. 468ff.] (see also [3]), except that in [4] the predicate Ord (x) says that "x is well-ordered with respect to ∈", while in our definition of Ord (x), "x is linearly ordered with respect to ∈". Note the following:…”

Section: Let Alsomentioning

confidence: 99%

Consequences of Vopěnka’s Principle over weak set theories

Tzouvaras

2016

Fund. Math.

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It is shown that Vopěnka's Principle (VP) can restore almost the entire ZF over a weak fragment of it. Namely, if EST is the theory consisting of the axioms of Extensionality, Empty Set, Pairing, Union, Cartesian Product, ∆0-Separation and Induction along ω, then EST + VP proves the axioms of Infinity, Replacement (thus also Separation) and Powerset. The result was motivated by previous ones (2014), as well as by H. Friedman's (2015), where a distinction is made among various forms of VP. As a corollary, EST + Foundation + VP = ZF + VP and EST + Foundation + AC + VP = ZFC + VP. Also, it is shown that the Foundation axiom is independent of ZF − {Foundation} + VP. It is open whether the Axiom of Choice is independent of ZF + VP. A very weak form of choice follows from VP, and some other similar forms of choice are introduced.

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“…The proper philosophical(Oppy 2009) as well as philosophical, mathematical, and semantical(Sondheimer, Rogerson 2006; Usó-Doménech Selva, Requena 2016;Luis, Moreno, Waldegg 1991) sense of Cantor's conception of infinity can be demonstrated as underlying the greatest contemporary mathematical problems(Stewart 2013). The logical and mathematical contradiction consists in the relation of the axiom of induction in arithmetic, on the one hand, and the axiom of infinity in set theory, on the other hand(Keyser 1903;Baratella, Ferro 1993).11 It is non-countable in general because "ω − 1" times repeated successively "n =" can be replaced by any infinite ordinal for "ω" as well as in both standard twin Peano arithmetics: "ω − 1" times repeated successively either "n + 1" or "n − 1".…”

mentioning

confidence: 99%

Hilbert arithmetic as a Pythagorean arithmetic: arithmetic as transcendental

Пенчев

2021

Preprint

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The paper considers a generalization of Peano arithmetic, Hilbert arithmetic as the basis of the world in a Pythagorean manner. Hilbert arithmetic unifies the foundations of mathematics (Peano arithmetic and set theory), foundations of physics (quantum mechanics and information), and philosophical transcendentalism (Husserl's phenomenology) into a formal theory and mathematical structure literally following Husserl's tracе of "philosophy as a rigorous science". In the pathway to that objective, Hilbert arithmetic identifies by itself information related to finite sets and series and quantum information referring to infinite one as both appearing in three "hypostases": correspondingly, mathematical, physical and ontological, each of which is able to generate a relevant science and area of cognition. Scientific transcendentalism is a falsifiable counterpart of philosophical transcendentalism. The underlying concept of the totality can be interpreted accordingly also mathematically, as consistent completeness, and physically, as the universe defined not empirically or experimentally, but as that ultimate wholeness containing its externality into itself.

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A theory of sets with the negation of the axiom of infinity

Cited by 11 publications

References 5 publications

ω-models of finite set theory

ω-models of finite set theory

Consequences of Vopěnka’s Principle over weak set theories

Hilbert arithmetic as a Pythagorean arithmetic: arithmetic as transcendental

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