Alternative Set Theories

Holmes, M. Randall; Förster, Thomas; Libert, Thierry

doi:10.1016/b978-0-444-51621-3.50008-6

Cited by 26 publications

(6 citation statements)

References 42 publications

(15 reference statements)

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“…The cardinal squaring principle has not yet been rigorously proved in our setting, although it seems to be a well-known fact. The main motivation for this paper is a remark about the cardinal squaring principle in [10]. The same remark is also stated in [6] and [5].…”

Section: Introductionmentioning

confidence: 76%

The Cardinal Squaring Principle and an Alternative Axiomatization of NFU

Adlešić,

Čačić

2023

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In this paper we rigorously prove the existence of type-level ordered pairs in Quine's New Foundations with atoms, augmented by the axiom of infinity and the axiom of choice (NFU+Inf+AC). The proof uses Tarski's theorem about choice, which is a theorem of NFU+Inf+AC. Therefore, we have a justification for proposing a new axiomatic extension of NFU, in order to obtain type-level ordered pairs almost from the beginning. This axiomatization is NFU+Inf+AC+Tarski, a conservative extension of NFU+Inf+AC.

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Section: Introductionmentioning

confidence: 76%

The Cardinal Squaring Principle and an Alternative Axiomatization of NFU

Adlešić,

Čačić

2023

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“…The idea was reworked (apparently independently) by W. Ackermann, and makes for a quite different theory from the standard one (it is considered to be the first genuine 'alternative set theory'). Ackermann's set theory, as explained in [24], is a theory of classes in which some classes are sets, and indeed the notion of set is indefinable-there is no simple definition of which classes are sets. It turns out, however, to be essentially the same theory as ZF.…”

Section: Inconsistent Sets and Proper Classesmentioning

confidence: 99%

“…Other set theories, such as New Foundations (NF) and the theory of semisets, treat the relationship between sets and classes in still different ways. The set theoretical proposals of P. Finsler, centered around his concept of "circle-free" sets, are regarded in principle as incoherent (see [24]). The idea was reworked (apparently independently) by W. Ackermann, and makes for a quite different theory from the standard one (it is considered to be the first genuine 'alternative set theory').…”

Section: Inconsistent Sets and Proper Classesmentioning

confidence: 99%

Paraconsistent Set Theory

Carnielli

Coniglio

2016

Paraconsistent Logic: Consistency, Contradiction and Negation

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Intuitively, a set is any collection of elements that satisfy a given property. By allowing arbitrary properties one finds a very flexible way of specifying sets-the method of abstraction. In this method a set is specified by giving the condition that an object must satisfy in order to belong to the set. In this way, the basic axiom known as the "Schema of Comprehension" or "Principle of Comprehension" proposed by Frege in 1893 in the course of his investigations (see [1]) was the following:If φ is a property (denoted by a first-order formula φ(x)) then there exists a set y formed by all elements satisfying property φ, that is, y = {x : φ(x)}.

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“…A copy can be found at https://drive.google.com/file/d/17JRj2orUVDw7lrBEmBS1K6OK06RP32Xa/view. (Holmes 2017) gives an abbreviated overview of its axioms. 8 Nonstandard models of Peano arithmetic contains infinite numbers.…”

Section: New Infinitary Mathematicsmentioning

confidence: 99%

“…A linear ordering that defines a countable set is wellordering. 15 Thus any two countable classes are isomorphic with respect to the ordering. 16…”

Section: Countable Classesmentioning

confidence: 99%

Infinity and continuum in the alternative set theory

Trlifajová

2021

Euro Jnl Phil Sci

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Alternative set theory was created by the Czech mathematician Petr Vopěnka in 1979 as an alternative to Cantor's set theory. Vopěnka criticised Cantor's approach for its loss of correspondence with the real world. Alternative set theory can be partially axiomatised and regarded as a nonstandard theory of natural numbers. However, its intention is much wider. It attempts to retain a correspondence between mathematical notions and phenomena of the natural world. Through infinity, Vopěnka grasps the phenomena of vagueness. Infinite sets are defined as sets containing proper semisets, i.e. vague parts of sets limited by the horizon. The new interpretation extends the field of applicability of mathematics and simultaneously indicates its limits. Compared to strict finitism and other attempts at a reduction of the infinite to the finite Vopěnka's theory reverses the process: he models the finite in the infinite.Vopěnka's Alternative set theory (AST) represents an attempt to present a new set theory on a phenomenal basis. It is "a mathematical-philosophical attempt in the study of the infinite". Both components are important and intertwined. This is not the only reason why the explanation of Vopěnka's theory is difficult. Its mathematical form has developed and changed several times. Vopěnka

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Alternative Set Theories

Cited by 26 publications

References 42 publications

The Cardinal Squaring Principle and an Alternative Axiomatization of NFU

The Cardinal Squaring Principle and an Alternative Axiomatization of NFU

Paraconsistent Set Theory

Infinity and continuum in the alternative set theory

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