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In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.[1]
^Zermelo: Untersuchungen über die Grundlagen der Mengenlehre, 1907, in: Mathematische Annalen 65 (1908), 261-281; Axiom des Unendlichen p. 266f.
of mathematics and philosophy that use it, the axiomofinfinity is one of the axiomsof Zermelo–Fraenkel set theory. It guarantees the existence of at...
sets. Among the axiomsof Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiomofinfinity, which guarantees...
axiomofinfinity for NF: ∅ ∉ N . {\displaystyle \varnothing \notin \mathbf {N} .} It may intuitively seem that one should be able to prove Infinity in...
Axiomof extensionality Axiomof empty set Axiomof pairing Axiomof union AxiomofinfinityAxiom schema of replacement Axiomof power set Axiomof regularity...
making it an axiom; by deriving it from a set-existence axiom (or logic) and the axiomof separation; by deriving it from the axiomofinfinity; or some other...
by axiomofinfinity, and is now included as part of it. Zermelo set theory does not include the axiomsof replacement and regularity. The axiomof replacement...
In mathematics, the axiomof regularity (also known as the axiomof foundation) is an axiomof Zermelo–Fraenkel set theory that states that every non-empty...
Baratella, Stefano; Ferro, Ruggero (1993). "A theory of sets with the negation of the axiomofinfinity". Mathematical Logic Quarterly. 39 (3): 338–352. doi:10...
relation of BIT, swapping its two arguments) models Zermelo–Fraenkel set theory Z F {\displaystyle {\mathsf {ZF}}} without the axiomofinfinity. Here,...
of any limit ordinal greater than ω requires the replacement axiom. The ordinal number ω·2 = ω + ω is the first such ordinal. The axiomofinfinity asserts...
the axiom of power set or from the axiomofinfinity. In the absence of some of the stronger ZFC axioms, the axiomof pairing can still, without loss, be...
cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiomof empty set, while...
of natural numbers (whose existence is postulated by the axiomofinfinity) is infinite. It is the only set that is directly required by the axioms to...
mathematical logic, the Peano axioms (/piˈɑːnoʊ/, [peˈaːno]), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers...
model of all of the axiomsof ZFC except infinity." Cohen 2008, p. 54, states: "The first really interesting axiom [of ZF set theory] is the Axiomof Infinity...
{\displaystyle \omega } exists. This is not an axiomofinfinity in the usual sense; if Infinity does not hold, the closure of ω {\displaystyle \omega } exists and...
basic axiomsof type theory, three further axioms that seemed to not be true as mere matters of logic, namely the axiomofinfinity, the axiomof choice...
of the natural numbers using the axiomofinfinity and axiom schema of specification. One variation of the principle of complete induction can be generalized...
existence of objects that are not explicitly built. This excludes, in particular, the use of the law of the excluded middle, the axiomofinfinity, and the...
the axiomof choice, abbreviated AC or AoC, is an axiomof set theory equivalent to the statement that a Cartesian product of a collection of non-empty...
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