Axiom of Infinity

The axiom of Zermelo-Fraenkel set theory which asserts the existence of a set containing all the natural numbers,

  exists x(emptyset in x ^  forall y in x(y^' in x)),

where  exists denotes exists, emptyset is the empty set,  ^ is logical AND,  forall means for all, and  in denotes "is an element of" (Enderton 1977). Following von Neumann, 0=emptyset, 1=0^'={0}, 2=1^'={0,1}, 3=2^'={0,1,2}, ....

See also

Zermelo-Fraenkel Set Theory

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Enderton, H. B. Elements of Set Theory. New York: Academic Press, 1977.Itô, K. (Ed.). "Zermelo-Fraenkel Set Theory." §33B in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, pp. 146-148, 1986.

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Axiom of Infinity

Cite this as:

Weisstein, Eric W. "Axiom of Infinity." From MathWorld--A Wolfram Web Resource.

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