axiom of infinity

There exists an infinite setMathworldPlanetmath.

The Axiom of InfinityMathworldPlanetmath is an axiom of Zermelo-Fraenkel set theoryMathworldPlanetmath. At first glance, this axiom seems to be ill-defined. How are we to know what constitutes an infinite set when we have not yet defined the notion of a finite setMathworldPlanetmath? However, once we have a theory of ordinal numbersMathworldPlanetmath in hand, the axiom makes sense.

Meanwhile, we can give a definition of finiteness that does not rely upon the concept of number. We do this by introducing the notion of an inductive setMathworldPlanetmath. A set S is said to be inductive if S and for every xS, x{x}S. We may then state the Axiom of Infinity as follows:

There exists an inductive set.

In symbols:


We shall then be able to prove that the following conditions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    There exists an inductive set.

  2. 2.

    There exists an infinite set.

  3. 3.

    The least nonzero limit ordinalMathworldPlanetmath, ω, is a set.

Title axiom of infinity
Canonical name AxiomOfInfinity
Date of creation 2013-03-22 13:43:52
Last modified on 2013-03-22 13:43:52
Owner Sabean (2546)
Last modified by Sabean (2546)
Numerical id 6
Author Sabean (2546)
Entry type Axiom
Classification msc 03E30
Synonym infinity