axiom of infinity
There exists an infinite set.
The Axiom of Infinity is an axiom of Zermelo-Fraenkel set theory
.
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 set
? However, once we have a theory of ordinal numbers
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
set. A set S is said to be inductive if ∅∈S
and for every x∈S, x∪{x}∈S. We may then state the
Axiom of Infinity as follows:
There exists an inductive set.
In symbols:
∃S[∅∈S∧(∀x∈S)[x∪{x}∈S]] |
We shall then be able to prove that the following conditions are equivalent:
-
1.
There exists an inductive set.
-
2.
There exists an infinite set.
-
3.
The least nonzero limit ordinal
, 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 |