Axiom of infinity

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory.

Formal statement

In the formal language of the Zermelo-Fraenkel axioms, the axiom reads::$exist\; mathbf\{I\}\; ,\; (\; empty\; in\; mathbf\{I\}\; ,\; and\; ,\; forall\; x\; in\; mathbf\{I\}\; ,\; (\; ,\; (\; x\; cup\; \{x\}\; )\; in\; mathbf\{I\}\; )\; )\; .$

In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I and such that whenever any "x" is a member of I, the set formed by taking the union of "x" with its singleton {"x"} is also a member of I. Such a set is sometimes called an inductive set.

Interpretation and consequences

This axiom is closely related to the standard construction of the naturals in set theory, in which the "successor" of "x" is defined as "x" ∪ {"x"}. If "x" is a set, then it follows from the other axioms of set theory that this successor is also a uniquely defined set. Successors are used to define the usual set-theoretic encoding of the natural numbers. In this encoding, zero is the empty set:

:0 = {}.

The number 1 is the successor of 0:

:1 = 0 ∪ {0} = {} ∪ {0} = {0}.

Likewise, 2 is the successor of 1:

:2 = 1 ∪ {1} = {0} ∪ {1} = {0,1},

and so on. A consequence of this definition is that every natural number is equal to the set of all preceding natural numbers.

This construction forms the natural numbers. However, the other axioms are insufficient to prove the existence of the set of all natural numbers. Therefore its existence is taken as an axiom — the axiom of infinity. This axiom asserts that there is a set I that contains 0 and is closed under the operation of taking the successor; that is, for each element of I, the successor of that element is also in I.

Thus the essence of the axiom is:

:There is a set, I, that includes all the natural numbers.

The axiom of infinity is also one of the von Neumann-Bernays-Gödel axioms.

Extracting the natural numbers from the infinite set

The infinite set I is a superset of the natural numbers. To show that the natural numbers themselves constitute a set, the axiom schema of specification can be applied to remove unwanted elements, leaving the set N of all natural numbers. This set is unique by the axiom of extensionality.

To extract the natural numbers, we need a definition of which sets are natural numbers. The natural numbers can be defined in a way which does not assume any axioms except the axiom of extensionality and the axiom of induction — a natural number is either zero or a successor and each of its elements is either zero or a successor of another of its elements. In formal language, the definition says:

:$forall\; n\; (n\; in\; mathbf\{N\}\; iff\; (\; [n\; =\; empty\; ,,or,,\; exist\; k\; (\; n\; =\; k\; cup\; \{k\}\; )]\; ,,and,,\; forall\; m\; in\; n\; [m\; =\; empty\; ,,or,,\; exist\; k\; in\; n\; (\; m\; =\; k\; cup\; \{k\}\; )]\; )).$

Or, even more formally:

:
::

Here, denotes the logical constant "false", so is a formula that holds only if "n" is the empty set.

Independence

If ZFC is consistent, then the axiom of infinity cannot be derived from the other axioms. Indeed, using the Von Neumann universe we can make a model of the axioms where the axiom of infinity is replaced by its negation. It is $V\_omega\; !$, the class of hereditarily finite sets, with the inherited element relation.

The cardinality of the set of natural numbers, aleph null ($aleph\_0$), has many of the properties of a large cardinal. Thus the axiom of infinity is sometimes regarded as the first "large cardinal axiom".

References

*Paul Halmos (1960) "Naive Set Theory". Princeton, NJ: D. Van Nostrand Company. Reprinted 1974 by Springer-Verlag. ISBN 0-387-90092-6.
*Thomas Jech (2003) "Set Theory: The Third Millennium Edition, Revised and Expanded". Springer-Verlag. ISBN 3-540-44085-2.
*Kenneth Kunen (1980) "Set Theory: An Introduction to Independence Proofs". Elsevier. ISBN 0-444-86839-9.