Axiom of infinity

In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:

${\displaystyle \exists \mathbf {I} \,(\emptyset \in \mathbf {I} \,\land \,\forall x\in \mathbf {I} \,(\,(x\cup \{x\})\in \mathbf {I} )).}$

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.

This axiom is closely related to the von Neumann construction of the natural numbers 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:

3 = {0,1,2} = { {}, {{}}, {{}, {{}}} };

4 = {0,1,2,3} = { {}, {{}}, { {}, {{}} }, { {}, {{}}, {{}, {{}}} } }.

A consequence of this definition is that every natural number is equal to the set of all preceding natural numbers. The count of elements in each set, at the top level, is the same as the represented natural number, and the nesting depth of the most deeply nested empty set {}, including its nesting in the set that represents the number of which it is a part, is also equal to the natural number that the set represents.

This construction forms the natural numbers. However, the other axioms are insufficient to prove the existence of the set of all natural numbers, ℕ₀. 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.

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:

${\displaystyle \forall n(n\in \mathbf {N} \iff ([n=\emptyset \,\,\lor \,\,\exists k(n=k\cup \{k\})]\,\,\land \,\,\forall m\in n[m=\emptyset \,\,\lor \,\,\exists k\in n(m=k\cup \{k\})])).}$

Or, even more formally:

${\displaystyle \forall n(n\in \mathbf {N} \iff ([\forall k(\lnot k\in n)\lor \exists k\forall j(j\in n\iff (j\in k\lor j=k))]\land }$

${\displaystyle \forall m(m\in n\Rightarrow [\forall k(\lnot k\in m)\lor \exists k(k\in n\land \forall j(j\in m\iff (j\in k\lor j=k)))]))).}$

Some old texts use an apparently weaker version of the axiom of infinity, to wit

${\displaystyle \exists x\,(\exists y\,(y\in x)\,\land \,\forall y(y\in x\,\rightarrow \,\exists z(z\in x\,\land \,y\subseteq z\,\land \,y\neq z)))\,.}$

This says that there is an element in x and for every element y of x there is another element of x which is a strict superset of y. This implies that x is an infinite set without saying much about its structure. However, with the help of the other axioms of ZF, we can show that this implies the existence of ω. First, if we take the powerset of any infinite set x, then that powerset will contain elements which are subsets of x of every finite cardinality (among other subsets of x). Proving the existence of those finite subsets may require either the axiom of separation or the axioms of pairing and union. Then we can apply the axiom of replacement to replace each element of that powerset of x by the initial ordinal number of the same cardinality (or zero, if there is no such ordinal). The result will be an infinite set of ordinals. Then we can apply the axiom of union to that to get an ordinal greater than or equal to ω.

The axiom of infinity cannot be proved from the other axioms of ZFC if they are consistent. (To see why, note that ZFC ${\displaystyle \vdash }$ Con(ZFC – Infinity) and use Gödel's Second incompleteness theorem.)

The negation of the axiom of infinity cannot be derived from the rest of the axioms of ZFC, if they are consistent. (This is tantamount to saying that ZFC is consistent, if the other axioms are consistent.) We believe this, but cannot prove it (if it is true).

Indeed, using the von Neumann universe, we can build a model of ZFC – Infinity + (¬Infinity). It is ${\displaystyle V_{\omega }\!}$, the class of hereditarily finite sets, with the inherited membership relation. Note that if the axiom of the empty set is not taken as a part of this system (since it can be derived from ZF + Infinity), then the empty domain also satisfies ZFC – Infinity + ¬Infinity, as all of its axioms are universally quantified, and thus trivially satisfied if no set exists.

The cardinality of the set of natural numbers, aleph null (${\displaystyle \aleph _{0}}$), has many of the properties of a large cardinal. Thus the axiom of infinity is sometimes regarded as the first large cardinal axiom, and conversely large cardinal axioms are sometimes called stronger axioms of infinity.

• Peano axioms
• Finitism

• 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.
• Hrbacek, Karel; Jech, Thomas (1999). Introduction to Set Theory (3 ed.). Marcel Dekker. ISBN 0-8247-7915-0.