In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory.
It was first published by Ernst Zermelo as part of his set theory in 1908.
[1] Using first-order logic primitive symbols, the axiom can be expressed as follows:[2]
In English, this sentence means: "there exists a set 𝐈 such that the empty set is an element of it, and for every element
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}.
Successors are used to define the usual set-theoretic encoding of the natural numbers.
In this encoding, zero is the empty set: The number 1 is the successor of 0: Likewise, 2 is the successor of 1: and so on: 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.
However, the other axioms are insufficient to prove the existence of the set of all natural numbers,
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.
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.
Informally, what we will do is take the intersection of all inductive sets.
More formally, we wish to prove the existence of a unique set
be an inductive set guaranteed by the Axiom of Infinity.
Then we use the axiom schema of specification to define our set
This definition is convenient because the principle of induction immediately follows: If
Both these methods produce systems that satisfy the axioms of second-order arithmetic, since the axiom of power set allows us to quantify over the power set of
Some old texts use an apparently weaker version of the axiom of infinity, to wit: This says that x is non-empty and for every element y of x there is another element z of x such that y is a subset of z and y is not equal to z.
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 that 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).
Then we can apply the axiom of union to that to get an ordinal greater than or equal to ω.
Con(ZFC − Infinity) and use Gödel's Second incompleteness theorem.)
Thus, ZFC implies neither the axiom of infinity nor its negation and is compatible with either.
Indeed, using the von Neumann universe, we can build a model of ZFC − Infinity + (¬Infinity).
, 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 (