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Inaxiomatic set theory and the branches ofmathematics andphilosophy that use it, theaxiom of infinity is one of theaxioms ofZermelo–Fraenkel set theory. It guarantees the existence of at least oneinfinite set, namely a set containing thenatural numbers. It was first published byErnst Zermelo as part of hisset theory in 1908.^[1]

Usingfirst-order logic primitive symbols, the axiom can be expressed as follows:^[2]

${\displaystyle \mathrm{\exists }\mathrm{I}(\mathrm{\exists }o(o\in \mathrm{I}\wedge \mathrm{\neg }\mathrm{\exists }n(n\in o))\wedge \mathrm{\forall }x(x\in \mathrm{I}\Rightarrow \mathrm{\exists }y(y\in \mathrm{I}\wedge \mathrm{\forall }a(a\in y\iff (a\in x\vee a=x))))).}$

In English, this sentence means: "there exists aset𝐈 such that theempty set is an element of it, and for every element${\displaystyle x}$ of𝐈, there exists an element${\displaystyle y}$ of𝐈 such that${\displaystyle x}$ is an element of${\displaystyle y}$, the elements of${\displaystyle x}$ are also elements of${\displaystyle y}$, and nothing else is an element of${\displaystyle y}$."

This sentence can be abbreviated in set-builder notation as:

${\displaystyle \mathrm{\exists }\mathrm{I}(\varnothing \in \mathrm{I}\wedge \mathrm{\forall }x(x\in \mathrm{I}\Rightarrow (x\cup \{x\})\in \mathrm{I})).}$

Some mathematicians may call a set built this way aninductive set.

This axiom is closely related to thevon Neumann construction of the natural numbers in set theory, in which thesuccessor ofx is defined asx ∪ {x}. Ifx 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 thenatural 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 ofall natural numbers,${\displaystyle {\mathbb{N}}_0}$. Therefore, its existence is taken as an axiom – the axiom of infinity. This axiom asserts that there is a setI that contains 0 and isclosed under the operation of taking the successor; that is, for each element ofI, the successor of that element is also inI.

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 thevon Neumann–Bernays–Gödel axioms.

The infinite setI is a superset of the natural numbers. To show that the natural numbers themselves constitute a set, theaxiom schema of specification can be applied to remove unwanted elements, leaving the setN of all natural numbers. This set is unique by theaxiom 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 that does not assume any axioms except theaxiom of extensionality and theaxiom 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:

${\displaystyle \mathrm{\forall }n(n\in \mathbf{N}⟺([n=\mathrm{\varnothing }\vee \mathrm{\exists }k(n=k\cup \{k\})]\wedge \mathrm{\forall }m\in n[m=\mathrm{\varnothing }\vee \mathrm{\exists }k\in n(m=k\cup \{k\})])).}$

Or, even more formally:

${\displaystyle \mathrm{\forall }n(n\in \mathbf{N}⟺([\mathrm{\forall }k(\mathrm{\neg }k\in n)\vee \mathrm{\exists }k\mathrm{\forall }j(j\in n⟺(j\in k\vee j=k))]\wedge }$

${\displaystyle \mathrm{\forall }m(m\in n\Rightarrow [\mathrm{\forall }k(\mathrm{\neg }k\in m)\vee \mathrm{\exists }k(k\in n\wedge \mathrm{\forall }j(j\in m⟺(j\in k\vee j=k)))]))).}$

An alternative method is the following. Let${\displaystyle \mathrm{\Phi }(x)}$ be the formula that says "x is inductive"; i.e.${\displaystyle \mathrm{\Phi }(x)=(\mathrm{\varnothing }\in x\wedge \mathrm{\forall }y(y\in x\to (y\cup \{y\}\in x)))}$. 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${\displaystyle W}$ such that

${\displaystyle \mathrm{\forall }x(x\in W\leftrightarrow \mathrm{\forall }I(\mathrm{\Phi }(I)\to x\in I)).}$ (*)

For existence, we will use the Axiom of Infinity combined with theAxiom schema of specification. Let${\displaystyle I}$ be an inductive set guaranteed by the Axiom of Infinity. Then we use the axiom schema of specification to define our set${\displaystyle W=\{x\in I:\mathrm{\forall }J(\mathrm{\Phi }(J)\to x\in J)\}}$ – i.e.${\displaystyle W}$ is the set of all elements of${\displaystyle I}$, which also happen to be elements of every other inductive set. This clearly satisfies the hypothesis of (*), since if${\displaystyle x\in W}$, then${\displaystyle x}$ is in every inductive set, and if${\displaystyle x}$ is in every inductive set, it is in particular in${\displaystyle I}$, so it must also be in${\displaystyle W}$.

For uniqueness, first note that any set that satisfies (*) is itself inductive, since 0 is in all inductive sets, and if an element${\displaystyle x}$ is in all inductive sets, then by the inductive property so is its successor. Thus if there were another set${\displaystyle W^{\prime }}$ that satisfied (*) we would have that${\displaystyle W^{\prime }\subseteq W}$ since${\displaystyle W}$ is inductive, and${\displaystyle W\subseteq W^{\prime }}$ since${\displaystyle W^{\prime }}$ is inductive. Thus${\displaystyle W=W^{\prime }}$. Let${\displaystyle \omega }$ denote this unique element.

This definition is convenient because theprinciple of induction immediately follows: If${\displaystyle I\subseteq \omega }$ is inductive, then also${\displaystyle \omega \subseteq I}$, so that${\displaystyle I=\omega }$.

Both these methods produce systems that satisfy the axioms ofsecond-order arithmetic, since theaxiom of power set allows us to quantify over thepower set of${\displaystyle \omega }$, as insecond-order logic. Thus they both completely determineisomorphic systems, and since they are isomorphic under theidentity map, they must in fact beequal.

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

${\displaystyle \mathrm{\exists }x(\mathrm{\exists }y(y\in x)\wedge \mathrm{\forall }y(y\in x\to \mathrm{\exists }z(z\in x\wedge y⊊z))).}$

This says thatx is non-empty and for every elementy ofx there is another elementz ofx such thaty is a subset ofz andy is not equal toz. This implies thatx 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 setx, then that powerset will contain elements that are subsets ofx of every finitecardinality (among other subsets ofx). 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 ofx by theinitialordinal 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${\displaystyle ⊢}$ Con(ZFC − Infinity) and use Gödel'sSecond 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.) Thus, ZFC implies neither the axiom of infinity nor its negation and is compatible with either.

Indeed, using thevon Neumann universe, we can build a model of ZFC − Infinity + (¬Infinity). It is${\displaystyle V_{\omega }}$, the class ofhereditarily 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 theempty 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 (${\displaystyle {\mathrm{\aleph }}_0}$), has many of the properties of alarge cardinal. Thus the axiom of infinity is sometimes regarded as the firstlarge cardinal axiom, and conversely large cardinal axioms are sometimes called^{[by whom?]} stronger axioms of infinity.

• Peano axioms
• Finitism

• Axioms of set theory
• Infinity

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