nLab axiom of infinity

Idea

In the foundations of mathematics, an axiom of infinity is any axiom that asserts that infinite sets exist. In set theory and set-level type theory, infinite sets cannot be constructed from finite sets, so their existence must be posited as an extra axiom. Further axioms, in this vein which assert the existence of even larger sets that cannot be constructed from smaller ones are called large cardinal axioms.

Statement

Natural numbers

One common form of the axiom of infinity says that the particular set $N$ of natural numbers exists. In material set theory this often takes the form of asserting that the von Neumann ordinal number $\omega$ exists, where $\omega$ is characterized as the smallest set such that $\emptyset\in\omega$ and whenever $a\in \omega$ then $a\cup \{a\}\in \omega$.

In dependent type theory, it is possible to define a Tarski universe $(V, \in)$ of pure sets which behaves as a material set theory. The universal type family of the Tarski universe is given by the type family $x:V \vdash \sum_{y:V} y \in x$. The axiom of infinity is given by the following inference rule:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{infinity}_V:\sum_{\omega:V} \left((\emptyset \in \omega) \times \prod_{x:V} (x \in \omega) \to \sum_{s:V} (s \in \omega)\right) \times \prod_{z:V} \left(\left((\emptyset \in z) \times \prod_{x:V} (x \in z) \to \sum_{s:V} (s \in z)\right) \to \prod_{x:V} (x \in \omega) \to (x \in z)\right)}$$

The axiom schema of induction is given by the following inference rule:

$$\frac{\Gamma, x:V \vdash \phi(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{induction}_V^{\phi(-)}:\left(\prod_{x:V} \mathrm{isProp}(\phi(x))\right) \to \sum_{\omega:V} \left((\emptyset \in \omega) \times \prod_{x:V} (x \in \omega) \to \sum_{s:V} (s \in \omega)\right) \times \left(\left(\phi(\emptyset) \times \prod_{x:V} (x \in \omega) \to (\phi(x) \to \sum_{s:V} \phi(s))\right) \to \prod_{x:V} (x \in \omega) \to \phi(x)\right)}$$

In structural set theory the usual form of the axiom of infinity is the existence of a natural numbers object.

In dependent type theory, the natural numbers type for a Tarski universe is given by the element

$$\mathrm{axinf}_U:\sum_{\mathbb{N}:U} \sum_{0:T(\mathbb{N})} \sum_{s:T(\mathbb{N}) \to T(\mathbb{N})} \prod_{C:T(\mathbb{N}) \to U} \prod_{c_0:T(C(0))} \prod_{c_s:\prod_{x:T(\mathbb{N})} T(C(x)) \to T(C(s(x)))} \sum_{c:\prod_{x:T(\mathbb{N})} T(C(x))} (c(0) =_{T(C(0))} c_0) \times \prod_{x:T(\mathbb{N})} (c(s(x)) =_{T(C(s(x)))} c_s(c(x)))$$

or

$$\mathrm{axinf}_U:\sum_{\mathbb{N}:U} \sum_{0:T(\mathbb{N})} \sum_{s:T(\mathbb{N}) \to T(\mathbb{N})} \prod_{C:U} \prod_{c_0:T(C)} \prod_{c_s:T(C) \to T(C)} \exists!c:T(\mathbb{N}) \to T(C).(f(0) =_{T(C)} c_0) \times \prod_{n:T(\mathbb{N})} c(s(n)) =_{T(C)} c_s(c(n))$$

which states that there is a natural numbers type in the universe.

There is an alternate way to express the axiom of infinity in a Tarski universe, as the axiom of resizing the set truncation of the type of finite types in $U$, since $\mathrm{isFinite}$ and set truncations are definable from the type of propositions in $U$, $\sum_{A:U} \mathrm{isProp}(A)$, but they are all usually large, and so have to be resized to be small:

$$\mathrm{axinf}_U:\sum_{\mathbb{N}:U} T(\mathbb{N}) \simeq \left[\sum_{A:U} \mathrm{isFinite}(T(A))\right]_0$$

Integers

Inductive definition

Instead of a defining the natural numbers via its induction principle, one can instead define the integers via its induction principle, and then use the fact that disjoint unions are disjoint and $\mathbb{Z} \cong \mathbb{Z} \uplus \mathbb{Z}$ to construct the natural numbers.

Second-order definition

Alternatively, one can assume a (trichotomous) ordered integral domain $\mathbb{Z}$, such that every (trichotomous) ordered integral subdomain of $\mathbb{Z}$ is equivalent to the improper subset of $\mathbb{Z}$. This defines the integers, since the integers are the initial (trichotomous) ordered integral domain and are strictly initial. Since the integers as defined automatically comes with a total order $\leq$ and a pseudo-order $\lt$, one can define the natural numbers as the set of non-negative integers.

Rational numbers

If one has power sets, one can assume an (trichotomous) ordered field $\mathbb{Q}$, such that every (trichotomous) ordered subfield of $\mathbb{Q}$ is equal to the improper subset of $\mathbb{Q}$. This defines the rational numbers, since the rational numbers are the initial (trichotomous) ordered field and are strictly initial. The rational numbers are automatically infinite, and one can construct the integers $\mathbb{Z}$ as the intersection of all ordered integral subdomains of $\mathbb{Q}$, and since the integers as defined automatically comes with a total order $\leq$ and a pseudo-order $\lt$, one can define the natural numbers as the set of non-negative integers.

Generalizations

In the form of an NNO, the axiom of infinity generalises to the existence of inductive types or W-types. These can be constructed from a NNO if power sets exist, but in predicative theories they can be added as additional axioms.

One could also posit the existence of the set of extended natural numbers instead of the set of natural numbers, as the set of extended natural numbers have countably infinite cardinality and is the categorical dual of the natural numbers in Set, a terminal coalgebra for the endofunctor $F(X) = 1 + X$ in Set. This generalises to the existence of coinductive types or M-types, which can be added as additional axioms.

One could also posit the existence of FinSet, the collection of finite sets. In dependent type theory this is a type of finite types, a universe $\mathcal{U}$ that satisfies the axiom of finiteness (see below).

Alternatives

Broadly speaking, finite mathematics is mathematics that does not use or need the axiom of infinity; a finitist is an extreme breed of constructivist that believes that mathematics is better without the axiom of infinity, or even that this axiom is false.

A more extreme case is to deny the axiom of infinity with an axiom of finiteness: every set is finite. There is one of these for every definition of ‘finite’ given on that page; here is the strongest stated directly in terms of set theory as an axiom of induction:

• Any property of sets that is invariant under isomorphism and holds for the empty set must hold for all sets if, whenever it holds for a set $X$, it holds for the disjoint union $X \uplus \{*\}$.

In material set theory, this is equivalent given the axiom of foundation (which guarantees that $X$ and $\{X\}$ are disjoint):

• Any property of sets that holds for the empty set must hold for all sets if, whenever it holds for a set $X$, it holds for the union $X \cup \{X\}$.

In higher categorical terms, the above axiom of finiteness could be stated as follows: Set is an initial algebra of the 2-endofunctor $F(X) \cong X \coprod 1$ in the (2,1)-category Grpd.

In dependent type theory, given a Tarski universe $(U, T)$ that is closed under the empty type, the unit type, and sum types, the axiom of finiteness for the universe states that

• For all type families $A:U \vdash C(A)$ such that $T(A) \simeq T(B)$ implies that $C(A) \simeq C(B)$, elements $c_0:C(\mathbb{0})$ and dependent functions $c_s:\prod_{A:U} C(A) \to C(A + \mathbb{1})$, there exists a unique dependent function $c:\prod_{A:U} C(A)$ such that $c(\mathbb{0}) =_{C(\mathbb{0})} c_0$ and for all $A:U$, $c(A + 1) =_{C(A + 1)} c_s(c(A))$.

In dependent type theory with dependent product types, dependent sum types, identity types, function extensionality, and a type of all propositions, the axiom of finiteness for the entire type theory is an axiom schema which states that given a type $A$, one could derive a witness that the type is a finite type:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{finWitn}_A:\mathrm{isFinite}(A)}$$

where

$$\mathrm{isFinite}(A) \equiv \begin{array}{c} \prod_{S:(A \to \mathrm{Prop}) \to \mathrm{Prop}} (((\lambda x:A.\bot) \in S) \times \prod_{P:A \to \mathrm{Prop}} \prod_{Q:A \to \mathrm{Prop}} (P \in S) \\ \times (\exists!x:A.x \in Q) \times (P \cap Q =_{A \to \mathrm{Prop}} \lambda x:A.\bot) \to (P \cup Q \in S)) \to ((\lambda x:A.\top) \in S) \end{array}$$

The membership relation and the subtype operations used above are defined in the nLab article on subtypes.

In particular, the axiom of finiteness for the entire type theory implies the principle of excluded middle for the type of all propositions, since the only finite propositions are the decidable propositions. Furthermore, the axiom of finiteness implies that the type theory is a set-level type theory because every finite type is an h-set.

Related concepts

• finite set, finite object

• Set, FinSet

References

• Wikipedia, Axiom of infinity

• Michael Shulman (2018). Comparing material and structural set theories. [arXiv:1808.05204&rbrack.

In relation to classifying toposes:

• Andreas Blass, Classifying topoi and the axiom of infinity, Algebra Universalis 26 (1989) 341-345 [doi:10.1007/BF01211840]

For constructing the natural numbers from the integers:

• Christian Sattler, Natural numbers from integers (pdf)

category: foundational axiom