symmetric monoidal (∞,1)-category of spectra
transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
An integer is a number that is a natural number or the negative of one.
The ring $\mathbb{Z}$ of all integers may defined as the free group on one generator or as the initial ring.
In keeping with a historical point of view in which integers are natural numbers with a sign attached, one may write
From an nPOV, one may consider this as follows: $\mathbb{Z}$ is a filtered colimit of sets
whereby $-n \in \mathbb{Z}$ is represented by the element $0$ in the $n^{th}$ copy of $\mathbb{N}$ appearing in this diagram (starting the count at the $0^{th}$ copy). The resulting induced map to the colimit
imparts a monoid (in fact a group) structure on $\mathbb{Z}$ descended from the monoid structure on $\mathbb{N} \times \mathbb{N}$; compare double-entry bookkeeping in medieval mathematics (partita doppia).
As a group, $\mathbb{Z}$ is abelian and is the Grothendieck group of the monoid (or semigroup) $\mathbb{N}$ of natural numbers.
The monoid of natural numbers is naturally even a rig – in fact the initial rig – and this multiplicative structure extends to $\mathbb{Z}$ to make it a ring – in fact the initial ring.
The integers form a commutative ring.
The integers have decidable equality and decidable order.
The integers are a strictly weakly ordered ring and a lattice-ordered ring.
The integers are a metric space and a normed space.
The integers are a Euclidean domain
The integers satisfy the Archimedean property, making it into an Archimedean integral domain.
The integers are in bijection with the natural numbers. Both the integers and the natural numbers are submonoids? of the rational numbers, with pointed monoid monomorphisms $i_\mathbb{Z}:\mathbb{Z} \hookrightarrow \mathbb{Q}$ and $i_\mathbb{N}:\mathbb{N} \hookrightarrow \mathbb{Q}$ preserving addition, zero, and one. Let us take the relation on the rational numbers
The relation
is a one-to-one correspondence, meaning that the integers are in bijection with the natural numbers.
Let $x:\mathbb{N} \to \mathbb{Z}$ be a sequence of integers, and let $b:\mathbb{Z}$ be an integer. Then, there is a limit relation defined as
This relation is a functional relation, making the integers a sequentially Hausdorff space.
A modulus of Cauchy convergence is a function $M:\mathbb{Z}_+ \to \mathbb{N}$ with a witness
$\mathbb{Z}$ is sequentially Cauchy complete if every sequence with a modulus of Cauchy convergence has a unique limit. But $\mathbb{Z}$ is sequentially Cauchy complete, because the only sequences with a unique limit are those sequences for which there exists a natural number $N:\mathbb{N}$ such that for all natural numbers $m \geq N$ and $n \geq N$, $x(m) = x(n)$. The sequence $x$ has many moduli of Cauchy convergence $M$, where the natural number $N$ is given by $N = M(1)$.
Since the integers are the initial Archimedean integral domain, the integers are also the initial sequentially Cauchy complete Archimedean integral domain. Every other sequentially Cauchy complete Archimedean integral domain is provably an ordered field and has the HoTT book real numbers $\mathbb{R}$ as an integral subdomain. That means, in the context of the limited principle of omniscience, the category of sequentially Cauchy complete Archimedean integral domains is equivalent to the walking arrow, with objects $\mathbb{Z}$ and $\mathbb{R}$ and homomorphism $h:\mathbb{Z} \to \mathbb{R}$.
Given a Tarski universe $(U, T)$ and a small type $A$ whose type reflection $T(A)$ is a directed set, let $x:T(A) \to \mathbb{Z}$ be a net of integers, and let $b:\mathbb{Z}$ be an integer. Then, there is a limit relation defined as
This relation is a functional relation, making the integers a Hausdorff space.
A modulus of $U$-Cauchy convergence is a function $M:\mathbb{Z}_+ \to T(A)$ with a witness
$\mathbb{Z}$ is $U$-Cauchy complete if every net with a modulus of $U$-Cauchy convergence has a unique limit. But $\mathbb{Z}$ is $U$-Cauchy complete, because the only $U$-nets with a unique limit are those nets for which there exists an element $N:T(A)$ such that for all elements $m \geq N$ and $n \geq N$, $x(m) = x(n)$. The net $x$ has many moduli of $U$-Cauchy convergence $M$, where the element $N$ is given by $N = M(1)$.
Since the integers are the initial Archimedean integral domain, the integers are also the initial $U$-Cauchy complete Archimedean integral domain. Every other $U$-Cauchy complete Archimedean integral domain is provably an ordered field and has the $U$-Dedekind real numbers $\mathbb{R}$ as an integral subdomain.
Let $\Omega$ be the type of all propositions, so that the foundations is impredicative. A Dedekind cut is an pair $(L, U)$ of predicates such that
There is a Dedekind cut for every integer $a:\mathbb{Z}$, given by $L_a(b) \coloneqq b \lt a$ and $U_a(b) \coloneqq a \lt b$. There are no other Dedekind cuts on the integers. Thus, the integers are Dedekind complete.
Since the integers are the initial Archimedean integral domain, the integers are also the initial Dedekind complete Archimedean integral domain. The only other Dedekind complete Archimedean integral domain is the Dedekind real numbers $\mathbb{R}$. That means, if there is a type of all propositions, the category of Dedekind complete Archimedean integral domains is equivalent to the walking arrow, with objects $\mathbb{Z}$ and $\mathbb{R}$ and homomorphism $h:\mathbb{Z} \to \mathbb{R}$.
The underlying sets $\mathbb{Z}$ and $\mathbb{N}$ are isomorphic. Some subcultures of mathematics (and not only set theorists) use the term ‘integer’ synonymously for a natural number. Computer scientists distinguish between ‘unsigned integers’ (natural numbers) and ‘signed integers’ (integers as described here). Translations can also cause confusion with the term ‘whole number’.
In number theory, one generalises integers to algebraic integers, an instance of the red herring principle. Accordingly, some number theorists will call the integers ‘rational integers’ to clarify; $\mathbb{Z}$ is the ring of integers in the number field $\mathbb{Q}$ of rational numbers. (Compare, for example, Gaussian integers and Gaussian numbers.)
The symbol ‘$\mathbb{Z}$’ derives from the German word ‘Zahlen’, which is a generic word for ‘numbers’. (Compare Dedekind's use of that word in the title of his famous book on the foundations of real numbers.)
geometry modeled on the formal dual, Spec(Z), of the ring of integers is arithmetic geometry
integers object in a topos
The first characterization of the integers as an ordered integral domain appeared in:
though the name “ordered integral domain” does not appear in the text.
History:
See also:
Formalization in univalent foundations of mathematics (homotopy type theory with the univalence axiom):
and specifically in Agda:
and specifically in cubical Agda
Last revised on February 23, 2024 at 19:53:49. See the history of this page for a list of all contributions to it.