nLab generalized Cauchy real number

Contents

Context

Analysis

Constructivism, Realizability, Computability

Contents

Idea

The Cauchy completion of the rational numbers, in the sense that all Cauchy nets in the rational numbers converge.

Definition

Let \mathbb{Q} be the rational numbers and let

+{x|0<x}\mathbb{Q}_{+} \coloneqq \{x \in \mathbb{Q} \vert 0 \lt x\}

be the set of positive rational numbers. Let II be a directed set and

C(I,){x I|ϵ +.NI.iI.jI.(iN)(jN)(|x ix j|<ϵ)}C(I, \mathbb{Q}) \coloneqq \{x \in \mathbb{Q}^I \vert \forall \epsilon \in \mathbb{Q}_{+}. \exists N \in I. \forall i \in I. \forall j \in I. (i \geq N) \wedge (j \geq N) \wedge (\vert x_i - x_j \vert \lt \epsilon)\}

is the set of all Cauchy nets with index set II and values in \mathbb{Q}. Let

CauchyNetsIn() IOb(DirectedSet 𝒰)C(I,)CauchyNetsIn(\mathbb{Q}) \coloneqq \bigcup_{I \in Ob(DirectedSet_\mathcal{U})} C(I, \mathbb{Q})

be the (large) set of all Cauchy nets in \mathbb{Q} in a universe 𝒰\mathcal{U}, where DirectedSet 𝒰DirectedSet_\mathcal{U} is the category of all directed sets in 𝒰\mathcal{U}.

Let the relation I,J\equiv_{I, J} in the Cartesian product C(I,)×C(J,)C(I, \mathbb{Q}) \times C(J, \mathbb{Q}) for directed sets II and JJ be defined as

a I,Jbϵ +,MI.iI.NJ.jJ.(iM)(jN)(|a ib j|<ϵ)a \equiv_{I, J} b \coloneqq \forall \epsilon \in \mathbb{Q}_{+}, \exists M \in I. \forall i \in I. \exists N \in J. \forall j \in J. (i \geq M) \wedge (j \geq N) \wedge (\vert a_i - b_j \vert \lt \epsilon)

Let a generalized Cauchy algebra be defined as a set AA with a function ιA CauchyNetsIn()\iota \in {A}^{CauchyNetsIn(\mathbb{Q})} such that

IOb(DirectedSet 𝒰).JOb(DirectedSet 𝒰).aC(I,).bC(J,).(a I,Jb)(ι(a)=ι(b))\forall I \in Ob(DirectedSet_\mathcal{U}). \forall J \in Ob(DirectedSet_\mathcal{U}). \forall a \in C(I, \mathbb{Q}). \forall b \in C(J, \mathbb{Q}). (a \equiv_{I, J} b) \implies (\iota(a) = \iota(b))

A generalized Cauchy algebra homomorphism is a function f:B Af:B^A between Cauchy algebras AA and BB such that

aC().f(ι A(a))=ι B(a)\forall a \in C(\mathbb{Q}). f(\iota_A(a)) = \iota_B(a)

The category of generalized Cauchy algebras is the category GCAlgGCAlg whose objects Ob(GCAlg)Ob(GCAlg) are generalized Cauchy algebras and whose morphisms Mor(GCAlg)Mor(GCAlg) are generalized Cauchy algebra homomorphisms. The set of generalized Cauchy real numbers, denoted \mathbb{R}, is defined as the initial object in the category of Cauchy algebras.

See also

Created on May 3, 2022 at 23:24:45. See the history of this page for a list of all contributions to it.