Homotopy Type Theory an axiomatization of the real numbers > history (Rev #6, changes)

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Idea

I am going to define this in terms of Archimedean ordered Q-algebras…

Definition

Commutative \mathbb{Q}-algebras

A commutative ring RR is a commutative \mathbb{Q}-algebra if there is a commutative ring homomorphism h:Rh:\mathbb{Q} \to R.

Totally ordered commutative rings

A commutative ring RR is a totally ordered commutative ring if it comes with a function max:R×RR\max:R \times R \to R such that

  • for all elements a:Ra:R, max(a,a)=a\max(a, a) = a

  • for all elements a:Ra:R and b:Rb:R, max(a,b)=max(b,a)\max(a, b) = \max(b, a)

  • for all elements a:Ra:R, b:Rb:R, and c:Rc:R, max(a,max(b,c))=max(max(a,b),c)\max(a, \max(b, c)) = \max(\max(a, b), c)

  • for all elements a:Ra:R and b:Rb:R, max(a,b)=b\max(a, b) = b implies that for all elements c:Rc:R, max(a+c,b+c)=b+c\max(a + c, b + c) = b + c

  • for all elements a:Ra:R and b:Rb:R, max(a,0)=a\max(a, 0) = a and max(b,0)=b\max(b, 0) = b implies max(ab,0)=ab\max(a \cdot b, 0) = a \cdot b

  • for all elements a:Ra:R and b:Rb:R, max(a,b)=a\max(a, b) = a or max(a,b)=b\max(a, b) = b

Totally ordered commutative \mathbb{Q}-algebras

Given totally ordered commutative rings RR and SS, a commutative ring homomorphism h:RSh:R \to S is monotonic if for all a:Ra:R and b:Rb:R, max(h(a),h(b))=h(max(a,b))\max(h(a), h(b)) = h(\max(a, b)).

A totally ordered commutative ring RR is a totally ordered commutative \mathbb{Q}-algebra if there is a monotonic commutative ring homomorphism h:Rh:\mathbb{Q} \to R.

Strictly ordered integral ring

A totally ordered commutative ring RR is a strictly ordered integral ring if it comes with a strict type order family<\lt such that

  • 0<10 \lt 1for all elements a:Ra:R and b:Rb:R, a<ba \lt b is a proposition
  • for all elements a:Ra:R , and b a : < R a b:R a \lt a , if is false0<a0 \lt a and 0<b0 \lt b, then 0<a+b0 \lt a + b
  • for all elements a:Ra:R , andb:Rb:R , if and0c < : a R 0 c:R \lt a , and if0a< b c 0 a \lt b c, then 0<a <b 0 \lt a \cdot \lt b or b<cb \lt c
  • for all elements a:Ra:R and b:Rb:R, if 0<max(a , <ba) 0 a \lt \max(a, b -a) is false and0<max(b , <ab) 0 b \lt \max(b, a -b) , is false, then0<max(a =b,ab) 0 a \lt = \max(a b \cdot b, -a \cdot b)
  • for all elements a:Ra:R and b:Rb:R, if a<ba \lt b, then b<ab \lt a is false.
  • 0<10 \lt 1
  • for all elements a:Ra:R and b:Rb:R, if 0<a0 \lt a and 0<b0 \lt b, then 0<a+b0 \lt a + b
  • for all elements a:Ra:R and b:Rb:R, if 0<a0 \lt a and 0<b0 \lt b, then 0<ab0 \lt a \cdot b
  • for all elements a:Ra:R and b:Rb:R, if 0<max(a,a)0 \lt \max(a, -a) and 0<max(b,b)0 \lt \max(b, -b), then 0<max(ab,ab)0 \lt \max(a \cdot b, -a \cdot b)

Strictly ordered integral \mathbb{Q}-algebras

Given strictly ordered integral rings RR and SS, a monotonic commutative ring homomorphism h:RSh:R \to S is strictly monotonic if for all a:Ra:R and b:Rb:R, a<ba \lt b implies h(a)<h(b)h(a) \lt h(b).

A strictly ordered integral ring RR is a strictly ordered integral \mathbb{Q}-algebra if there is a strictly monotonic commutative ring homomorphism h:Rh:\mathbb{Q} \to R.

Archimedean ordered integral \mathbb{Q}-algebras

A strictly ordered integral \mathbb{Q}-algebra AA is an Archimedean ordered integral \mathbb{Q}-algebra if for all elements a:Aa:A and b:Ab:A, if a<ba \lt b, then there merely exists a rational number q:q:\mathbb{Q} such that a<h(q)a \lt h(q) and h(q)<bh(q) \lt b.

Sequentially Cauchy complete Archimedean ordered integral \mathbb{Q}-algebra

Let AA be an Archimedean ordered integral \mathbb{Q}-algebra and let

A + a:A0<aA_{+} \coloneqq \sum_{a:A} 0 \lt a

be the positive elements in AA. AA is sequentially Cauchy complete if every Cauchy sequence in AA converges:

isCauchy(x)ϵA +.NI.iI.jI.(iN)(jN)(|x ix j|<ϵ)isCauchy(x) \coloneqq \forall \epsilon \in A_{+}. \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)
isLimit(x,l)ϵA +.NI.iI.(iN)(|x il|<ϵ)isLimit(x, l) \coloneqq \forall \epsilon \in A_{+}. \exists N \in I. \forall i \in I. (i \geq N) \to (\vert x_i - l \vert \lt \epsilon)
x:A.isCauchy(x)lA.isLimit(x,l)\forall x: \mathbb{N} \to A. isCauchy(x) \wedge \exists l \in A. isLimit(x, l)

See also

Revision on June 16, 2022 at 23:49:57 by Anonymous?. See the history of this page for a list of all contributions to it.