Editing Constructive analysis (section)

====Cauchy reals====
In some frameworks of analysis, the name ''real numbers'' is given to such well-behaved sequences or rationals, and relations such as <math>x\cong y</math> are called the ''equality or real numbers''. Note, however, that there are properties which can distinguish between two <math>\cong</math>-related reals.

In contrast, in a set theory that models the naturals <math>{\mathbb N}</math> and validates the existence of even classically uncountable function spaces (and certainly [[Constructive set theory#Constructive Zermelo–Fraenkel|say <math>{\mathsf{CZF}}</math>]] or even <math>{\mathsf{ZFC}}</math>) the numbers equivalent with respect to "<math>\cong</math>" in <math>{\mathbb Q}^{\mathbb N}</math> may be collected into a set and then this is called the ''[[Construction of the real numbers#Construction from Cauchy sequences|Cauchy real number]]''. In that language, regular rational sequences are degraded to a mere representative of a Cauchy real. Equality of those reals is then given by the equality of sets, which is governed by the set theoretical [[axiom of extensionality]]. An upshot is that the set theory will prove properties for the reals, i.e. for this class of sets, expressed using the logical equality. Constructive reals in the presence of appropriate choice axioms will be Cauchy-complete but not automatically order-complete.<ref>Robert S. Lubarsky, [https://arxiv.org/pdf/1510.00639.pdf ''On the Cauchy Completeness of the Constructive Cauchy Reals''], July 2015</ref>