# Constructive Mathematics without Choice

```@inproceedings{Richman2001ConstructiveMW,
title={Constructive Mathematics without Choice},
author={Fred Richman},
year={2001}
}```
What becomes of constructive mathematics without the axiom of (countable) choice? Using illustrations from a variety of areas, it is argued that it becomes better.
27 Citations
• Mathematics
• 2018
We present a method for completing a premetric space, in the sense introduced by F. Richman in the context of constructive Mathematics without countable choice.
• Ulrich Berger
• Mathematics
Mathesis Universalis, Computability and Proof
• 2019
This essay describes an approach to constructive mathematics based on abstract i.e. axiomatic mathematics. Rather than insisting on structures to be explicitly constructed, constructivity is defined
• Mathematics
LFCS
• 2018
It is shown that several weakenings of the Cauchy condition are all equivalent under the assumption of countable choice, and to what extent choice is necessary.
This conjecture is substantiated by re-examining some basic tools of mathematical analysis from a choice-free constructive point of view, starting from Dedekind cuts as an appropriate notion of real numbers.
A constructive algebraic integration theory is presented that is constructive in the sense of Bishop, however it avoids the axiom of countable, or dependent, choice and can be interpreted in any topos.
• Mathematics
Math. Log. Q.
• 2018
Bishop's Lemma is a centrepiece in the development of constructive analysis. We show that 1.its proof requires some form of the axiom of choice; and that 2.the completeness requirement in Bishop's
Even after yet another grand conjecture has been proved or refuted, any omniscience principle that had trivially settled this question is just as little acceptable as before. The significance of the
• Mathematics
• 2010
AbstractTo prove Kronecker’s density theorem in Bishop-style constructive analysis one needs to define an irrational number as a real number that is bounded away from each rational number. In fact,