Constructive Mathematics without Choice

  title={Constructive Mathematics without Choice},
  author={Fred Richman},
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. 

Completion of premetric spaces

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.

On the Constructive and Computational Content of Abstract Mathematics

  • 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

Notions of Cauchyness and Metastability

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.

Elementary Choiceless Constructive Analysis

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.

Constructive algebraic integration theory without choice

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.

Bishop's Lemma

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

Unique existence, approximate solutions, and countable choice

Too simple solutions of hard problems

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

Kronecker’s density theorem and irrational numbers in constructive reverse mathematics

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,



The fundamental theorem of algebra: a constructive development without choice

Is it reasonable to do constructive mathematics without the axiom of countable choice? Serious schools of constructive mathematics all assume it one way or another, but the arguments for it are not

A weak countable choice principle

A weak choice principle is introduced that is implied by both countable choice and the law of excluded middle. This principle suffices to prove that metric independence is the same as linear

Constructing roots of polynomials over the complex numbers

Constructive proofs of the Fundamental Theorem of Algebra are known since 1924, when L. E. J. Brouwer, B. de Loor, and H. Weyl showed that nonconstant monic polynomials over the complex numbers have

A Course in Constructive Algebra

I. Sets.- 1. Constructive vs. classical mathematics.- 2. Sets, subsets and functions.- 3. Choice.- 4. Categories.- 5. Partially ordered sets and lattices.- 6. Well-founded sets and ordinals.- Notes.-

Flat dimension and the Hilbert syzygy theorem

  • New Zealand J. Math
  • 1997