# Cantor-Bernstein implies Excluded Middle

@article{Pradic2019CantorBernsteinIE, title={Cantor-Bernstein implies Excluded Middle}, author={Pierre Pradic and Chad E. Brown}, journal={ArXiv}, year={2019}, volume={abs/1904.09193} }

We prove in constructive logic that the statement of the Cantor-Bernstein theorem implies excluded middle. This establishes that the Cantor-Bernstein theorem can only be proven assuming the full power of classical logic. The key ingredient is a theorem of Mart\'in Escard\'o stating that quantification over a particular subset of the Cantor space $2^{\mathbb{N}}$, the so-called one-point compactification of $\mathbb{N}$, preserves decidable predicates.

## 7 Citations

The Cantor–Schröder–Bernstein Theorem for $$\infty $$-groupoids

- Mathematics
- 2020

We show that the Cantor-Schroder-Bernstein Theorem for homotopy types, or $\infty$-groupoids holds in the following form: For any two types, if each one is embedded into the other, then they are…

The Cantor–Schröder–Bernstein Theorem for -groupoids

- Mathematics
- 2021

We show that the Cantor–Schröder–Bernstein Theorem for homotopy types, or ∞groupoids, holds in the following form: For any two types, if each one is embedded into the other, then they are equivalent.…

Computational Back-And-Forth Arguments in Constructive Type Theory

- MathematicsITP
- 2022

The back-and-forth method is a well-known technique to establish isomorphisms of countable structures. In this proof pearl, we formalise this method abstractly in the framework of constructive type…

Infinite Omniscient Sets in Constructive Mathematics

- Mathematics, Computer Science
- 2020

It is proved that N∞ the subset of all descending sequences in 2N, satisfies this omniscience principle and then it is shown how this can be generalised to many other subsets of 2N.

The generalised continuum hypothesis implies the axiom of choice in Coq

- EconomicsCPP
- 2021

Two Coq mechanisations of Sierpinski's result that the generalised continuum hypothesis (GCH) implies the axiom of choice (AC) are discussed and compared, concerning type-theoretic formulations of GCH and AC.

Countable sets versus sets that are countable in reverse mathematics

- MathematicsComput.
- 2022

The program Reverse Mathematics (RM for short) seeks to identify the axioms necessary to prove theorems of ordinary mathematics, usually working in the language of second-order arithmetic L 2 . A…

Division by Two, in Homotopy Type Theory

- MathematicsFSCD
- 2022

Natural numbers are isomorphism classes of finite sets and one can look for operations on sets which, after quotienting, allow recovering traditional arithmetic operations. Moreover, from a…

## References

SHOWING 1-8 OF 8 REFERENCES

The Cantor–Schröder–Bernstein Theorem for -groupoids

- Mathematics
- 2021

We show that the Cantor–Schröder–Bernstein Theorem for homotopy types, or ∞groupoids, holds in the following form: For any two types, if each one is embedded into the other, then they are equivalent.…

THOUGHTS ON THE CANTOR-BERNSTEIN THEOREM

- Mathematics
- 1986

The usual proofs of the well-known set-theoretical theorem “Given one-one maps f: A → B and g:B → A, there exists a one-one onto map h:A → B” actually produce a map h:A → B contained in the relation…

Infinite sets that Satisfy the Principle of Omniscience in any Variety of Constructive Mathematics

- MathematicsThe Journal of Symbolic Logic
- 2013

Abstract We show that there are plenty of infinite sets that satisfy the omniscience principle, in a minimalistic setting for constructive mathematics that is compatible with classical mathematics. A…

Set Theory

- Computer ScienceParadoxes and Inconsistent Mathematics
- 2021

The study of set algebra provides a solid background to understanding of probability and statistics, which are important business decision-making tools.

Foundations of Constructive Analysis

- Mathematics
- 2012

This article has no associated abstract. (fix it)

Proofs of the Cantor-Bernstein Theorem: A Mathematical Excursion

- Mathematics
- 2015

Preface. - Part I: Cantor and Dedekind.- Cantor's CBT proof for sets of the power of (II).- Generalizing Cantor's CBT proof.- CBT in Cantor's 1878 Beitrag.- The theory of inconsistent sets.-…