• Corpus ID: 125943205

# Cantor-Bernstein implies Excluded Middle

@article{Pradic2019CantorBernsteinIE,
title={Cantor-Bernstein implies Excluded Middle},
journal={ArXiv},
year={2019},
volume={abs/1904.09193}
}
• Published 19 April 2019
• Mathematics
• ArXiv
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.
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## References

SHOWING 1-8 OF 8 REFERENCES
The Cantor–Schröder–Bernstein Theorem for -groupoids
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
• M. Escardó
• Mathematics
The 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
• F. Stephan
• Computer Science