• Corpus ID: 125943205

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. 
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