This paper argues that, insofar as we doubt the bivalence of the
Continuum Hypothesis or the truth of the Axiom of Choice, we should
also doubt the consistency of third-order arithmetic, both the
classical and intuitionistic versions.
Underlying this argument is the following philosophical view.
Mathematical belief springs from certain intuitions, each of which
can be either accepted or doubted in its entirety, but not
half-accepted. Therefore, our beliefs about reality, bivalence,
c…
Read moreThis paper argues that, insofar as we doubt the bivalence of the
Continuum Hypothesis or the truth of the Axiom of Choice, we should
also doubt the consistency of third-order arithmetic, both the
classical and intuitionistic versions.
Underlying this argument is the following philosophical view.
Mathematical belief springs from certain intuitions, each of which
can be either accepted or doubted in its entirety, but not
half-accepted. Therefore, our beliefs about reality, bivalence,
choice and consistency should all be aligned.