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, choice and consistency should all be aligned.

中文翻译：

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数学怀疑的代价

本文认为，就我们怀疑连续统假设的二价性或选择公理的真实性而言，我们也应该怀疑三阶算术的一致性，无论是经典版本还是直觉版本。这一论点的基础是以下哲学观点。数学信念源于某些直觉，每个直觉都可以被完全接受或怀疑，但不能半信半疑。因此，我们对现实、二元性、选择和一致性的信念都应该保持一致。