The Yale Undergraduate Research Journal


This paper analyzes two natural-looking arguments that seek to leverage Gödel’s first incompleteness theorem for and against intuitionism, concluding in both cases that the argument is unsound because it equivocates on the meaning of “proof,” which differs between formalism and intuitionism. I argue that this difference explains why “proof” has definite extension for the formalist but not for the intuitionist. Sections 1-3 summarize various philosophies of mathematics and Gödel’s result. Section 4 argues that, because the Gödel sentence of a formal system is provable to the intuitionist, they are neither aided nor attacked by Gödel’s first incompleteness theorem. Section 5 concludes that the intuitionist’s notion of proof is indefinitely extensible.