The proof of the reals being uncountable depends on the idea that one can build a number that depends on being able to make an infinite number of choices, each of which depends on the absolute truth or falseness of a statement.
But what happens if we open it up to have statements be true, false, or currently unknown? That is we develop a system of mathematics that could be in principle done inside of a Turing machine?
Then Cantor's diagonalization argument falls apart because of all of those "currently unknown" options. See https://news.ycombinator.com/item?id=13843725 for a previous explanation that I gave of this.
But what happens if we open it up to have statements be true, false, or currently unknown? That is we develop a system of mathematics that could be in principle done inside of a Turing machine?
Then Cantor's diagonalization argument falls apart because of all of those "currently unknown" options. See https://news.ycombinator.com/item?id=13843725 for a previous explanation that I gave of this.