Comment by j-pb

The incompleteness proof comes to a russel paradox like contradiction, caused by a sentence of the form "I am not proovable." encoded via goedel numbering onto peano atrithmetic.

But proof by contradiction itself is problematic, because it relies on the law of the excluded middle, which only holds in classical two valued logic.

If you construct math from the top down rather than from the bottom up, then it is natural that it also has a multi valued logic.

In fact, it also would have infinitely valued logic. Infinite sentences, infinite theorems.

Such math is non expressible for us, because we rely on discrete descriptions.

But that doesn't preclude its existence.

Why should it though? What evidence do we have, if we can't express it or fit it into our existing mathematical framework.

To illustrate, think of Goedelsz approach and turn it backwards for a second. Instead of taking predicate logic and assigning each sentence a natural number, imagine that predicate/classical logic is a different view on the natural numbers. Now that means that there might also be logical interpretation of the reals, the hyperreals and so on. (We could do the same with Alephs, in fact they might be the more fundamental objects.)