To be fair - there are statements in math that are independent of the axioms. For those statements, the universe you find yourself in can pick either version (true OR false) and still be consistent.

See also: noneuclidian geometry and axiom of choice.

Well if the fundamental constants or hidden variables of the universe are shifting because of his comment then it can change the outcome.

Depends how fundamental the variables are. If we can code a sim for a topos[1], why can’t we be in such a sim?

1. https://arxiv.org/pdf/1012.5647

unless mechanism behind our universe dynamically alters our logic on the fly to be artificially self-consistent