I think you are missing the point. Kronecker does not deny that irrational numbers exist, he states his belief that the fundamental number system which is inher

I think you are missing the point. Kronecker does not deny that irrational numbers exist, he states his belief that the fundamental number system which is inherent to reality are the natural numbers. That (by divine authority) are the irreducable thing which other number system must take from. E.g. the rationals are pairs of natural numbers, the reals cauchy sequences of rational numbers etc. > I'm sure there are other such proofs. The reals are uncountable.