There is a lot of discussion of this subject, but I think we are actually gaining progress at these miraclous years by with lead of our wild nearly outside-of-the-box thinking professor.

I have rethought his message and the problem of real numbers, are in fact, the pure rational numbers he himself praises, which I personlly call fake-numbers. A pure rational number is, in fact a pair of natural numbers glued together with ill-defined operator "÷" when the result is not a single natural number. Let us think simple and classical example of square root of two. sqrt(2)=(2)^(1÷2). Number triplet 2,1,2 are well defined. "^" is also well defined forward operand in natural numbers, with only exemption of 0^0 (which actually is becouse of ÷ operator, cannot divide by zero, 0^0=0^(1-1)=0÷0.). The only possible breaking of logic then lies solely in the operator "÷". It produces a backward opereting inverse problem. A never ending algorithm.

Tl;dr: ÷ is as fake as sqrt if the result doesn't product complete and one single natural number.