Incommensurability of Mathematics

Posy, C., 1992, noted that commensurability is false (found irrational numbers); if commensurability were true, (1+1)=2 must equal p/q where p and q are natural numbers. Proof that Commensurability is False (first recorded use of reductio ad absurdum proof):
1) Assume 2=p/q, p and q {set of natural numbers}
2) Reduce p/q to lowest terms (can be done with any fraction)
3) p=q2 p2=2q2
4) Thus, p2 is an even number (it is equal to twice an integer)
5) p2 = p is also even (all even sqaures' roots are even)
6) So, p=2k, k {set of natural numbers}
7) (2k)2=2q2 4k2=2q2 2k2=q2
8) Thus, q2 is even, so q is even
9) Both q and p are even, thus they share the common factor of 2; this however, contradicts step #2, that p/q is reduced to lowest terms, thus our assumption that 2=p/q is false.