[ska]

I think you are mixing up Godel (incompleteness of axiomatic systems) and Cantor (uncountable reals).

Timing wise Cantors proof was done the year before Lebesque was born; Cantor was a generation before Lebesgue, and Fourier a generation before that iirc. Godel's is a generation younger than Lebesgue. Lesbesgue and Borel were working at the same time - Godel was very young when he published his incompleteness theorem in the 1930s.