i think it's no coincidence Godel's proof of uncountable reals, comes around the same time as Lebesgue integration. As mathematicians started exploring what the serious use of Fourier series
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.
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.