Mathematics and Set Theory:: A Perspective from Gödel's Speedup Theorem

Since Gödel’s Incompleteness Theorems were published in 1931, not a few mathematicians have been trying to do mathematics in a framework as weak as possible to remain in a “safe” terrain. While the Incompleteness Theorems do not offer any direct motivations for exploring the terrae incognitae of the alarmingly general and consistency-wise strong settings like the full ZFC or even ZFC with large large cardinals etc., Gödel’s Speedup Theorem, a sort of a variant of the Incompleteness Theorems, in contrast, seems to provide positive reasons for studying mathematics in these powerful extended frameworks in spite of the peril called the (in)consistency strength. In this article of purely expository character, we will examine a version of the Speedup Theorem with a detailed proof and discuss the impact of the Speedup Theorem on the whole mathematics.