The document discusses Godel's incompleteness theorems and Hilbert's program. It provides background on key figures like Hilbert, Godel, Russell and Cantor. It then explains Hilbert's program to formalize all of mathematics and prove its consistency. Godel showed that any theory capable of elementary arithmetic cannot be both consistent and complete. Specifically, for any formal theory T including basic arithmetic truths, T includes a statement of its own consistency if and only if T is inconsistent.
