Doing Number Theory in Weak Systems of Arithmetic

摘要

Although Godel's Theorem shows that even ZFC is incomplete for unsolvability of diophantine equations, nothing explicit of any real interest to number theorists has ever been shown to be unprovable. I will consider various important statements about solvability modulo all prime powers, and exhibit a wide class which get decided by PA (first order Peano Arithmetic) using serious algebraic geometry inside nonstandard models of PA. So although PA is often misrepresented as very weak, it is rather strong for basic results of 20th century number theory.