  We present a coherent collection of finite mathematical theorems some of
which can only be proved by going well beyond the usual axioms for mathematics.
The proofs of these theorems illustrate in clear terms how one uses the well
studied higher infinities of abstract set theory called large cardinals in an
essential way in order to derive results in the context of the natural numbers.
The findings raise the specific issue of what consitutes a valid mathematical
proof and the general issue of objectivity in mathematics in a down to earth
way.
  Large cardinal axioms, which go beyond the usual axioms for mathematics, have
been commonly used in abstract set theory since the 1960's. We believe that the
results reported on here are the early stages of an evolutionary process in
which new axioms for mathematics will be commonly used in an essential way in
the more concrete parts of mathematics.
