We suggest a new approach for
addressing the problem of establishing an axiomatic foundation
for large cardinals. An axiom asserting the existence of a large cardinal
can naturally be viewed as a strong Axiom of Infinity. However,
it has not been clear on the basis of our knowledge of ω itself,
or of generally agreed upon intuitions
about the true nature of the mathematical universe, what the right
strengthening of the Axiom of Infinity is—which large cardinals
ought to be derivable? It was shown in the 1960s by Lawvere that the
existence of an infinite set is equivalent to the existence of
a certain kind of structure-preserving transformation from V to itself,
not isomorphic to the identity.
We use Lawvere's transformation, rather than ω, as a starting
point for a reasonably natural sequence of strengthenings and refinements,
leading to a proposed strong Axiom of Infinity. A first refinement
was discussed in later work by Trnková—Blass, showing that
if the preservation properties of Lawvere's tranformation are strengthened
to the point of requiring it to be an exact functor, such a transformation is provably
equivalent to the existence of a measurable cardinal. We propose to push
the preservation properties as far as possible, short of inconsistency.
The resulting transformation V→V is strong enough to account for
virtually all large cardinals, but is at the same time a natural generalization
of an assertion about transformations V→V known to be equivalent
to the Axiom of Infinity.