Algebra Seminar talk
2024-06-07
Farmer Schlutzenberg
Large cardinals, inner models and the Axiom of Choice
Abstract:
The usual axioms of mathematics (ZFC) are incomplete,
leaving even basic properties of simply definable sets of real numbers
undecided (for example, their Lebesgue measurability).
Large cardinal axioms posit the existence of "large" infinite sets with
transcendent properties. They form a natural hierarchy of extensions
of ZFC. A key insight, which arose about half a century ago, was that
large cardinals have profound implications for the nature of the real
numbers, deciding various questions left unanswered by ZFC. This
phenomenon is intertwined with the existence of canonical
structures, or inner models, which satisfy ZFC + large cardinal axioms.
These inner models are highly orderly, satisfying strong versions of
the Axiom of Choice and the Continuum Hypothesis, for example. But
in contrast to this, recent work suggests the coherence (in ZF without
Choice) of large cardinals so strong that they violate the Axiom of
Choice.