Using the axiom of choice one can construct set of reals which are
pathological in some sense. Similar constructions can be produce such
"pathological" subsets of any non trivial Polish space (= a complete
separable metric space).
A "pathological set" can be a non measurable set , a set which does
not have the property of Baire (namely it is not a Borel set modulo a
rst category set).
A subset of the innite subsets of natural numbers,
can be considered to be "pathological" if it is a counter example to
innitary Ramsey theorem. Namely there does not exists an innite
set of natural numbers such that all it subsets are in our sets or all its
innite subsets are not in the set.
A subset of the Baire space A can be considered to be "patho-
logical" if the innite game GA is not determined. The game GA is
an innite game where two players alternate picking natural numbers,
forming an infinite sequence of natural numbers. The first
player wins the round if the resulting sequence is in A. The game is
determined if one of the players has a winning strategy.
A prevailing paradigm in Descriptive Set Theory is that sets that
has a "simple description" should not be pathological. Evidence for
this maxim is the fact that Borel sets are not pathological in any of the
senses described above.In these talks we shall present a notion of "super
regularity" for subsets of a Polish space, the family of universally Baire
sets. This family of sets generalizes the family of Borel sets and form
a sigma algebra. We shall prove some regularity properties of the family of
universally Baire sets , such as their measurability with respect to any
regular Borel measure, the fact that they have an innitary Ramsey
property etc.
Some of these proofs require the use of large cardinals. A typical
example is the closure of universally Baire sets under continuous images
Though we shall not be able to provide the full set theoretic basis for
these facts , we shall try to explain how the assumption of the existence
of large cardinals can have a deep impact on the structure of denable
sets of reals and of other Polish spaces.