Woodin Cardinal
In
mathematical logic
, a
Woodin cardinal
is a
cardinal number
κ such that for all
f
: κ → κ
there exists
α < κ with
f
α
⊆ α
and an
elementary embedding
j
:
V
→
M
from
V
into a
transitive inner model
M
with critical point α and
V
j(f)(α)
⊆
M
.
Woodin cardinals are important in
descriptive set theory
. Existence of infinitely many Woodin cardinals implies
projective determinacy
, which in turn implies that every projective set is
measurable
, has the
Baire property
(differs from an open set by a
meager set
, that is, a set which is a countable union of nowhere dense sets), and the
perfect subset property
(is either countable or contains a
perfect
subset).
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