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In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number such that for all functions , there exists a cardinal with and an elementary embedding from the Von Neumann universe into a transitive inner model with critical point and .
An equivalent definition is this: is Woodin if and only if is strongly inaccessible and for all there exists a which is --strong.
being --strong means that for all ordinals , there exist a which is an elementary embedding with critical point , , and . (See also strong cardinal.)
A Woodin cardinal is preceded by a stationary set of measurable cardinals, and thus it is a Mahlo cardinal. However, the first Woodin cardinal is not even weakly compact.
In set theory, a Woodincardinal (named for W. Hugh Woodin) is a cardinal number λ {\displaystyle \lambda } such that for all functions f : λ → λ {\displaystyle...
there is a Woodincardinal with a measurable cardinal above it, then Π12 determinacy holds. More generally, if there are n Woodincardinals with a measurable...
populated place in Coconino County Woodincardinal This disambiguation page lists articles associated with the title Woodin. If an internal link led you here...
measurable limit λ of both Woodincardinals and cardinals strong up to λ. If V has Woodincardinals but not cardinals strong past a Woodin one, then under appropriate...
Woodin, W. Hugh (2001). "The continuum hypothesis, part II". Notices of the American Mathematical Society. 48 (7): 681–690. "Large Cardinals and Determinacy"...
In set theory, Berkeley cardinals are certain large cardinals suggested by Hugh Woodin in a seminar at the University of California, Berkeley in about...
and Woodincardinals in consistency strength. However, the least strong cardinal is larger than the least superstrong cardinal. Every strong cardinal is...
2^{\kappa }>\kappa ^{+}} holds for every infinite cardinal κ {\displaystyle \kappa } . Later Woodin extended this by showing the consistency of 2 κ =...
John R. Steel and W. Hugh Woodin concluded a long line of research. Assuming the existence of some uncountable cardinal numbers analogous to ℵ0, they...
large cardinals suggested by Woodin. List of large cardinal properties Jensen, Ronald (1995), "Inner Models and Large Cardinals", The Bulletin of Symbolic...
large cardinal strength comes from PFA, and currently the best lower bound is a bit below the existence of a Woodincardinal that is a limit of Woodin cardinals...
cardinals (such as "ZFC plus there is a measurable cardinal", "ZFC plus there are infinitely many Woodincardinals") will prove that Grothendieck universes exist...
This box: view talk edit Borel hierarchy J. Steel, "What is... a Woodincardinal?". Notices of the American Mathematical Society vol. 54, no. 9 (2007)...
with ZFC. PD follows from certain large cardinal axioms, such as the existence of infinitely many Woodincardinals. PD implies that all projective sets are...
provided that a sufficiently strong large cardinal axiom is consistent (the existence of infinitely many Woodincardinals). Quine's system of axiomatic set theory...
calculus of partition sequences, changing cofinalities, and a question of Woodin", Transactions of the American Mathematical Society, 352 (3): 969–1003,...
model, one that contains all large cardinals. Woodin's Ω-conjecture: if there is a proper class of Woodincardinals, then Ω-logic satisfies an analogue...
has the same cardinality as x, and y ∈ KDJ. (If 0# does not exist, then KDJ = L.) If the core model K exists (and has no Woodincardinals), then If K has...
the University of California, Berkeley under the supervision of W. Hugh Woodin, with a dissertation entitled Lifting and Extending Measures by Forcing;...
reflection properties, on huge cardinals, on Vopěnka's principle, on extendible cardinals, on strong cardinals, and on Woodincardinals. The book concludes with...
MR 0281602 R. B. Jensen, Manuscript on fine structure, inner model theory, and the core model below one Woodincardinal (pp. 22--31). Accessed 2022-12-07...
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