In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a subcompact cardinal is a certain kind of
large cardinal
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least Î ...
number.
A
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
''κ'' is subcompact if and only if for every ''A'' ⊂ ''H''(''κ''
+) there is a non-trivial
elementary embedding In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences.
If ''N'' is a substructure of ''M'', one often ...
j:(''H''(''μ''
+), ''B'') → (''H''(''κ''
+), ''A'') (where ''H''(''κ''
+) is the set of all sets of
cardinality hereditarily less than ''κ''+) with
critical point ''μ'' and ''j''(''μ'') = ''κ''.
Analogously, ''κ'' is a quasicompact cardinal if and only if for every ''A'' ⊂ ''H''(''κ''
+) there is a non-trivial elementary embedding ''j'':(''H''(''κ''
+), ''A'') → (''H''(''μ''
+), ''B'') with critical point ''κ'' and ''j''(''κ'') = ''μ''.
''H''(''λ'') consists of all sets whose transitive closure has cardinality less than ''λ''.
Every quasicompact cardinal is subcompact. Quasicompactness is a strengthening of subcompactness in that it projects large cardinal properties upwards. The relationship is analogous to that of
extendible versus
supercompact cardinal In set theory, a supercompact cardinal is a type of large cardinal. They display a variety of reflection properties.
Formal definition
If ''λ'' is any ordinal, ''κ'' is ''λ''-supercompact means that there exists an elementary ...
s. Quasicompactness may be viewed as a strengthened or "boldface" version of 1-extendibility. Existence of subcompact cardinals implies existence of many 1-extendible cardinals, and hence many
superstrong cardinal In mathematics, a cardinal number κ is called superstrong if and only if there exists an elementary embedding ''j'' : ''V'' → ''M'' from ''V'' into a transitive inner model ''M'' with critical point κ and V_ ⊆ ''M''.
Similarl ...
s. Existence of a 2
''κ''-supercompact cardinal ''κ'' implies existence of many quasicompact cardinals.
Subcompact cardinals are noteworthy as the least large cardinals implying a failure of the
square principle In mathematical set theory, a square principle is a combinatorial principle asserting the existence of a cohering sequence of
short closed unbounded (club) sets so that no one (long) club set coheres with them all. As such they may be viewed as a ...
. If κ is subcompact, then the square principle fails at κ. Canonical inner models at the level of subcompact cardinals satisfy the square principle at all but subcompact cardinals. (Existence of such models has not yet been proved, but in any case the square principle can be forced for weaker cardinals.)
Quasicompactness is one of the strongest large cardinal properties that can be witnessed by current inner models that do not use long extenders. For current inner models, the elementary embeddings included are determined by their effect on ''P''(''κ'') (as computed at the stage the embedding is included), where κ is the critical point. This prevents them from witnessing even a ''κ''
+ strongly compact cardinal In set theory, a branch of mathematics, a strongly compact cardinal is a certain kind of large cardinal.
A cardinal κ is strongly compact if and only if every κ-complete filter can be extended to a κ-complete ultrafilter.
Strongly compact cardi ...
''κ''.
Subcompact and quasicompact cardinals were defined by
Ronald Jensen
Ronald Björn Jensen (born April 1, 1936) is an American mathematician who lives in Germany, primarily known for his work in mathematical logic and set theory.
Career
Jensen completed a BA in economics at American University in 1959, and a Ph.D. ...
.
References
*"Square in Core Models" in the September 2001 issue of the Bulletin of Symbolic Logic
Large cardinals
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