Strongly compact cardinal
Encyclopedia
In mathematical
set theory
, a strongly compact cardinal is a certain kind of large cardinal number
; their existence can neither be proven nor disproven from the standard axioms of set theory.
A cardinal κ is strongly compact if and only if every κ-complete filter can be extended to a κ complete ultrafilter.
Strongly compact cardinals were originally defined in terms of infinitary logic
, where logical operators are allowed to take infinitely many operands. The logic on a regular cardinal
κ is defined by requiring the number of operands for each operator to be less than κ; then κ is strongly compact if its logic satisfies an analog of the compactness
property of finitary logic.
Specifically, a statement which follows from some other collection of statements should also follow from some subcollection having cardinality less than κ.
The property of strong compactness may be weakened by only requiring this compactness property to hold when the original collection of statements has cardinality below a certain cardinal λ; we may then refer to λ-compactness. A cardinal is weakly compact
if and only if it is κ-compact; this was the original definition of that concept.
Strong compactness implies measurability
, and is implied by supercompactness. Given that the relevant cardinals exist, it is consistent with ZFC either that the first measurable cardinal is strongly compact, or that the first strongly compact cardinal is supercompact; these cannot both be true, however. A measurable limit of strongly compact cardinals is strongly compact, but the least such limit is not supercompact.
The consistency strength of strong compactness is strictly above that of a Woodin cardinal
. Some set theorists conjecture that existence of a strongly compact cardinal is equiconsistent with that of a supercompact cardinal. However, a proof is unlikely until a canonical inner model theory for supercompact cardinals is developed.
Extendibility
is a second-order analog of strong compactness.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, a strongly compact cardinal is a certain kind of large cardinal number
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
; their existence can neither be proven nor disproven from the standard axioms of set theory.
A cardinal κ is strongly compact if and only if every κ-complete filter can be extended to a κ complete ultrafilter.
Strongly compact cardinals were originally defined in terms of infinitary logic
Infinitary logic
An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. Some infinitary logics may have different properties from those of standard first-order logic. In particular, infinitary logics may fail to be compact or complete. Notions of compactness and...
, where logical operators are allowed to take infinitely many operands. The logic on a regular cardinal
Regular cardinal
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. So, crudely speaking, a regular cardinal is one which cannot be broken into a smaller collection of smaller parts....
κ is defined by requiring the number of operands for each operator to be less than κ; then κ is strongly compact if its logic satisfies an analog of the compactness
Compactness theorem
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model...
property of finitary logic.
Specifically, a statement which follows from some other collection of statements should also follow from some subcollection having cardinality less than κ.
The property of strong compactness may be weakened by only requiring this compactness property to hold when the original collection of statements has cardinality below a certain cardinal λ; we may then refer to λ-compactness. A cardinal is weakly compact
Weakly compact cardinal
In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; weakly compact cardinals are large cardinals, meaning that their existence can not be proven from the standard axioms of set theory....
if and only if it is κ-compact; this was the original definition of that concept.
Strong compactness implies measurability
Measurable cardinal
- Measurable :Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ...
, and is implied by supercompactness. Given that the relevant cardinals exist, it is consistent with ZFC either that the first measurable cardinal is strongly compact, or that the first strongly compact cardinal is supercompact; these cannot both be true, however. A measurable limit of strongly compact cardinals is strongly compact, but the least such limit is not supercompact.
The consistency strength of strong compactness is strictly above that of a Woodin cardinal
Woodin cardinal
In set theory, a Woodin cardinal is a cardinal number λ such that for all functionsthere exists a cardinal κ In set theory, a Woodin cardinal is a cardinal number λ such that for all functions...
. Some set theorists conjecture that existence of a strongly compact cardinal is equiconsistent with that of a supercompact cardinal. However, a proof is unlikely until a canonical inner model theory for supercompact cardinals is developed.
Extendibility
Extendible cardinal
In mathematics, extendible cardinals are large cardinals introduced by , who was partly motivated by reflection principles. A cardinal number κ is called η-extendible if for some λ there is a nontrivial elementary embedding j ofinto...
is a second-order analog of strong compactness.