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 ...

, an unfoldable 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 � ...

number. Formally, 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 λ-unfoldable if and only if for every

transitive model In mathematical set theory, a transitive model is a model of set theory that is standard and transitive. Standard means that the membership relation is the usual one, and transitive means that the model is a transitive set or class. Examples *An i ...

''M'' of cardinality κ of ZFC-minus-

power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...

such that κ is in ''M'' and ''M'' contains all its sequences of length less than 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'' of ''M'' into a transitive model with the critical point of ''j'' being κ and ''j''(κ) ≥ λ. A cardinal is unfoldable if and only if it is an λ-unfoldable for all ordinals λ. A

κ is strongly λ-unfoldable if and only if for every

''M'' of cardinality κ of ZFC-minus-

such that κ is in ''M'' and ''M'' contains all its sequences of length less than κ, there is a non-trivial

''j'' of ''M'' into a transitive model "N" with the critical point of ''j'' being κ, ''j''(κ) ≥ λ, and V(λ) is a subset of ''N''.

Without loss of generality ''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicate ...

, we can demand also that ''N'' contains all its sequences of length λ. Likewise, a cardinal is strongly unfoldable if and only if it is strongly λ-unfoldable for all λ. These properties are essentially weaker versions of

strong Strong may refer to: Education * The Strong, an educational institution in Rochester, New York, United States * Strong Hall (Lawrence, Kansas), an administrative hall of the University of Kansas * Strong School, New Haven, Connecticut, United Sta ...

and supercompact cardinals, consistent with

V = L The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as ''V'' = ''L'', where ''V'' and ''L'' denote the von Neumann universe and the constructib ...

. Many theorems related to these cardinals have generalizations to their unfoldable or strongly unfoldable counterparts. For example, the existence of a strongly unfoldable implies the consistency of a slightly weaker version of the

proper forcing axiom In the mathematical field of set theory, the proper forcing axiom (''PFA'') is a significant strengthening of Martin's axiom, where forcings with the countable chain condition (ccc) are replaced by proper forcings. Statement A forcing or parti ...

Relations between large cardinal properties

Assuming V = L, the least unfoldable cardinal is greater than the least indescribable cardinal.^p.14 Assuming a Ramsey cardinal exists, it is less than the least Ramsey cardinal.^p.3 A

Ramsey cardinal In mathematics, a Ramsey cardinal is a certain kind of large cardinal number introduced by and named after Frank P. Ramsey, whose theorem establishes that ω enjoys a certain property that Ramsey cardinals generalize to the uncountable case. Let ...

is unfoldable and will be strongly unfoldable in L. It may fail to be strongly unfoldable in V, however. In L, any unfoldable cardinal is strongly unfoldable; thus unfoldable and strongly unfoldable have the same

consistency strength In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In this case, they are, roughly speaking, "as consistent as each other". In general, it is not po ...

. A cardinal k is κ-strongly unfoldable, and κ-unfoldable, if and only if it is weakly compact. A κ+ω-unfoldable cardinal is indescribable and preceded by a stationary set of totally indescribable cardinals.

References

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Citations

Large cardinals {{settheory-stub