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 ...

, subtle cardinals and ethereal cardinals are closely related kinds 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. A cardinal ''κ'' is called subtle if for every closed and unbounded ''C'' ⊂ ''κ'' and for every sequence ''A'' of length ''κ'' for which element number ''δ'' (for an arbitrary ''δ''), ''A''_''δ'' ⊂ ''δ'', there exist ''α'', ''β'', belonging to ''C'', with ''α'' < ''β'', such that ''A''_''α'' = ''A''_''β'' ∩ ''α''. A cardinal ''κ'' is called ethereal if for every closed and unbounded ''C'' ⊂ ''κ'' and for every sequence ''A'' of length ''κ'' for which element number ''δ'' (for an arbitrary ''δ''), ''A''_''δ'' ⊂ ''δ'' and ''A''_''δ'' has the same cardinal as ''δ'', there exist ''α'', ''β'', belonging to ''C'', with ''α'' < ''β'', such that card(''α'') = card(''A''_''β'' ∩ ''A''_''α''). Subtle cardinals were introduced by . Ethereal cardinals were introduced by . Any subtle cardinal is ethereal, and any strongly inaccessible ethereal cardinal is subtle.

Theorem

There is a subtle cardinal ≤ ''κ'' if and only if every transitive set ''S'' of cardinality ''κ'' contains ''x'' and ''y'' such that ''x'' is a proper subset of ''y'' and ''x'' ≠ Ø and ''x'' ≠ . An infinite ordinal ''κ'' is subtle if and only if for every ''λ'' < ''κ'', every transitive set ''S'' of cardinality ''κ'' includes a chain (under inclusion) of order type ''λ''.

References

* * * Large cardinals {{settheory-stub

Theorem

See also

References