Jónsson cardinal

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In set theory, a Jónsson cardinal (named after Bjarni Jónsson) is a certain kind of large cardinal number.

An uncountable cardinal number κ is said to be Jónsson if for every function f: [κ] < ω → κ there is a set H of order type κ such that for each n, f restricted to n-element subsets of H omits at least one value in κ.

Every Rowbottom cardinal is Jónsson. By a theorem of Kleinberg, the theories ZFC+“there is a Rowbottom cardinal” and ZFC+“there is a Jónsson cardinal” are equiconsistent.

In general, Jónsson cardinals need not be large cardinals in the usual sense: they can be singular. Using the axiom of choice, a lot of small cardinals (the \aleph_n, for instance) can be proved to be not Jónsson. Results like this need the axiom of choice, however: The axiom of determinacy does imply that for every positive natural number n, the cardinal \aleph_n is Jónsson.

[edit] References

  • Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings, 2nd ed, Springer. ISBN 3540003843.