Rowbottom cardinal explained
In set theory, a Rowbottom cardinal, introduced by, is a certain kind of large cardinal number.
is said to be
-Rowbottom if for every function
f: [κ]
<ω → λ (where λ < κ) there is a set
H of order type
that is quasi-
homogeneous for
f, i.e., for every
n, the
f-image of the set of
n-element subsets of
H has <
elements.
is
Rowbottom if it is
- Rowbottom.
Every Ramsey cardinal is Rowbottom, and 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, Rowbottom cardinals need not be large cardinals in the usual sense: Rowbottom cardinals could be singular. It is an open question whether ZFC + “
is Rowbottom” is consistent. If it is, it has much higher consistency strength than the existence of a Rowbottom cardinal. The
axiom of determinacy does imply that
is Rowbottom (but contradicts the
axiom of choice).
References
- Book: Kanamori, Akihiro. Akihiro Kanamori. 2003. Springer. The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. The Higher Infinite . 2nd. 3-540-00384-3.
