In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, a Rowbottom cardinal, introduced by , 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.
An
uncountable
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
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 ...
is said to be ''
- Rowbottom'' if for every function ''f'':
kappa;sup><ω → λ (where λ < κ) there is a set ''H'' of order type
that is quasi-
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
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 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 Rowbottom, and every Rowbottom cardinal is
Jónsson Jónsson is a surname of Icelandic origin, meaning ''son of Jón''. In Icelandic names, the name is not strictly a surname, but a patronymic. The name refers to:
* Arnar Jónsson (actor) (born 1943), Icelandic actor
* Arnar Jónsson (basketball) (bo ...
. 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 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 ...
s in the usual sense: Rowbottom cardinals could be
singular
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular homology
* SINGULAR, an open source Computer Algebra System (CAS)
* Singular or sounder, a group of boar, ...
. 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
In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of ...
does imply that
is Rowbottom (but contradicts the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
).
References
*
*
Large cardinals
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