In
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 ...
, a Ramsey 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 Î ...
number introduced by and named after
Frank P. Ramsey
Frank Plumpton Ramsey (; 22 February 1903 – 19 January 1930) was a British philosopher, mathematician, and economist who made major contributions to all three fields before his death at the age of 26. He was a close friend of Ludwig Wittgenste ...
, whose
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
establishes that
ω enjoys a certain property that Ramsey cardinals generalize to the
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 ...
case.
Let
'κ''sup><ω denote the set of all finite subsets of ''κ''. 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 called Ramsey if, for every function
:''f'':
'κ''sup><ω →
there is a set ''A'' of cardinality ''κ'' that is
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''. That is, for every ''n'', the function ''f'' is
constant on the subsets of cardinality ''n'' from ''A''. A cardinal ''κ'' is called ineffably Ramsey if ''A'' can be chosen to be a
stationary subset of ''κ''. A cardinal ''κ'' is called virtually Ramsey if for every function
:''f'':
'κ''sup><ω →
there is ''C'', a closed and unbounded subset of ''κ'', so that for every ''λ'' in ''C'' of uncountable
cofinality
In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''.
This definition of cofinality relies on the axiom of choice, as it uses the ...
, there is an unbounded subset of ''λ'' that is homogenous for ''f''; slightly weaker is the notion of almost Ramsey where homogenous sets for ''f'' are required of order type ''λ'', for every ''λ'' < ''κ''.
The existence of any of these kinds of Ramsey cardinal is sufficient to prove the existence of
0#, or indeed that every set with
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
less than ''κ'' has a
sharp
Sharp or SHARP may refer to:
Acronyms
* SHARP (helmet ratings) (Safety Helmet Assessment and Rating Programme), a British motorcycle helmet safety rating scheme
* Self Help Addiction Recovery Program, a charitable organisation founded in 19 ...
.
Every
measurable cardinal
In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal , or more generally on any set. For a cardinal , it can be described as a subdivisio ...
is a Ramsey cardinal, and every Ramsey cardinal is a
Rowbottom cardinal In set theory, a Rowbottom cardinal, introduced by , is a certain kind of large cardinal number.
An uncountable cardinal number \kappa is said to be ''\lambda- Rowbottom'' if for every function ''f'': kappa;sup><ω → λ (whe ...
.
A property intermediate in strength between Ramseyness and
measurability is existence of a ''κ''-complete normal non-principal
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
''I'' on ''κ'' such that for every and for every function
:''f'':
'κ''sup><ω →
there is a set ''B'' ⊂ ''A'' not in ''I'' that is homogeneous for ''f''. This is strictly stronger than ''κ'' being ineffably Ramsey.
The existence of a Ramsey cardinal implies the existence of
0# and this in turn implies the falsity of the
Axiom of Constructibility
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 construc ...
of
Kurt Gödel
Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imme ...
.
References
*
*
*
Large cardinals
Ramsey theory
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