In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by Erdős & Tarski (1961); weakly compact cardinals are large cardinals, meaning that their existence can not be proven from the standard axioms of set theory.
Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function f: 2 → {0, 1} there is a set of cardinality κ that is homogeneous for f. In this context, 2 means the set of 2-element subsets of κ, and a subset S of κ is homogeneous for f if and only if either all of 2 maps to 0 or all of it maps to 1.
The name "weakly compact" refers to the fact that if a cardinal is weakly compact then a certain related infinitary language satisfies a version of the compactness theorem; see below.
Weakly compact cardinals are Mahlo cardinals, and the set of Mahlo cardinals less than a given weakly compact cardinal is stationary.
Some authors use a weaker definition of weakly compact cardinals, such as one of the conditions below with the condition of inaccessibility dropped.
Read more about Weakly Compact Cardinal: Equivalent Formulations
Famous quotes containing the words weakly, compact and/or cardinal:
“If a weakly mortal is to do anything in the world besides eat the bread thereof, there must be a determined subordination of the whole nature to the one aim—no trifling with time, which is passing, with strength which is only too limited.”
—Beatrice Potter Webb (1858–1943)
“What compact mean you to have with us?
Will you be pricked in number of our friends,
Or shall we on, and not depend on you?”
—William Shakespeare (1564–1616)
“The Poles do not know how to hate, thank God.”
—Stefan, Cardinal Wyszynski (1901–1981)