Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...

without the

axiom of choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...

, a strong partition cardinal is an

uncountable In mathematics, an uncountable set, informally, 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 number is larger tha ...

well-ordered In mathematics, a well-order (or well-ordering or well-order relation) on a set is a total ordering on with the property that every non-empty subset of has a least element in this ordering. The set together with the ordering is then called a ...

cardinal Cardinal or The Cardinal most commonly refers to * Cardinalidae, a family of North and South American birds **''Cardinalis'', genus of three species in the family Cardinalidae ***Northern cardinal, ''Cardinalis cardinalis'', the common cardinal of ...

k

such that every partition of the set

k

of size

k

subsets of

k

into less than

k

pieces has a homogeneous set of size

k

. The existence of strong partition cardinals contradicts the axiom of choice. 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 o ...

implies that ℵ₁ is a strong partition cardinal.

References

* *. Cardinal numbers {{settheory-stub