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 Jónsson cardinal (named after

Bjarni Jónsson Bjarni Jónsson (February 15, 1920 – September 30, 2016) was an Icelandic mathematician and logician working in universal algebra, lattice theory, model theory and set theory. He was emeritus distinguished professor of mathematics at Vanderbilt ...

) 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 ...

κ is said to be ''Jónsson'' if for every function ''f'': �sup><ω → κ there is a set ''H'' of order type κ such that for each ''n'', ''f'' restricted to ''n''-element subsets of ''H'' omits at least one value in κ. Every

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 ...

is Jónsson. By a theorem of Eugene M. Kleinberg, the theories ZFC + “there is a

” and ZFC + “there is a Jónsson cardinal” are equiconsistent.

William Mitchell William Mitchell may refer to: People Media and the arts * William Mitchell (sculptor) (1925–2020), English sculptor and muralist * William Frederick Mitchell (1845–1914), British naval artist * William M. Mitchell, American writer, ministe ...

proved, with the help of the Dodd-Jensen

core model In set theory, the core model is a definable inner model of the von Neumann universe, universe of all Set (mathematics), sets. Even though set theorists refer to "the core model", it is not a uniquely identified mathematical object. Rather, it is a ...

that the consistency of the existence of a Jónsson cardinal implies the consistency of the existence of a

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 ...

, so that the existence of Jónsson cardinals and the existence of Ramsey cardinals are equiconsistent.Mitchell, William J.: "Jonsson Cardinals, Erdos Cardinals and the Core Model", Journal of Symbolic Logic 64(3):1065-1086, 1999. In general, Jónsson cardinals need not be large cardinals in the usual sense: they can 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, ...

. But the existence of a singular Jónsson cardinal is equiconsistent to the existence of a

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 ...

. Using 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 ...

, a lot of small cardinals (the

\aleph_n

, for instance) can be proved to be not Jónsson. Results like this need the axiom of choice, however: 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 for every positive natural number ''n'', the cardinal

\aleph_n

is Jónsson. A Jónsson algebra is an algebra with no proper subalgebras of the same cardinality. (They are unrelated to

Jónsson–Tarski algebra In mathematics, a Jónsson–Tarski algebra or Cantor algebra is an algebraic structure encoding a bijection from an infinite set onto the Cartesian product, product . They were introduced by . , named them after Georg Cantor because of Cantor's ...

s). Here an algebra means a model for a language with a countable number of function symbols, in other words a set with a countable number of functions from finite products of the set to itself. A cardinal is a Jónsson cardinal if and only if there are no Jónsson algebras of that cardinality. The existence of Jónsson functions shows that if algebras are allowed to have infinitary operations, then there are no analogues of Jónsson cardinals.

References

* * Large cardinals {{settheory-stub