mathematics, limit cardinals are certain cardinal numbers. A cardinal number λ is a weak limit cardinal if λ is neither a successor cardinal nor zero. This...

set theory, a cardinal number is a strongly inaccessible cardinal if it is uncountable, regular, and a strong limit cardinal. A cardinal is a weakly inaccessible...

a limit ordinal. Using von Neumann cardinal assignment, every infinite cardinal number is also a limit ordinal. Various other ways to define limit ordinals...

a successor cardinal. Cardinals that are not successor cardinals are called limit cardinals; and by the above definition, if λ is a limit ordinal, then...

means "strongly Mahlo cardinal", though the cardinals originally considered by Mahlo were weakly Mahlo cardinals. If κ is a limit ordinal and the set of...

successor cardinal or a limit cardinal. Some cardinalities cannot be proven to be equal to any particular aleph, for instance the cardinality of the continuum...

is a strong limit cardinal, which completes the proof of its inaccessibility. Although it follows from ZFC that every measurable cardinal is inaccessible...

theory, a worldly cardinal is a cardinal κ such that the rank Vκ is a model of Zermelo–Fraenkel set theory. A strong limit cardinal κ is worldly if and...

limit 1.  A (weak) limit cardinal is a cardinal, usually assumed to be nonzero, that is not the successor κ+ of another cardinal κ 2.  A strong limit...

cardinal deacons, 50 cardinal priests, and six cardinal bishops. Popes respected that limit until Pope John XXIII increased the number of cardinals several...

the two lower orders of cardinals, besides having all six such cardinals being over the age limit for a conclave. Each cardinal is assigned a titular church...

be a singular cardinal. According to Mitchell (1992), the singular cardinals hypothesis is: If κ is any singular strong limit cardinal, then 2κ = κ+....

uncountable strong limit cardinal is not satisfied in such a model; thus the existence of ℶω (the smallest uncountable strong limit cardinal) cannot be proved...

{\displaystyle \lambda } is a limit ordinal. The cardinal ℶ 0 = ℵ 0 {\displaystyle \beth _{0}=\aleph _{0}} is the cardinality of any countably infinite set...

theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They were introduced by the mathematician...

successor cardinal or a limit cardinal, and either a regular cardinal or a singular cardinal. A great many sets studied in mathematics have cardinality equal...

\omega _{n}} for natural numbers n (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the ω n {\displaystyle \omega...

principle at a singular strong limit cardinal—in fact, at all singular cardinals and all regular successor cardinals—it implies that the axiom of determinacy...

larger than all finite numbers. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the...

 2005–2013) created 90 cardinals in five consistories. With three of those consistories, he respected the limit on the number of cardinal electors set at 120...

natural numbers n {\displaystyle n} (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the ω n {\displaystyle \omega...

In set theory, a strongly compact cardinal is a certain kind of large cardinal. An uncountable cardinal κ is strongly compact if and only if every κ-complete...

cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by Paul Erdős and András Hajnal (1958). A cardinal κ...

XVI, the 120 limit at the conclave was not surpassed. On 30 April 2025, the general congregation of cardinals confirmed that all cardinal electors present...

the cardinality of x. Then for any universe U, c(U) is either zero or strongly inaccessible. Assuming it is non-zero, it is a strong limit cardinal because...

Lac Cardinal is a lake in northwestern Alberta, Canada. It is located at the southern limit of Mackenzie Highway, near Grimshaw. Lac Cardinal has a total...

singular cardinal λ {\displaystyle \lambda } ? Does the generalized continuum hypothesis imply the existence of an ℵ2-Suslin tree? If ℵω is a strong limit cardinal...

of Francis' consistories increased the number of cardinal electors from at or less than the set limit of 120 to a number higher than 120, as high as 140...

measurable cardinals below κ which is regular, and thus κ is a limit of κ-many measurable cardinals. Strong cardinals also lie below superstrong cardinals and...

cardinals remain initial in L {\displaystyle L} . Regular ordinals remain regular in L {\displaystyle L} . Weak limit cardinals become strong limit cardinals...