In mathematics, an iterable cardinal is a type of large cardinal introduced by Gitman (2011), and Sharpe and Welch (2011), and further studied by Gitman and Welch (2011). Sharpe and Welch defined a cardinal κ to be iterable if every subset of κ is contained in a weak κ-model M for which there exists an M-ultrafilter on κ which allows for wellfounded iterations by ultrapowers of arbitrary length.
Gitman gave a finer notion, where a cardinal κ is defined to be α-iterable
if ultrapower iterations only of length α are required to wellfounded. (By standard arguments iterability is equivalent to ω1-iterability.)
In mathematics, an iterablecardinal is a type of large cardinal introduced by Gitman (2011), and Sharpe and Welch (2011), and further studied by Gitman...
that of 1-iterablecardinals, which in turn is below remarkable cardinals, which in turn is below ω-Erdős cardinals. A list of large cardinal axioms by...
Woodin cardinals). K is maximal, universal, and fully iterable. This implies that for every iterable extender model M (called a mouse), there is an elementary...
In mathematics, a Mahlo cardinal is a certain kind of large cardinal number. Mahlo cardinals were first described by Paul Mahlo (1911, 1912, 1913). As...
larger than all finite numbers. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the...
In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number λ {\displaystyle \lambda } such that for all functions f : λ → λ {\displaystyle...
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...
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...
Huge cardinals were introduced by Kenneth Kunen (1978). In what follows, j n {\displaystyle j^{n}} refers to the n {\displaystyle n} -th iterate of the...
showed that iterated forcing can construct models where Martin's axiom holds and the continuum is any given regular cardinal. In iterated forcing, one...
the mathematical discipline of set theory, a cardinal characteristic of the continuum is an infinite cardinal number that may consistently lie strictly between...
worldly cardinals (equivalently, a limit of κ worldly cardinals). The least worldly κ and λ with Vκ ≺Σ2 Vλ (this is higher than even a κ-fold iteration of...
numbers are distinct from cardinal numbers, which measure the size of sets. Although the distinction between ordinals and cardinals is not always apparent...
A cardinal invariant is a property of the real line measured by a cardinal number. For example, a well-studied invariant is the smallest cardinality of...
\dots ,n-1} , and one-sided difference at the endpoints of the data set. A cardinal spline, sometimes called a canonical spline, is obtained if m k = ( 1 −...
{\displaystyle y_{2}(x)=4x-3} Where T n ( y ) {\displaystyle T_{n}(y)} is the nth cardinal function of the chebyshev polynomials of the first kind with input argument...
induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Let P ( α ) {\displaystyle...
parallel for the equirectangular projection, sinc is the unnormalized cardinal sine function, and α = arccos ( cos φ cos λ 2 ) . {\displaystyle...
viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity...
(it is without interior points) and an uncountable set (of the same cardinality as the real numbers). Like F ( f ) {\displaystyle \operatorname {F}...
5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number. It has garnered...
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