ⓘ Iterable cardinal. In mathematics, an iterable cardinal is a type of large cardinal introduced by Gitman, and Sharpe and Welch, and further studied by Gitman an ..

ⓘ Iterable cardinal

In mathematics, an iterable cardinal is a type of large cardinal introduced by Gitman, and Sharpe and Welch, and further studied by Gitman and Welch. 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.

• that of 1 - iterable cardinals which in turn is below remarkable cardinals which in turn is below ω - Erdos cardinals A list of large cardinal axioms by
• 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
• totally ineffable cardinals remarkable cardinals α - Erdos cardinals for countable α 0 not a cardinal γ - iterable γ - Erdos cardinals for uncountable
• 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
• whose elements are in M. Huge cardinals were introduced by Kenneth Kunen  1978 In what follows, jn refers to the n - th iterate of the elementary embedding
• 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
• 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
• the mathematical discipline of set theory, a cardinal characteristic of the continuum is an infinite cardinal number that may consistently lie strictly between
• showed that iterated forcing can construct models where Martin s axiom holds and the continuum is any given regular cardinal In iterated forcing, one
• Cardinal is the second studio album by American rock band Pinegrove, released February 12, 2016 on Run for Cover. Pinegrove formed in Montclair, New Jersey
• an added condition of iterability referring to the existence of wellfounded iterated ultrapowers a mouse is then an iterable premouse. The notion of
• alternative to large cardinal axioms. The Fundamental Theorem of Proper Forcing, due to Shelah, states that any countable support iteration of proper forcings