WebThis is a list of some countable ordinals in increasing order. 0, the least ordinal. 1, the least successor ordinal. \ (\omega\), 1st transfinite ordinal. \ (\omega+1\), 1st transfinite successor ordinal. \ (\omega2\), 2nd transfinite limit ordinal. \ (\omega2+1\), 1st successor ordinal after the 2nd transfinite limit ordinal. A cardinal is inaccessible if and only if it is Π n-indescribable for all positive integers n, equivalently iff it is Π 2-indescribable, equivalently if it is Σ 1-indescribable. Π 1-indescribable cardinals are the same as weakly compact cardinals. If V=L, then for a natural number n>0, an uncountable cardinal is Π n-indescribable iff it's (n+1)-stationary.
Inaccessible Cardinal -- from Wolfram MathWorld
WebAn inaccessible cardinal is to ZFC as omega is to PA; the only way to reason that the infinite exists using arithmetic is to 'intuit' it must due to there being no largest natural. However, it requires an additional axiom to assert the existence of the infinite. Same goes for inaccessibles compared to ZFC. The entirety of the universe of ZFC ... WebJan 9, 2024 · 1 Answer. There are two kinds of inaccessible cardinals, weakly inaccessibles and strongly inaccessibles. κ is weakly inaccessible if it is a regular limit cardinal. κ is strongly inaccessible if it is a regular strong limit cardinal, that is, if κ is weakly inaccessible and 2 α < κ for all α < κ. Assuming the Generalized Continuum ... songs of green pheasant
inaccessible cardinal in nLab
Web1.3 Inaccessible cardinals An uncountable limit cardinal that is regular is called weakly inaccessible. A weakly inaccessible cardinal is strongly inaccessible if < implies 2 < . … WebIn set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal κ is strongly inaccessible if it is uncountable, it is not a sum of fewer than κ cardinals smaller than κ, and α < κ {\displaystyle \alpha <\kappa } implies 2 α < κ {\displaystyle 2^{\alpha … WebApr 2, 2010 · Here the problem about inaccessible cardinals has a metamathematical or metalogical setting. Tarski’s student Hanf proved that a very large class of inaccessible … songs of grief and remembrance