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Berkeley Cardinal
In set theory, Berkeley cardinals are certain large cardinals suggested by Hugh Woodin in a seminar at the University of California, Berkeley in about 1992. A Berkeley cardinal is a cardinal ''κ'' in a model of Zermelo–Fraenkel set theory with the property that for every transitive set ''M'' that includes ''κ'' and α < κ, there is a nontrivial Elementary equivalence, elementary embedding of ''M'' into ''M'' with α < Critical point (set theory), critical point < ''κ''. Berkeley cardinals are a strictly stronger cardinal axiom than Reinhardt cardinals, implying that they are not compatible with the axiom of choice. A weakening of being a Berkeley cardinal is that for every binary relation ''R'' on ''V''''κ'', there is a nontrivial elementary embedding of (''V''''κ'', ''R'') into itself. This implies that we have elementary : ''j''1, ''j''2, ''j''3, ... : ''j''1: (''V''''κ'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of '' naive set theory''. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set theory is commonly employed as a foundational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
