Existence of A Proper Class of Inaccessibles
There are many important axioms in set theory which assert the existence of a proper class of cardinals which satisfy a predicate of interest. In the case of inaccessibility, the corresponding axiom is the assertion that for every cardinal μ, there is an inaccessible cardinal κ which is strictly larger, μ < κ. Thus this axiom guarantees the existence of an infinite tower of inaccessible cardinals (and may occasionally be referred to as the inaccessible cardinal axiom). As is the case for the existence of any inaccessible cardinal, the inaccessible cardinal axiom is unprovable from the axioms of ZFC. Assuming ZFC, the inaccessible cardinal axiom is equivalent to the universe axiom of Grothendieck and Verdier: every set is contained in a Grothendieck universe. The axioms of ZFC along with the universe axiom (or equivalently the inaccessible cardinal axiom) are denoted ZFCU (which could be confused with ZFC with urelements). This axiomatic system is useful to prove for example that every category has an appropriate Yoneda embedding.
This is a relatively weak large cardinal axiom since it amounts to saying that ∞ is 1-inaccessible in the language of the next section, where ∞ denotes the least ordinal not in V, i.e. the class of all ordinals in your model.
Read more about this topic: Inaccessible Cardinal
Famous quotes containing the words existence, proper and/or class:
“Creation destroys as it goes, throws down one tree for the rise of another. But ideal mankind would abolish death, multiply itself million upon million, rear up city upon city, save every parasite alive, until the accumulation of mere existence is swollen to a horror.”
—D.H. (David Herbert)
“Discipline isn’t just punishing, forcing compliance or stamping out bad behavior. Rather, discipline has to do with teaching proper deportment, caring about others, controlling oneself and putting someone else’s wishes before one’s own when the occasion calls for it.”
—Lawrence Balter (20th century)
“... in the fierce competition of modern society the only class left in the country possessing leisure is that of women supported in easy circumstances by husband or father, and it is to this class we must look for the maintenance of cultivated and refined tastes, for that value and pursuit of knowledge and of art for their own sakes which can alone save society from degenerating into a huge machine for making money, and gratifying the love of sensual luxury.”
—Mrs. H. O. Ward (1824–1899)