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 of, existence, proper and/or class:
“The very existence of society depends on the fact that every member of it tacitly admits he is not the exclusive possessor of himself, and that he admits the claim of the polity of which he forms a part, to act, to some extent, as his master.”
—Thomas Henry Huxley (1825–95)
“An “unemployed” existence is a worse negation of life than death itself.”
—José Ortega Y Gasset (1883–1955)
“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)
“Ours is the old, old story of every uprising race or class or order. The work of elevation must be wrought by ourselves or not at all.”
—Frances Power Cobbe (1822–1904)