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 a, existence of, existence, proper and/or class:
“‘Tis going, I own, like the Knight of the Woeful Countenance, in quest of melancholy adventures—but I know not how it is, but I am never so perfectly conscious of the existence of a soul within me, as when I am entangled in them.”
—Laurence Sterne (1713–1768)
“Analysis brings no curative powers in its train; it merely makes us conscious of the existence of an evil, which, oddly enough, is consciousness.”
—Henry Miller (1891–1980)
“I create my social existence by earning and spending.”
—Mason Cooley (b. 1927)
“I thought it altogether proper that I should take a brief furlough from official duties at Washington to mingle with you here to-day as a comrade, because every President of the United States must realize that the strength of the Government, its defence in war, the army that is to muster under its banner when our Nation is assailed, is to be found here in the masses of our people.”
—Benjamin Harrison (1833–1901)
“Money couldn’t buy friends, but you got a better class of enemy.”
—Spike Milligan (b. 1918)