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 government at all, infers inequality. The citizen who is preferred to office becomes the superior to those who are not, so long as he is the repository of power, and the child inherits the wealth of the parent as a controlling law of society.”
—James Fenimore Cooper (1789–1851)
“This seems to be advanced as the surest basis for our belief in the existence of gods, that there is no race so uncivilized, no one in the world so barbarous that his mind has no inkling of a belief in gods.”
—Marcus Tullius Cicero (106–43 B.C.)
“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)
“You must drop all your democracy. You must not believe in “the people.” One class is no better than another. It must be a case of Wisdom, or Truth. Let the working classes be working classes. That is the truth. There must be an aristocracy of people who have wisdom, and there must be a Ruler: a Kaiser: no Presidents and democracies.”
—D.H. (David Herbert)