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)
“It’s [motherhood] the biggest on-the-job- training program in existence today.”
—Erma Bombeck (20th century)
“Television programming for children need not be saccharine or insipid in order to give to violence its proper balance in the scheme of things.... But as an endless diet for the sake of excitement and sensation in stories whose plots are vehicles for killing and torture and little more, it is not healthy for young children. Unfamiliar as yet with the full story of human response, they are being misled when they are offered perversion before they have fully learned what is sound.”
—Dorothy H. Cohen (20th century)
“There is a struggle between the Oriental and the Occidental in every nation; some who would be forever contemplating the sun, and some who are hastening toward the sunset. The former class says to the latter, When you have reached the sunset, you will be no nearer to the sun. To which the latter replies, But we so prolong the day.”
—Henry David Thoreau (1817–1862)