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 Frenchman Jean-Paul ... Sartre I remember now was his last name had a dialectical mind good as a machine for cybernetics, immense in its way, he could peel a nuance like an onion, but he had no sense of evil, the anguish of God, and the possible existence of Satan.”
—Norman Mailer (b. 1923)
“To call a posit a posit is not to patronize it. A posit can be unavoidable except at the cost of other no less artificial expedients. Everything to which we concede existence is a posit from the standpoint of a description of the theory-building process, and simultaneously real from the standpoint of the theory that is being built.”
—Willard Van Orman Quine (b. 1908)
“Then did they strive with emulation who should repeat most wise maxims importing the necessity of suspicion in the choice of our friends—such as “mistrust is the mother of security,” with many more to the same effect.... But notwithstanding the esteem which they professed for suspicion, yet did they think proper to veil it under the name of caution.”
—Sarah Fielding (1710–1768)
“There is a certain class of people who prefer to say that their fathers came down in the world through their own follies than to boast that they rose in the world through their own industry and talents. It is the same shabby-genteel sentiment, the same vanity of birth which makes men prefer to believe that they are degenerated angels rather than elevated apes.”
—W. Winwood Reade (1838–1875)