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:
“It would strike me as ridiculous to want to doubt the existence of Napoleon; but if someone doubted the existence of the earth 150 years ago, perhaps I should be more willing to listen, for now he is doubting our whole system of evidence.”
—Ludwig Wittgenstein (1889–1951)
“Each of us, even the lowliest and most insignificant among us, was uprooted from his innermost existence by the almost constant volcanic upheavals visited upon our European soil and, as one of countless human beings, I can’t claim any special place for myself except that, as an Austrian, a Jew, writer, humanist and pacifist, I have always been precisely in those places where the effects of the thrusts were most violent.”
—Stefan Zweig (18811942)
“Our life is a faint tracing on the surface of mystery, like the idle, curved tunnels of leaf miners on the face of a leaf. We must somehow take a wider view, look at the whole landscape, really see it, and describe what’s going on here. Then we can at least wail the right question into the swaddling band of darkness, or, if it comes to that, choir the proper praise.”
—Annie Dillard (b. 1945)
“To-day women constitute the only class of sane people excluded from the franchise ...”
—Mary Putnam Jacobi (1842–1906)