The Continuum Hypothesis
The famous continuum hypothesis asserts that is also the second aleph number . In other words, the continuum hypothesis states that there is no set whose cardinality lies strictly between and
This statement is now known to be independent of the axioms of Zermelo–Fraenkel set theory with the axiom of choice (ZFC). That is, both the hypothesis and its negation are consistent with these axioms. In fact, for every nonzero natural number n, the equality = is independent of ZFC. (The case is the continuum hypothesis.) The same is true for most other alephs, although in some cases equality can be ruled out by König's theorem on the grounds of cofinality, e.g., In particular, could be either or, where is the first uncountable ordinal, so it could be either a successor cardinal or a limit cardinal, and either a regular cardinal or a singular cardinal.
Read more about this topic: Cardinality Of The Continuum
Famous quotes containing the words continuum and/or hypothesis:
“The further jazz moves away from the stark blue continuum and the collective realities of Afro-American and American life, the more it moves into academic concert-hall lifelessness, which can be replicated by any middle class showing off its music lessons.”
—Imamu Amiri Baraka (b. 1934)
“Oversimplified, Mercier’s Hypothesis would run like this: “Wit is always absurd and true, humor absurd and untrue.””
—Vivian Mercier (b. 1919)