# proof:If CH holds, then well-order as with .

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A basic reference is Godel's ``What is Cantor's Continuum Problem?", from1947 with a 1963 supplement, reprinted in Benacerraf and Putnam's collectionPhilosophy of Mathematics.

## Continuum Hypothesis | Mathematical Proof | Axiom

### Continuum hypothesis - Stanford Encyclopedia of …

Forexample, the following have adjacent numbers that (before rounding)differ by a factor of the square root of 2 (1.414...), the "halfoctave", i.e., the exponential growth in clock age is a little over 41%per step:** Interpolating again (before rounding) produces thefollowing "quarter octave" series (as with the above, one series isbased on an exact 1, and the other on an exact 10):**** Another interpolation and rounding produces the "eighthoctave" series, with horizontally adjacent numbers differing by afactor of the eighth root of 2. **

### Continuum Hypothesis - proof attempt - Google Groups

Again, axioms of definable determinacy and large cardinal axiomsimply this version of CH for richer notions of definability. Forexample, if AD^{(ℝ)} holds then this version of CHholds for all sets of real numbers in (ℝ). And if there is aproper class of Woodin cardinals then this version of CH holds for alluniversally Baire sets of reals.

## The continuum hypothesis is a famous problem of set theory ..

Gödel and Cohen's negative results are not universally acceptedas disposing of the hypothesis, and Hilbert's problem remains anactive topic of contemporary research (see Woodin 2001a).

## The Continuum Hypothesis for Borelian Sets | …

The continuum hypothesis is closely related to many statementsin , point set and . As a result of itsindependence, many substantial in those fields havesubsequently been shown to be independent as well.

## Continuum hypothesis | Nature of Mathematics

Correspondences canbe used to compare the sizes of much larger collections than six goats—includinginfinite collections. The rule is that, if a correspondence exists between twocollections, then they have the same size. If not, then one must be bigger. Forexample, the collection of all natural numbers {1,2,3,4,…} contains thecollection of all multiples of five {5,10,15,20,…}. At first glance, this seemsto indicate that the collection of natural numbers is larger than thecollection of multiples of five. But in fact they are equal in size: everynatural number can be paired uniquely with a multiple of five such that nonumber in either collection remains unpaired. One such correspondence wouldinvolve the number 1 pairing with 5, 2 with 10, and so on.