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

Axioms of determinacy restrict the class of sets of reals that might be counterexamples to CH.

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 fundamental importance of the logarithm has suggested theterm

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.

Like the semantic relation, this quasi-syntactic proof relation isrobust under large cardinal assumptions:

Gödel, K.: Consistency of the Continuum Hypothesis. …

This Σ2-statement is invariant under set forcingand hence is one adherents to the generic multiverse view of truthmust deem determinate. Moreover, the key arguments above go throughwith this Σ2-statement instead of the ΩConjecture. The person taking this second line of response would thusalso have to maintain that this statement is false. But there issubstantial evidence that this statement is true. The reasonis that there is no known example of a Σ2-statementthat is invariant under set forcing relative to large cardinal axiomsand which cannot be settled by large cardinal axioms. (Such astatement would be a candidate for an absolutely undecidablestatement.) So it is reasonable to expect that this statement isresolved by large cardinal axioms. However, recent advances in innermodel theory—in particular, those in Woodin (2010)—provideevidence that no large cardinal axiom can refute thisstatement. Putting everything together: It is very likely that thisstatement is in fact true ; so this line of response is notpromising.

Freiling's proof, does not invoke large cardinals orintense infinitary combinatorics to make the point that CH impliescounter-intuitive propositions.

I am putting the proof below despite it not being ..

Put another way: forthere to be a proof of the Continuum Hypothesis, it would have to be true inall models of set theory, which it isn’t. Similarly, for the Hypothesis to bedisproven, it would have to remain invalid in all models of set theory, whichit also isn’t.

For example, a year adds 10% to the life of aten-year-old, but only 5% to that of a twenty-year-old.

Continuum Hypothesis | The Book of Threes

Gödel believed that CH is false and that his proof that CH is only shows that theZermelo-Frankel axioms are defective. Gödel was a and therefore hadno problems with asserting the truth and falsehood of statementsindependent of their provability. Cohen, though a , also tendedtowards rejecting CH.