The continuum hypothesis was advanced by Georg Cantor in 1878,[1] and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900.
This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940.
Cantor believed the continuum hypothesis to be true and for many years tried in vain to prove it.
[3] It became the first on David Hilbert's list of important open questions that was presented at the International Congress of Mathematicians in the year 1900 in Paris.
[2] The second half of the independence of the continuum hypothesis – i.e., unprovability of the nonexistence of an intermediate-sized set – was proved in 1963 by Paul Cohen.
However, this intuitive analysis is flawed since it does not take into account the fact that all three sets are infinite.
His proofs, however, give no indication of the extent to which the cardinality of the integers is less than that of the real numbers.
Assuming the axiom of choice, there is a unique smallest cardinal number
[6][7] The independence of the continuum hypothesis (CH) from Zermelo–Fraenkel set theory (ZF) follows from combined work of Kurt Gödel and Paul Cohen.
Gödel[8][2] showed that CH cannot be disproved from ZF, even if the axiom of choice (AC) is adopted, i.e. from ZFC.
Gödel's proof shows that both CH and AC hold in the constructible universe
Cohen[4][9] showed that CH cannot be proven from the ZFC axioms, completing the overall independence proof.
To prove his result, Cohen developed the method of forcing, which has become a standard tool in set theory.
Further research has shown that CH is independent of all known large cardinal axioms in the context of ZFC.
The continuum hypothesis is closely related to many statements in analysis, point set topology and measure theory.
However, Gödel and Cohen's negative results are not universally accepted as disposing of all interest in the continuum hypothesis.
The continuum hypothesis and the axiom of choice were among the first genuinely mathematical statements shown to be independent of ZF set theory.
Gödel was a Platonist and therefore had no problems with asserting the truth and falsehood of statements independent of their provability.
Parallel arguments were made for and against the axiom of constructibility, which implies CH.
More recently, Matthew Foreman has pointed out that ontological maximalism can actually be used to argue in favor of CH, because among models that have the same reals, models with "more" sets of reals have a better chance of satisfying CH.
[15] Another viewpoint is that the conception of set is not specific enough to determine whether CH is true or false.
This viewpoint was advanced as early as 1923 by Skolem, even before Gödel's first incompleteness theorem.
In order to argue against this viewpoint, it would be sufficient to demonstrate new axioms that are supported by intuition and resolve CH in one direction or another.
In 1986, Chris Freiling[17] presented an argument against CH by showing that the negation of CH is equivalent to Freiling's axiom of symmetry, a statement derived by arguing from particular intuitions about probabilities.
[18][19] A difficult argument against CH developed by W. Hugh Woodin has attracted considerable attention since the year 2000.
However, Woodin stated in the 2010s that he now instead believes CH to be true, based on his belief in his new "ultimate L" conjecture.
Peter Koellner wrote a critical commentary on Feferman's article.
[25] In a related vein, Saharon Shelah wrote that he does "not agree with the pure Platonic view that the interesting problems in set theory can be decided, that we just have to discover the additional axiom.
To prove this, Sierpiński showed GCH implies that every cardinality n is smaller than some aleph number, and thus can be ordered.
[29] Kurt Gödel showed that GCH is a consequence of ZF + V=L (the axiom that every set is constructible relative to the ordinals), and is therefore consistent with ZFC.