In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: The name of the hypothesis comes from the term continuum for the real numbers. In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: , or even shorter with beth numbers: .
Georg Cantor advanced the continuum hypothesis in 1878. It became one of the most studied problems in set theory, and establishing its truth or falsehood was the first of Hilbert's problems presented in 1900. The answer to this problem is independent of ZFC. This means that ZFC's axioms can neither prove nor disprove the continuum hypothesis, meaning either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen, complementing work by Kurt Gödel in 1940.
The generalized continuum hypothesis states that for every ordinal .
History Cantor first introduced the continuum hypothesis in his 1878 paper "Ein Beitrag zur Mannigfaltigkeitslehre", in a form now called the weak continuum hypothesis which is equivalent to the standard formulation under the then-undeveloped axiom of choice. He initially presented the weak continuum hypothesis as a theorem, but did not give a proof and later became uncertain of it. On 25 October 1882, Cantor wrote to his correspondent Gösta Mittag-Leffler, and formulated the continuum hypothesis (CH) in its modern form and gave his belief he could prove it. By the 1890s, many mathematicians in Germany and France were aware of the problem. Cantor tried for many years to prove CH but never succeeded.
It became the first on David Hilbert's list of important open questions that was presented at the International Congress of Mathematicians (ICM) in 1900 in Paris. At that point, axiomatic set theory was not yet formulated.
Many erroneous proofs and disproofs of CH were given. As early as 1884, Paul Tannery claimed to prove CH, but the proof was erroneous. In 1890, Beppo Levi claimed to have a proof from assuming that every subset of the real numbers has the property of Baire, but no proof was ever published. At the 1904 ICM in Heidelberg, Gyula Kőnig announced he had disproven CH, drawing wide attention that even reached the Grand Duke of Baden Frederick I via Felix Klein, but a flaw in the proof was found the next day. Felix Bernstein also published an attempted proof of CH in 1904, but it was wrong and attracted little response. In 1926, Hilbert claimed to have solved the continuum hypothesis and gave a proof sketch, but this was also incorrect, although it influenced later ideas in recursion theory.
In 1906, Kőnig revised part of his attempted CH disproof and established Kőnig's theorem, which, using the concept of cofinality introduced in 1908 by Felix Hausdorff, shows that cannot equal for with cofinality . For example, . Hausdorff formulated the question of whether for all ordinals , which Alfred Tarski named the generalized continuum hypothesis in 1925. In 1923, Thoralf Skolem conjectured that CH could not be settled by the axioms of Zermelo set theory.
Kurt Gödel proved in 1940 that the negation of the continuum hypothesis, i.e., the existence of a set with intermediate cardinality, could not be proved in standard set theory. 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.
Cardinality of infinite sets
Two sets are said to have the same cardinality or cardinal number if there exists a bijection (a one-to-one correspondence) between them. Intuitively, for two sets and to have the same cardinality means that it is possible to "pair off" elements of with elements of in such a fashion that every element of is paired off with exactly one element of and vice versa. Hence, the set has the same cardinality as despite the sets themselves containing different elements.
With infinite sets such as the set of integers or rational numbers, the existence of a bijection between two sets becomes more difficult to demonstrate. The rational numbers seemingly form a counterexample to the continuum hypothesis: the integers form a proper subset of the rationals, which themselves form a proper subset of the reals, so intuitively, there are more rational numbers than integers and more real numbers than rational numbers. However, this intuitive analysis is flawed since it does not take into account the fact that all three sets are infinite. Perhaps more importantly, it in fact conflates the concept of "size" of the set with the order or topological structure placed on it. In fact, it turns out the rational numbers can actually be placed in one-to-one correspondence with the integers, and therefore the set of rational numbers is the same size (cardinality) as the set of integers: they are both countable sets.For a proof sketch, see Countability of the Rationals
Cantor gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers (see Cantor's first uncountability proof and Cantor's diagonal argument). His proofs, however, give no indication of the extent to which the cardinality of the integers is less than that of the real numbers. Cantor proposed the continuum hypothesis as a possible solution to this question.
In simple terms, the Continuum Hypothesis (CH) states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. That is, every infinite set of real numbers can either be mapped one-to-one into the integers or the real numbers can be mapped one-to-one into . Since the real numbers are equinumerous with the powerset of the integers, i.e. , CH can be restated as follows:
Assuming the axiom of choice, there is a unique smallest cardinal number greater than , and the continuum hypothesis is in turn equivalent to the equality .Asaf Karagila (https://math.stackexchange.com/users/622/asaf-karagila), How to formulate continuum hypothesis without the axiom of choice?, URL (version: 2017-04-13): https://math.stackexchange.com/q/404813
Independence from ZFC
The independence of the continuum hypothesis (CH) from Zermelo–Fraenkel set theory (ZF) follows from the work of Gödel and Cohen.
Gödel showed that CH cannot be disproved from ZF, even if the axiom of choice (AC) is adopted, i.e. from ZFC. His proof shows that both CH and AC hold in the constructible universe , an inner model of ZF set theory, assuming only the axioms of ZF. The existence of an inner model of ZF in which additional axioms hold shows that the additional axioms are (relatively) consistent with ZF, provided ZF itself is consistent. The latter condition cannot be proved in ZF itself, due to Gödel's incompleteness theorems, but is widely believed to be true and can be proved in stronger set theories.
Cohen showed that CH cannot be proven from the ZFC axioms, completing the overall independence proof. To prove his result, he developed the method of forcing, which has become a standard tool in set theory. Essentially, this method begins with a model of ZF in which CH holds and constructs another model that contains more sets than the original in a way that CH does not hold in the new model. Cohen was awarded the Fields Medal in 1966 for his proof.
Cohen's independence proof shows that CH is independent of ZFC. Further research has shown that CH is independent of all known large cardinal axioms in the context of ZFC. Moreover, it has been shown that the cardinality of the continuum can be any cardinal consistent with Kőnig's theorem. A result of Solovay, proved shortly after Cohen's result on the independence of the continuum hypothesis, shows that in any model of ZFC, if is a cardinal of uncountable cofinality, then there is a forcing extension in which . However, per Kőnig's theorem, it is not consistent to assume is or or any cardinal with cofinality .
The continuum hypothesis is closely related to many statements in analysis, point-set topology and measure theory. As a result of its independence, many substantial conjectures in those fields have subsequently been shown to be independent as well.
The independence from ZFC means that proving or disproving the CH within ZFC is impossible. But Gödel's and Cohen's negative results are not universally accepted as disposing of all interest in the continuum hypothesis. The continuum hypothesis remains an active topic of research: see Woodin and Koellner for an overview of the current research status.
The continuum hypothesis and the axiom of choice were among the first genuinely mathematical statements shown to be independent of ZF set theory. The existence of some statements independent of ZFC had already been known more than two decades earlier: for example, assuming good soundness properties and the consistency of ZFC, Gödel's incompleteness theorems, published in 1931, establish that there is a formal statement Con(ZFC) (one for each appropriate Gödel numbering scheme) expressing the consistency of ZFC that is also independent of it. The latter independence result indeed holds for many theories.
Arguments for and against the continuum hypothesis
Gödel believed that CH is false, and that his proof that CH is consistent with ZFC only shows that the Zermelo–Fraenkel axioms do not adequately characterize the universe of sets. Gödel was a Platonist and therefore believed it was meaningful to assert the truth and falsehood of statements independent of their provability. Cohen, though a formalist, also tended toward rejecting CH.
Historically, mathematicians who favored a "rich" and "large" universe of sets were against CH, while those favoring a "neat" and "controllable" universe favored it. 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.
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. Skolem argued on the basis of what is now known as Skolem's paradox, and it was later supported by the independence of CH from the axioms of ZFC since these axioms are enough to establish the elementary properties of sets and cardinalities. 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. Although the axiom of constructibility does resolve CH, it is not generally considered intuitively true any more than CH is generally considered false.
At least two other axioms have been proposed that have implications for the continuum hypothesis, although they have not found wide acceptance in the mathematical community. In 1986, Chris Freiling 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. Freiling believes this axiom is "intuitively clear" but others disagree.