Continuum Hypothesis

The continuum hypothesis (CH) is a statement in set theory asserting that no set has a cardinality strictly greater than that of the natural numbers and strictly smaller than that of the real numbers. Its standard symbolic formulation is $2^{\aleph_0}=\aleph_1$. The combined work of Kurt Gödel and Paul Cohen established that CH is independent of Zermelo–Fraenkel set theory with the axiom of choice (ZFC): if ZFC is consistent, its axioms can neither prove nor refute CH. (arxiv.org)

Mathematical formulation

Two sets have the same cardinality when a bijection pairs their elements one-to-one. The cardinality of the natural numbers is denoted by $\aleph_0$. An infinite countable set has this cardinality; the real numbers form an uncountable set. The cardinality of the real line, called the cardinality of the continuum, is written $\mathfrak c$, with

Here $2^{\aleph_0}$ denotes the cardinality of the power set of the natural numbers, equivalently the set of infinite binary sequences. (arxiv.org)

The symbol $\aleph_1$ denotes the least uncountable cardinal. It is the cardinality of the set of all countable ordinals. CH therefore states

Within ZFC, this is equivalent to saying that every infinite subset of $\mathbb R$ is either countably infinite or has the same cardinality as $\mathbb R$. The equivalence uses the well-ordering of sets supplied by the axiom of choice; without choice, formulations concerning intermediate sizes require additional care. (virtualmath1.stanford.edu)

Historical development

Georg Cantor proposed the hypothesis in 1878, initially in the form that every infinite subset of the continuum is either countable or equinumerous with the continuum. His subsequent work on well-ordering led to the formulation identifying the continuum with the cardinality of the countable ordinals. The hypothesis became a central problem in the developing theory of infinite sets. (virtualmath1.stanford.edu)

In 1900, David Hilbert placed the continuum problem first in his influential list of mathematical problems. The eventual resolution did not establish CH or its negation from the accepted axioms. Instead, it established that those axioms do not determine the answer. (arxiv.org)

Kurt Gödel announced his relative-consistency result in 1938 and developed it in a 1940 monograph. Cohen obtained the complementary result in 1963, publishing papers in December 1963 and January 1964. These results introduced methods that became fundamental tools for studying the scope and limitations of axiomatic set theory. (pmc.ncbi.nlm.nih.gov)

Gödel’s constructible universe

Gödel’s argument uses the constructible universe, denoted by $L$. Its construction proceeds through ordinal stages: successor stages collect subsets definable over the preceding stage, while limit stages take unions of earlier stages. This produces an inner universe in which the axioms of ZFC and the generalized continuum hypothesis hold. (pmc.ncbi.nlm.nih.gov)

The logical conclusion is a relative consistency statement:

Thus, if ZFC contains no contradiction, adding CH cannot introduce one. This shows that ZFC cannot prove the negation of CH. It does not establish that all sets belong to $L$, or that CH follows from ZFC. The additional axiom $V=L$, asserting that every set is constructible, does imply CH. (pmc.ncbi.nlm.nih.gov)

Cohen’s forcing method

Cohen proved the consistency of the opposite answer by introducing forcing. Forcing constructs an extension of a model of set theory by adjoining a suitably generic object. A partially ordered set of conditions specifies partial information about that object; genericity ensures that the extension meets the requirements needed to preserve the axioms. (pubmed.ncbi.nlm.nih.gov)

To obtain a failure of CH, forcing can add sufficiently many distinct subsets of the natural numbers while preserving the relevant cardinals. Since such subsets encode real numbers, the extension has more than $\aleph_1$ reals. Cardinal preservation is essential: merely adjoining objects is insufficient if the construction also changes which cardinal is the least uncountable one. (pmc.ncbi.nlm.nih.gov)

Cohen’s result gives

Together with Gödel’s theorem, this establishes the independence of CH from ZFC. Forcing subsequently became a general method for constructing models with different set-theoretic properties, rather than a technique confined to the continuum problem. (pubmed.ncbi.nlm.nih.gov)

What independence means

Independence concerns provability within a specified formal system. Assuming ZFC is consistent, there is no ZFC proof of CH and no ZFC proof of its negation. In the language of model theory, models satisfying ZFC can give different answers to the continuum problem. (arxiv.org)

This does not mean that CH and its negation hold simultaneously in one model. Nor does it prevent stronger axioms from determining an answer. It identifies a limitation of ZFC, not a prohibition on further mathematical investigation. The mathematical issue beyond independence is therefore which additional principles to adopt, and how to justify them independently of the desired answer to CH. (arxiv.org)

Generalization and foundational interpretations

The generalized continuum hypothesis (GCH) extends CH to every infinite cardinal:

where $\kappa^+$ is the least cardinal greater than $\kappa$. CH is its instance at $\kappa=\aleph_0$. Gödel’s constructible universe satisfies GCH, while Cohen’s construction of a model where CH fails also establishes that GCH cannot be proved in consistent ZFC. (pmc.ncbi.nlm.nih.gov)

The independence theorem leaves room for different foundational interpretations. A universe approach treats set theory as describing one intended universe and seeks additional principles that determine its structure. A multiverse approach emphasizes the range of set-theoretic universes and studies how CH behaves among them. Joel David Hamkins has defended the latter interpretation, arguing that the diversity revealed by forcing is central to understanding the continuum problem. These interpretations differ philosophically without disputing the independence theorem itself. (arxiv.org)

References

1. Independence of the Continuum Hypothesis: an Intuitive Introductionarxiv.org
2. The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesispmc.ncbi.nlm.nih.gov
3. The Consistency of the Axiom of Choice and of the Generalized Continuum-hypothesis with the Axioms of Set Theorybooks.google.com
4. THE INDEPENDENCE OF THE CONTINUUM HYPOTHESISpubmed.ncbi.nlm.nih.gov
5. THE INDEPENDENCE OF THE CONTINUUM HYPOTHESIS, IIpmc.ncbi.nlm.nih.gov
6. CH is Indefinitevirtualmath1.stanford.edu
7. The set-theoretic multiversearxiv.org