kb/data/en.wikipedia.org/wiki/Continuum_hypothesis-0.md

title	chunk	source	category	tags	date_saved	instance
Continuum hypothesis	1/5	https://en.wikipedia.org/wiki/Continuum_hypothesis	reference	science, encyclopedia	2026-05-05T09:59:15.765779+00:00	kb-cron

In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states:

There is no set whose cardinality is strictly between that of the integers and the real numbers. 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:

      2
      
        
          ℵ
          
            0
          
        
      
    
    =
    
      ℵ
      
        1
      
    
  

{\displaystyle 2^{\aleph _{0}}=\aleph _{1}}

, or even shorter with beth numbers:

      ℶ
      
        1
      
    
    =
    
      ℵ
      
        1
      
    
  

{\displaystyle \beth _{1}=\aleph _{1}}

. The continuum hypothesis was advanced by Georg Cantor 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 23 problems presented in 1900. The answer to this problem is independent of ZFC. This means the axioms of ZFC 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 earlier work by Kurt Gödel in 1940. The generalized continuum hypothesis states that

      ℵ
      
        α
        +
        1
      
    
    =
    
      2
      
        
          ℵ
          
            α
          
        
      
    
  

{\displaystyle \aleph _{\alpha +1}=2^{\aleph _{\alpha }}}

for every ordinal

    α
  

{\displaystyle \alpha }

.

== History == The continuum hypothesis was first introduced by Georg Cantor 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. Cantor 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 the year 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 Baire property, 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 by using the concept of cofinality introduced in 1908 by Felix Hausdorff, shows that result that

      2
      
        
          ℵ
          
            0
          
        
      
    
  

{\displaystyle 2^{\aleph _{0}}}

cannot equal

      ℵ
      
        α
      
    
  

{\displaystyle \aleph _{\alpha }}

for

    α
  

{\displaystyle \alpha }

with cofinality

    ω
  

{\displaystyle \omega }

. For example,

      2
      
        
          ℵ
          
            0
          
        
      
    
    ≠
    
      ℵ
      
        ω
      
    
  

{\displaystyle 2^{\aleph _{0}}\neq \aleph _{\omega }}

. Hausdorff formulated the question of whether

      2
      
        
          ℵ
          
            α
          
        
      
    
    =
    
      ℵ
      
        α
      
    
    +
    1
  

{\displaystyle 2^{\aleph _{\alpha }}=\aleph _{\alpha }+1}

for all ordinals

    α
  

{\displaystyle \alpha }

, which would be named the generalized continuum hypothesis by Alfred Tarski 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 ==