kb/data/en.wikipedia.org/wiki/Continuum_hypothesis-3.md

title	chunk	source	category	tags	date_saved	instance
Continuum hypothesis	4/5	https://en.wikipedia.org/wiki/Continuum_hypothesis	reference	science, encyclopedia	2026-05-05T09:59:15.765779+00:00	kb-cron

== 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 had no problems with asserting the truth and falsehood of statements independent of their provability. Cohen, though a formalist, also tended towards 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 CH. 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. 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. Although the axiom of constructibility does resolve CH, it is not generally considered to be intuitively true any more than CH is generally considered to be false. At least two other axioms have been proposed that have implications for the continuum hypothesis, although these axioms have not currently 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 have disagreed. A difficult argument against CH developed by W. Hugh Woodin has attracted considerable attention since the year 2000. Foreman does not reject Woodin's argument outright but urges caution. Woodin proposed a new hypothesis that he labeled the "(*)-axiom", or "Star axiom". The Star axiom would imply that

      2
      
        
          ℵ
          
            0
          
        
      
    
  

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

is

      ℵ
      
        2
      
    
  

{\displaystyle \aleph _{2}}

, thus falsifying CH. The Star axiom was bolstered by an independent May 2021 proof showing the Star axiom can be derived from a variation of Martin's maximum. 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. Solomon Feferman argued that CH is not a definite mathematical problem. He proposed a theory of "definiteness" using a semi-intuitionistic subsystem of ZF that accepts classical logic for bounded quantifiers but uses intuitionistic logic for unbounded ones, and suggested that a proposition

    ϕ
  

{\displaystyle \phi }

is mathematically "definite" if the semi-intuitionistic theory can prove

    (
    ϕ
    ∨
    ¬
    ϕ
    )
  

{\displaystyle (\phi \lor \neg \phi )}

. He conjectured that CH is not definite according to this notion, and proposed that CH should, therefore, be considered not to have a truth value. Peter Koellner wrote a critical commentary on Feferman's article. Joel David Hamkins proposes a multiverse approach to set theory and argues that "the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and, as a result, it can no longer be settled in the manner formerly hoped for". 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. My mental picture is that we have many possible set theories, all conforming to ZFC".

== Generalized continuum hypothesis == The generalized continuum hypothesis (GCH) states that if an infinite set's cardinality lies between that of an infinite set S and that of the power set

        P
      
    
    (
    S
    )
  

{\displaystyle {\mathcal {P}}(S)}

of S, then it has the same cardinality as either S or

        P
      
    
    (
    S
    )
  

{\displaystyle {\mathcal {P}}(S)}

. That is, for any infinite cardinal

    λ
  

{\displaystyle \lambda }

there is no cardinal

    κ
  

{\displaystyle \kappa }

such that

    λ
    <
    κ
    <
    
      2
      
        λ
      
    
  

{\displaystyle \lambda <\kappa <2^{\lambda }}

. GCH is equivalent to: