Continuum hypothesis: Encyclopedia II - Continuum hypothesis - Arguments pro and con

Continuum hypothesis - Arguments pro and con

Gödel believed strongly that CH is false. To him, his independence proof only showed that the prevalent set of axioms was defective. Gödel was a platonist and therefore had no problems with asserting truth and falsehood of statements independent of their provability. Cohen, however, was a formalist, but even he tended towards rejecting CH.

Historically, mathematicians who favor a "rich" and "large" universe of sets were against CH, while those favoring a "neat" and "controllable" universe favored CH. More recently, some experts (e.g. Matthew Foreman) have pointed out that ontological maximalism can actually be taken as a point in favor of CH, given that between models that have all the same reals, it's the one with more sets of reals that has more chance of satisfying CH. See (Maddy, p. 500).

Chris Freiling in 1986 presented an argument against CH: he showed that the negation of CH is equivalent to a statement about probabilities which he calls "intuitively true", but others have disagreed.

A difficult argument developed by W. Hugh Woodin, against CH, has attracted considerable attention since about the year 2000. See the references in Notices of the AMS. The Foreman reference does not reject Woodin's argument outright but urges caution.

Adapted from the Wikipedia article "Arguments pro and con", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki