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Many set theorists yearn for a definitive solution of the continuum problem, what I call a dream solution, one by which we settle the continuum hypothesis (CH) on the basis of a new fundamental principle of set theory, a missing axiom, widely regarded as true, which determines the truth value of CH. In an earlier article, I have described the dream solution template as proceeding in two steps: first, one introduces the new set-theoretic principle, considered obviously true for sets in the same way that many mathematicians find the axiom of choice or the axiom of replacement to be true; and second, one proves the CH or its negation from this new axiom and the other axioms of set theory. Such a situation would resemble Zermelo’s proof of the ponderous well-order principle on the basis of the comparatively natural axiom of choice and the other Zermelo axioms. If achieved, a dream solution to the continuum problem would be remarkable, a cause for celebration.
In this article, however, I argue that a dream solution of CH has become impossible to achieve. Specifically, what I claim is that our extensive experience in the set-theoretic worlds in which CH is true and others in which CH is false prevents us from looking upon any statement settling CH as being obviously true. We simply have had too much experience by now with the contrary situation. Even if set theorists initially find a proposed new principle to be a natural, obvious truth, nevertheless once it is learned that the principle settles CH, then this preliminary judgement will evaporate in the face of deep experience with the contrary, and set-theorists will look upon the statement merely as an intriguing generalization or curious formulation of CH or $\neg$CH, rather than as a new fundamental truth. In short, success in the second step of the dream solution will inevitably undermine success in the first step.
This article is based upon an argument I gave during the course of a three-lecture tutorial on set-theoretic geology at the summer school Set Theory and Higher-Order Logic: Foundational Issues and Mathematical Development, at the University of London, Birkbeck in August 2011. Much of the article is adapted from and expands upon the corresponding section of material in my article The set-theoretic multiverse.