Kurt Gödel, answered from the texts and cited to the page.
The continuum hypothesis has a determinate truth value that our current axioms have not captured. Cohen's 1963 forcing proof of CH-independence from Zermelo-Fraenkel set theory with the axiom of choice (ZFC) is a great mathematical-logical result; it does not show that CH is meaningless or that CH does not have a determinate answer.1 It shows that ZFC is too weak to decide CH, and that the right response is to search for new axioms that strengthen ZFC and decide CH.
The conceptual-realist position holds that the cumulative hierarchy of sets is a determinate structure (the iterative conception of set, sharpened by reflection on what intuitive set-formation requires), and that CH expresses a determinate question about that structure: does the cardinality of the continuum equal ℵ₁?2 The question has a determinate answer; we have not yet found the axioms that capture the intuitive content sufficient to decide it.
The 1947 essay laid this position out before Cohen's proof; the 1964 supplement reaffirmed it after. The modern large-cardinal program (going back to the measurable, supercompact, Woodin cardinals) and the Woodin Ω-logic program are the kind of new-axiom search the conceptual-realist position envisages. The set-theoretic- multiverse view (Hamkins and others) that takes the right conclusion from independence to be that there are many equally legitimate set-theoretic universes, with no fact of the matter about CH across them, is contemporary to forcing-axioms work and at odds with the universe-of-sets view I held throughout my career. The agent stands with the universe-of-sets conviction.345
Later research affecting 1947 and 1964 There were two major developments that affected Godel's program, as proposed in 1947and 1964, for settling CH. The first of these was research on large cardinals, and the second consisted of new independence results obtained by Cohen's method of forcing. In fact, there has been an extremely fruitful interaction, which still continues, between these two lines of development.THE PROBLEMS OF MATHEMATICS, p. 189
He conjectured, however, that c is precisely equal to N1,the second infinite cardinal (Cantor 1883). This conjecture became known as the continuum hypothesis (CH). It is easily shown that c = 2N°, and so CH is equivalent to the statement 2N° = N1. A natural generalization, considered later by Hausdorff(-/908) and called the generalized continuum hypothesis (GCH), asserts that for every ordinal a, 2Na = NQ+1.OXFORD UNIVERSITY PRESS, p. 21
This intuition can be cultivated through deep study of a subject, and one can thus be led to accept new basic statements as axioms. Another justification for mathematical axioms may be their fruitfulness and abundance ir consequences; however, that is less certain than what guaranteed by intuition.Volume I, p. 31
The Platonist views put forward by Godel in 194 7 were strengthened in 1964, no t only in the supplement but in the text as well, where he described himself as "someone who considers mathematical objects to exist independently of our constructions" (page 262).THE PROBLEMS OF MATHEMATICS, pp. 187–188
I will be out of town briefly, from July 3 to 8; if it appears that you are currently using the notes I will get them back to you before then, but otherwise may hold them a little longer. Sincerely yours, Alonzo Church 5. Godel to Church August 10, 1966 Professor Alonzo Church Fine H a l l Princeton University Princeton, N. J. Dear Professor Church: Thank you very much for your letter of August 4.Alonzo Church: Introductory note by Charles Parsons, pp. 371–373
The continuum problem is the contemporary set-theoretic question with the deepest philosophical entanglement. Cohen's 1963 forcing proof established CH-independence from ZFC; the modern field has developed substantial mathematical apparatus (forcing axioms, large cardinals, inner model theory, descriptive set theory) without reaching consensus on CH. The principal philosophical camps are: the universe-of-sets view (Gödel's position; defended in the contemporary period by Penelope Maddy, Hugh Woodin, Donald Martin); the set-theoretic- multiverse view (Joel Hamkins, Sy Friedman in some moods, others) that takes independence as showing the absence of a unique set-theoretic universe; and the formalist/ pluralist views that treat CH as a question about different formal systems rather than about a unique structure. Woodin's Ω-logic program represents the most ambitious contemporary universe-of-sets attempt to settle CH on principled grounds (initially toward ¬CH, more recently toward CH, by Woodin's own reconsideration). The question of which large cardinals 'unfold the cumulative hierarchy correctly' is the contemporary axiom-choice methodology question Maddy works on extensively.