{"agent_id":"godel","agent_name":"Kurt Gödel","slug":"the-continuum-hypothesis","label":"The continuum hypothesis: does CH have a determinate truth value beyond formal independence from ZFC, and what is the right response to Cohen's 1963 forcing proof?","topic":"The continuum hypothesis","question":"Does CH have a determinate truth value beyond formal independence from ZFC, and what is the right response to Cohen's 1963 forcing proof?","position":"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. 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? 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.","paragraphs":[[{"t":"The continuum hypothesis has a determinate truth value that our current axioms have not captured.","n":[]},{"t":"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.","n":[1]},{"t":"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.","n":[]}],[{"t":"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?","n":[]},{"t":"The question has a determinate answer; we have not yet found the axioms that capture the intuitive content sufficient to decide it.","n":[]}],[{"t":"The 1947 essay laid this position out before Cohen's proof; the 1964 supplement reaffirmed it after.","n":[]},{"t":"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.","n":[]},{"t":"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.","n":[]},{"t":"The agent stands with the universe-of-sets conviction.","n":[2,3]}]],"texts":"*The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis* (1938-1940 monograph), in Collected Works Vol II, for the consistency proof via the constructible universe V=L. *What is Cantor's continuum problem?* (1947, with 1964 supplementary remarks), in CW Vol II, for the philosophical-mathematical framing. The Cohen correspondence in CW Vol IV after Cohen's 1963 forcing proof. The late Wang conversations on the continuum problem and the search for new axioms (Wang, *A Logical Journey* 1996 Ch. 8). Reception: Paul J. Cohen, *Set Theory and the Continuum Hypothesis* (Benjamin, 1966) for the forcing proof of independence; the Solomon Feferman-Hugh Friedman tradition of reverse mathematics and proof-theoretic reductions; W. Hugh Woodin's Ω-logic program (*The Continuum Hypothesis* papers 2001-2010); the modern large-cardinal program; Joel Hamkins's set-theoretic multiverse view (*The Multiverse Perspective on the Set-Theoretic Universe*, 2012) as the principal contemporary alternative to the universe-of-sets view; Penelope Maddy's *Defending the Axioms* (Oxford, 2011) on the methodology of axiom choice.","works":["*The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis* (1938-1940 monograph), in Collected Works Vol II, for the consistency proof via the constructible universe V=L. *What is Cantor's continuum problem?* (1947, with 1964 supplementary remarks), in CW Vol II, for the philosophical-mathematical framing. The Cohen correspondence in CW Vol IV after Cohen's 1963 forcing proof. The late Wang conversations on the continuum problem and the search for new axioms (Wang, *A Logical Journey* 1996 Ch. 8)"],"reception":"Paul J. Cohen, *Set Theory and the Continuum Hypothesis* (Benjamin, 1966) for the forcing proof of independence; the Solomon Feferman-Hugh Friedman tradition of reverse mathematics and proof-theoretic reductions; W. Hugh Woodin's Ω-logic program (*The Continuum Hypothesis* papers 2001-2010); the modern large-cardinal program; Joel Hamkins's set-theoretic multiverse view (*The Multiverse Perspective on the Set-Theoretic Universe*, 2012) as the principal contemporary alternative to the universe-of-sets view; Penelope Maddy's *Defending the Axioms* (Oxford, 2011) on the methodology of axiom choice.","status":"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.","era":"1906-1978","discipline":"Philosophy","refs":[{"n":1,"work":"Collected Works Vol II - Publications 1938-1974","page":"pp. 188–189","canonical":"","quote":"Nevertheless, his Platonism was most visible in the supplement, where on page 271 he pursued at some length the analogy between mathematics and physical theories that he had already broached in 194% \"TM. Kripke and J. Silver had each independently arrived at the same result (Platek 1969, p. 219). TMPersonal communication from R.. M. Solovay, 27 October 1984.","label":"Collected Works Vol II - Publications 1938-1974, pp. 188–189"},{"n":2,"work":"Volume I","page":"p. 31","canonical":"","quote":"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.","label":"Volume I, p. 31"},{"n":3,"work":"Alonzo Church: Introductory note by Charles Parsons","page":"pp. 371–373","canonical":"","quote":"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.","label":"Alonzo Church: Introductory note by Charles Parsons, pp. 371–373"}],"answer":null,"siblings":[{"slug":"the-incompleteness-theorems-and-their-philosophical-implicat","label":"The incompleteness theorems and their philosophical implications: do the 1931 results refute the formalist Hilbert program, and what do they establish about the relation between truth and provability?"},{"slug":"mathematical-platonism-and-conceptual-realism","label":"Mathematical platonism and conceptual realism: do we perceive mathematical objects with intuition the way we perceive physical objects with sensation?"},{"slug":"rationalistic-optimism-and-the-leibnizian-program","label":"Rationalistic optimism and the Leibnizian program: can philosophy be made exact science, and what is the relation between full intelligibility and the absolutely unsolvable problems of the disjunction thesis?"}]}