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The Continuum Hypothesis, Part I, Volume 48, Number 6 fea-woodin.qxp 6/6/01 4:39 PM Page 567 The Continuum Hypothesis, Part I W. Hugh Woodin Introduction of these axioms and related issues, see [Kanamori, Arguably the most famous formally unsolvable 1994]. problem of mathematics is Hilbert’s first prob- The independence of a proposition φ from the axioms of set theory is the arithmetic statement: lem: ZFC does not prove φ, and ZFC does not prove ¬φ. Cantor’s Continuum Hypothesis: Suppose that Of course, if ZFC is inconsistent, then ZFC proves X ⊆ R is an uncountable set. Then there exists a bi- anything, so independence can be established only jection π : X → R. by assuming at the very least that ZFC is consis- tent. Sometimes, as we shall see, even stronger as- This problem belongs to an ever-increasing list sumptions are necessary. of problems known to be unsolvable from the The first result concerning the Continuum (usual) axioms of set theory. Hypothesis, CH, was obtained by Gödel. However, some of these problems have now Theorem (Gödel). Assume ZFC is consistent. Then been solved. But what does this actually mean? so is ZFC + CH. Could the Continuum Hypothesis be similarly solved? These questions are the subject of this ar- The modern era of set theory began with Cohen’s ticle, and the discussion will involve ingredients discovery of the method of forcing and his appli- from many of the current areas of set theoretical cation of this new method to show: investigation. Most notably, both Large Cardinal Theorem (Cohen). Assume ZFC is consistent. Then Axioms and Determinacy Axioms play central roles. so is ZFC + “CH is false”. For the problem of the Continuum Hypothesis, I shall focus on one specific approach which has de- I briefly discuss the methodology for estab- veloped over the last few years. This should not lishing that a proposition is unsolvable, reviewing be misinterpreted as a claim that this is the only some basic notions from mathematical logic. It is approach or even that it is the best approach. How- customary to work within set theory, though ulti- mately the theorems, fundamentally arithmetic ever, it does illustrate how the various, quite dis- statements, can be proved in number theory. tinct, lines of investigation in modern set theory L(ˆ=, ∈ˆ) denotes the formal language for set the- can collectively yield new, potentially fundamen- ory; this is a countable collection of formulas. The tal, insights into questions as basic as that of the formulas of L(ˆ=, ∈ˆ) with no unquantified occur- Continuum Hypothesis. rences of variables are sentences. Both =ˆ and ∈ˆ are The generally accepted axioms for set theory— simply symbols of this formal language with no but I would call these the twentieth-century other a priori significance. choice—are the Zermelo-Fraenkel Axioms together From elementary logic one has the notion of a with the Axiom of Choice, ZFC. For a discussion structure for this language. This is a pair M = M,E , where M is a nonempty set and W. Hugh Woodin is professor of mathematics at the E ⊆ M × M is a binary relation on the set M. If φ University of California, Berkeley. His e-mail address is is a sentence in the language L(ˆ=, ∈ˆ), then the [email protected]. structure M is a model of φ, written “M|= φ” if JUNE/JULY 2001 NOTICES OF THE AMS 567 fea-woodin.qxp 6/6/01 4:39 PM Page 568 the sentence is true when interpreted as an as- for it is equal to the set {[j]∼∞ | (j,i) ∈ E0}, which sertion within the structure M,E , where the sym- evidently has cardinality at most i. bol ∈ˆ is interpreted by the binary relation E and Building New Models of ZFC =ˆ is interpreted by equality on M. Of course, one Is there a model of ZFC? A consequence of Gödel’s could consider structures of the form M,E,∼ , Second Incompleteness Theorem is that one can- where ∼ is an equivalence relation on M intended not hope to prove the existence of a model of ZFC as the interpretation of =ˆ, but then one could pass working just from the axioms of ZFC. Nevertheless, to the quotient structure M/ ∼,E/∼ . So nothing one can still study the problem of building new is really gained by this attempt at generality. (hopefully interesting) models of ZFC from given A theory is a set of sentences, and for a given models of ZFC. theory T, I write “M,E |= T” to indicate that Gödel solved the substructure problem in 1938, M,E |= φ for each sentence φ ∈ T. showing that if M,E |= ZFC, then there exists ∗ ⊆ ZFC denotes a specific (infinite) theory. A model M M such that ∗ ∗ ∗ of ZFC is simply a structure M,E such that M ,E∩ (M × M ) |= ZFC + CH. | M,E = ZFC. Over 25 years later Cohen, arguably the Galois This can be defined quite naturally without re- of set theory, solved the extension problem. The weakest version of Cohen’s extension theorem is course to formal logic. For example, one of the ax- actually formally equivalent to the statement of ioms of ZFC is the Axiom of Extensionality, which Cohen’s theorem given at the beginning of this ar- is formally expressible as ticle. This weak version simply asserts that if | ∀x1∀x2(x1=ˆx2 ↔∀x3(x3∈ˆx1 ↔ x3∈ˆx2)) M,E = ZFC, then there exists a structure ∗∗ ∗∗ and which informally is just the assertion that two M ,E |= ZFC + “CH is false’’, sets are equal if they have the same elements. such that M ⊆ M∗∗ and such that Thus E = E∗∗ ∩ (M × M). Cohen’s method has turned out to be quite pow- M,E |= “Axiom of Extensionality’’ erful: it and its generalizations are the basic tools if and only if for all a ∈ M and for all b ∈ M, if for establishing independence. Moreover, essen- tially no other effective method for building ex- {c ∈ M | (c,a) ∈ E} = {c ∈ M | (c,b) ∈ E}, tensions of models of ZFC is known. An important point is that neither Cohen’s then a = b. method of extension nor Gödel’s method of re- Therefore R,< |= “Axiom of Extensionality’’ ; striction affects the arithmetic statements true in but if we define E ⊂ N × N by the structures, so the intuition of a true model of E = {(n, m) | For some prime p, number theory remains unchallenged. It seems that most mathematicians do believe n+1| n+2 } p m and p m , that arithmetic statements are either true or false. then N,E |= “Axiom of Extensionality’’ . No generalization of Cohen’s method has yet been Continuing by examining the remaining axioms, discovered to challenge this view. But this is not one can develop naturally the notion of a model to say that such a generalization will never be of ZFC. found. The empirical completeness of arithmetic cou- One can fairly easily define a model which sat- pled with an obvious failure of completeness for isfies all of the axioms of ZFC except for the cru- set theory has led some to speculate that the phe- cial Axiom of Infinity. For example, let E denote 0 nomenon of independence is fundamental, in par- the binary relation on N just specified, and define ticular that the continuum problem is inherently an equivalence relation ∼ by i ∼ j if 0 0 vague with no solution. It is in this view a ques- {k | (k, i) ∈ E } = {k | (k, j) ∈ E } . Define a binary 0 0 tion that is fundamentally devoid of meaning, anal- ∈ ∈ relation E1 by (i,k) E1 if (j,k) E0 for some ogous to asking, “What is the color of π?” ∼ ∼ ∼ j 0 i, and define 1 from E1 as 0 was defined Wherein lies the truth? I shall begin by de- from E0. Continue by induction to define increas- scribing some classical questions of Second Order ∼ ∈ N ∈ N ing sequences n: n and En : n . Let Number Theory—this is the theory of the integers ∼∞= ∪{∼n| n ∈ N}, together with all sets of integers—which are also not solvable from ZFC. Here I maintain there is a and let E∞ = ∪{En | n ∈ N}. The quotient struc- solution: there are axioms for Second Order Num- ture N/ ∼∞,E∞/ ∼∞ satisfies all of the axioms ber Theory which provide a theory as canonical as of ZFC except the Axiom of Infinity. The Axiom of that of number theory. These relatively new axioms Infinity fails to hold, since for each i ∈ N the set provide insights to Second Order Number Theory of equivalence classes {[j]∼∞ | (j,i) ∈ E∞} is finite, which transcend those provided by even ZFC. 568 NOTICES OF THE AMS VOLUME 48, NUMBER 6 fea-woodin.qxp 6/6/01 4:39 PM Page 569 Can these axioms be extended to more compli- cardinals, as are ω and ω1. The assertion that a cated sets in order to solve the continuum prob- set X has cardinality ℵ1 is the assertion that there lem? This question will be the focus of the second exists a bijection of X with ω1. Similarly, X has car- ℵ part of this article. dinality 2 0 , or c, if there is a bijection of X with Some Preliminaries P(N), powerset of N, which is the set of all subsets For purposes of this article it is convenient to work of N. within set theory. This can initially be conceptu- The ordinals measure height in the universe of ally confusing, for we shall work within set theory sets.
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