The Continuum Hypothesis

First published Wed May 22, 2013

The continuum hypothesis (CH) is one of the most central openproblems in set theory, one that is important for both mathematicaland philosophical reasons.

The problem actually arose with the birth of set theory; indeed,in many respects it stimulated the birth of set theory. In 1874 Cantorhad shown that there is a one-to-one correspondence between thenatural numbers and the algebraic numbers. More surprisingly, heshowed that there is no one-to-one correspondence between the naturalnumbers and the real numbers. Taking the existence of a one-to-onecorrespondence as a criterion for when two sets have the same size(something he certainly did by 1878), this result shows that there ismore than one level of infinity and thus gave birth to the higherinfinite in mathematics. Cantor immediately tried to determine whetherthere were any infinite sets of real numbers that wereofintermediate size, that is, whether there was an infiniteset of real numbers that could not be put into one-to-onecorrespondence with the natural numbers and could not be put intoone-to-one correspondence with the real numbers. Thecontinuumhypothesis (under one formulation) is simply the statement thatthere is no such set of real numbers. It was through his attempt toprove this hypothesis that led Cantor do develop set theory into asophisticated branch of mathematics.^[1]

Despite his efforts Cantor could not resolve CH. The problempersisted and was considered so important by Hilbert that he placed itfirst on his famous list of open problems to be faced by the20^th century. Hilbert also struggled to resolve CH,again without success. Ultimately, this lack of progress was explainedby the combined results of Gödel and Cohen, which together showedthat CHcannot be resolved on the basis of the axioms thatmathematicians were employing; in modern terms, CH is independent ofZermelo-Fraenkel set theory extended with the Axiom of Choice (ZFC).

This independence result was quickly followed by many others. Theindependence techniques were so powerful that set theorists soon foundthemselves preoccupied with the meta-theoretic enterprise of provingthat certain fundamental statements couldnot be proved orrefuted within ZFC. The question then arose as to whether there wereways to settle the independent statements. The community ofmathematicians and philosophers of mathematics was largely divided onthis question. Thepluralists (like Cohen) maintained thatthe independence results effectively settled the question by showingthat ithad no answer. On this view, one could adopt asystem in which, say CH was an axiom and one could adopt a system inwhich ¬CH was an axiom and that was the end of thematter—there was no question as to which of two incompatibleextensions was the “correct”one. Thenon-pluralists (like Gödel) held that theindependence results merely indicated the paucity of our means forcircumscribing mathematical truth. On this view, what was needed werenew axioms, axioms that are both justified and sufficient for thetask. Gödel actually went further in proposing candidates for newaxioms—large cardinal axioms—and he conjectured that theywould settle CH.

Gödel's program for large cardinal axioms proved to beremarkably successful. Over the course of the next 30 years it wasshown that large cardinal axioms settle many of the questions thatwere shown to be independent during the era of independence. However,CH was left untouched. The situation turned out to be rather ironicsince in the end it was shown (in a sense that can be made precise)that although the standard large cardinal axioms effectively settleall question of complexity strictly below that of CH, they cannot (byresults of Levy and Solovay and others) settle CH itself. Thus, inchoosing CH as a test case for his program, Gödel put his fingerprecisely on the point where it fails. It is for this reason that CHcontinues to play a central role in the search for new axioms.

In this entry we shall give an overview of the major approaches tosettling CH and we shall discuss some of the major foundationalframeworks which maintain that CH does not have an answer. The subjectis a large one and we have had to sacrifice full comprehensiveness intwo dimensions. First, we have not been able to discuss the majorphilosophical issues that are lying in the background. For this thereader is directed to the entry “Large Cardinals and Determinacy”, which contains a general discussion of theindependence results, the nature of axioms, the nature ofjustification, and the successes of large cardinal axioms in the realm“below CH”. Second, we have not been able to discuss everyapproach to CH that is in the literature. Instead we have restrictedourselves to those approaches that appear most promising from aphilosophical point of view and where the mathematics has beendeveloped to a sufficiently advanced state. In the approaches we shalldiscuss—forcing axioms, inner model theory, quasi-largecardinals—the mathematics has been pressed to a very advancedstage over the course of 40 years. And this has made our task somewhatdifficult. We have tried to keep the discussion as accessible aspossible and we have placed the more technical items in theendnotes. But the reader should bear in mind that we are presenting abird's eye view and that for a higher resolution at any pointthe reader should dip into the suggested readings that appear at theend of each section.^[2]

There are really two kinds of approaches to newaxioms—thelocal approach and theglobalapproach. On the local approach one seeks axioms that answer questionsconcerning a specifiable fragment of the universe, suchasV_ω+1 orV_ω+2, whereCH lies. On the global approach one seeks axioms that attempt toilluminate theentire structure of the universe of sets. Theglobal approach is clearly much more challenging. In this entry weshall start with the local approach and toward the end we shallbriefly touch upon the global approach.

Here is an overview of the entry: Section 1 surveys theindependence results in cardinal arithmetic, covering both the case ofregular cardinals (where CH lies) and singular cardinals. Section 2considers approaches to CH where one successively verifies a hierarchyof approximations to CH, each of which is an “effective”version of CH. This approach led to the remarkable discovery of Woodinthat it is possible (in the presence of large cardinals) to have aneffective failure of CH, thereby showing, that the effective failureof CH is as intractable (with respect to large cardinal axioms) as CHitself. Section 3 continues with the developments that stemmed fromthis discovery. The centerpiece of the discussion is the discovery ofa “canonical” model in which CH fails. This formed thebasis of a network of results that was collectively presented byWoodin as a case for the failure of CH. To present this case in themost streamlined form we introduce the strong logicΩ-logic. Section 4 takes up the competing foundational view thatthere is no solution to CH. This view is sharpened in terms ofthegeneric multiverse conception of truth and that view isthen scrutinized. Section 5 continues the assessment of the case for¬CH by investigating a parallel case for CH. In the remaining twosections we turn to the global approach to new axioms and here weshall be much briefer. Section 6 discusses the approach through innermodel theory. Section 7 discusses the approach through quasi-largecardinal axioms.

1. Independence in Cardinal Arithmetic

In this section we shall discuss the independence results incardinal arithmetic. First, we shall treat of the case of regularcardinals, where CH lies and where very little is determined in thecontext of ZFC. Second, for the sake of comprehensiveness, we shalldiscuss the case of singular cardinals, where much more can beestablished in the context of ZFC.

1.1 Regular Cardinals

The addition and multiplication of infinite cardinal numbers istrivial: For infinite cardinals κ and λ,

κ + λ = κ ⋅ λ = max{κ,λ}.

The situation becomes interesting when one turns to exponentiationand the attempt to compute κ^λ for infinitecardinals.

During the dawn of set theory Cantor showed that for everycardinal κ,

2^κ > κ.

There is no mystery about the size of 2ⁿ forfiniten. The first natural question then is where2^ℵ₀ is located in the aleph-hierarchy: Is itℵ₁, ℵ₂, …, ℵ₁₇or something much larger?

The cardinal 2^ℵ₀ is important sinceit is the size of the continuum (the set of real numbers). Cantor'sfamouscontinuum hypothesis (CH) is the statement that2^ℵ₀ =ℵ₁. This is a special case of thegeneralizedcontinuum hypothesis (GCH) which asserts that for all α,2^ℵ_α =ℵ_α+1. One virtue of GCH is that it gives acomplete solution to the problem of computingκ^λ for infinite cardinals: Assuming GCH, ifκ ≤ λ then κ^λ =λ⁺; if cf(κ) ≤ λ ≤ κ thenκ^λ = κ⁺; and if λ <cf(κ) then κ^λ = κ.

Very little progress was made on CH and GCH. In fact, in the earlyera of set theory the only other piece of progress beyondCantor's result that 2^κ > κ (andthe trivial result that if κ ≤ λ then2^κ ≤ 2^λ) was König'sresult that cf(2^κ) > κ. The explanationfor the lack of progress was provided by the independence results inset theory:

Theorem 1.1 (Gödel1938a, 1938b).: Assume that ZFCis consistent. ThenZFC + CHand ZFC + GCHare consistent.

To prove this Gödel invented the method ofinnermodels —he showed that CH and GCH held in the minimal innermodelL of ZFC. Cohen then complemented this result:

Theorem 1.2 (Cohen1963).: Assume that ZFCisconsistent. Then ZFC + ¬CHand ZFC +¬GCHare consistent.

He did this by inventing the method ofouter models andshowing that CH failed in ageneric extensionV^B ofV. The combinedresults of Gödel and Cohen thus demonstrate that assuming theconsistency of ZFC, it is in principle impossible to settle either CHor GCH in ZFC.

In the Fall of 1963 Easton completed the picture by showing thatfor infinite regular cardinals κ theonly constraintson the function κ ↦ 2^κ that are provable inZFC are the trivial constraint and the results of Cantor andKönig:

Theorem 1.3 (Easton1963).

Assume that ZFCisconsistent. SupposeFis a(definable class)function defined on infiniteregular cardinals such that

if κ ≤ λthenF(κ) ≤F(λ),
F(κ) > κ, and
cf(F(κ)) > κ.

Then ZFC + “For all infinite regularcardinals κ, 2^κ =F(κ)”is consistent.

Thus, set theorists had pushed the cardinal arithmetic of regularcardinals as far as it could be pushed within the confines ofZFC.

1.2 Singular Cardinals

The case of cardinal arithmetic on singular cardinals is much moresubtle. For the sake of completeness we pause to briefly discuss thisbefore proceeding with the continuum hypothesis.

It was generally believed that, as in the case for regularcardinals, the behaviour of the functionκ ↦ 2^κ would be relatively unconstrainedwithin the setting of ZFC. But then Silver proved the followingremarkable result:^[3]

Theorem 1.4 (Silver 1974).: Ifℵ_δis a singular cardinal of uncountablecofinality, then, if GCHholds belowℵ_δ, then GCHholdsat ℵ_δ.

It turns out that (by a deep result of Magidor, published in 1977)GCH can first fail at ℵ_ω (assuming theconsistency of a supercompact cardinal). Silver's theorem showsthat it cannot first fail at ℵ_ω₁andthis is provable in ZFC.

This raises the question of whether one can “control”the size of 2^ℵ_δ with a weaker assumptionthan that ℵ_δ is a singular cardinal ofuncountable cofinality such that GCH holds belowℵ_δ. The natural hypothesis to consider isthat ℵ_δ is a singular cardinal of uncountablecofinality which isa strong limit cardinal, that is, thatfor all α < ℵ_δ,2^α < ℵ_δ. In 1975Galvin and Hajnal proved (among other things) that under this weakerassumption there is indeed a bound:

Theorem 1.5 (Galvin and Hajnal1975).: Ifℵ_δis a singular strong limit cardinal ofuncountable cofinality then
2^ℵ_δ <ℵ_{(|δ|^cf(δ))⁺}.

It is possible that there is a jump—in fact, Woodin showed(again assuming large cardinals) that it is possible that for allκ, 2^κ = κ⁺⁺. What the abovetheorem shows is that in ZFC there is a provable bound on how big thejump can be.

The next question is whether a similar situation prevails withsingular cardinals of countable cofinality. In 1978 Shelah showed thatthis is indeed the case. To fix ideas let us concentrate onℵ_ω.

Theorem 1.6 (Shelah 1978).: Ifℵ_ωis a strong limit cardinalthen
2^ℵ_ω <ℵ_{(2^ℵ₀)⁺}.

One drawback of this result is that the bound is sensitive to theactual size of 2^ℵ₀, which can be anything belowℵ_ω. Remarkably Shelah was later able toremedy this with the development of his pcf (possible cofinalities)theory. One very quotable result from this theory is thefollowing:

Theorem 1.7(Shelah 1982).: Ifℵ_ωis a strong limit cardinal then(regardless of the size of2^ℵ₀)
2^ℵ_ω <ℵ_ω₄.

In summary, although the continuum function at regular cardinalsis relatively unconstrained in ZFC, the continuum function at singularcardinals is (provably in ZFC) constrained in significant ways by thebehaviour of the continuum function on the smaller cardinals.

Further Reading: For more cardinal arithmetic see Jech(2003). For more on the case of singular cardinals and pcf theory seeAbraham & Magidor (2010) and Holz, Steffens & Weitz(1999).

2. Definable Versions of the Continuum Hypothesis and its Negation

Let us return to the continuum function on regular cardinals andconcentrate on the simplest case, the size of2^ℵ₀. One of Cantor's original approachesto CH was by investigating “simple” sets of realnumbers (see Hallett (1984), pp. 3–5 and §2.3(b)). One of the first results in this direction is theCantor-Bendixson theorem that every infinite closed set is eithercountable or contains a perfect subset, in which case it has the samecardinality as the set of reals. In other words, CH holds (in thisformulation) when one restricts one's attention to closed setsof reals. In general, questions about “definable” sets ofreals are more tractable than questions about arbitrary sets of realsand this suggests looking at definable versions of the continuumhypothesis.

2.1 Three Versions

There are three different formulations of the continuumhypothesis—theinterpolant version,thewell-ordering version, and thesurjectionversion. These versions are all equivalent to one another in ZFC butwe shall be imposing a definability constraint and in this case therecan be interesting differences (our discussion followsMartin (1976)). There is really a hierarchy of notionsof definability—ranging up through the Borel hierarchy, theprojective hierarchy, the hierarchy inL(ℝ), and, moregenerally, the hierarchy of universally Baire sets—and so eachof these three general versions is really a hierarchy of versions,each corresponding to a given level of the hierarchy ofdefinability (for a discussion of the hierarchy ofdefinability see§2.2.1 and §4.6 of the entry “Large Cardinals and Determinacy”).

2.1.1 Interpolant Version

The first formulation of CH is that there isnointerpolant, that is, there is no infinite setAof real numbers such that the cardinality ofA is strictlybetween that of the natural numbers and the real numbers. To obtaindefinable versions one simply asserts that there is no“definable” interpolant and this leads to a hierarchy ofdefinable interpolant versions, depending on which notion ofdefinability one employs. More precisely, for a given pointclassΓ in the hierarchy of definable sets of reals, the correspondingdefinable interpolant version of CH asserts that there is nointerpolant in Γ.

The Cantor-Bendixson theorem shows that there is no interpolant inΓ in the case where Γ is the pointclass of closed sets,thus verifying this version of CH. This was improved by Suslin whoshowed that this version of CH holds for Γ where Γ is theclass of Σ̰11 sets. One cannot go much further within ZFC—to prove strongerversions one must bring in stronger assumptions. It turns out thataxioms of definable determinacy and large cardinal axioms achievethis. For example, results of Kechris and Martin show that ifΔ̰1n-determinacyholds then this version of CH holds for the pointclass ofΣ̰1n+1sets. Going further, if one assumes AD^L(ℝ) thenthis version of CH holds for all sets of real numbers appearing inL(ℝ). Since these hypotheses follow from large cardinal axiomsone also has that stronger and stronger large cardinal assumptionssecure stronger and stronger versions of this version of the effectivecontinuum hypothesis. Indeed large cardinal axioms imply that thisversion of CH holds forall sets of reals in the definabilityhierarchy we are considering; more precisely, if there is a properclass of Woodin cardinals then this version of CH holds for alluniversally Baire sets of reals.

2.1.2 Well-ordering Version

The second formulation of CH asserts that every well-ordering ofthe reals has order type less than ℵ₂. For a givenpointclass Γ in the hierarchy, the corresponding definablewell-ordering version of CH asserts that every well-ordering (coded bya set) in Γ has order type less than ℵ₂.

Again, axioms of definable determinacy and large cardinal axiomsimply this version of CH for richer notions of definability. Forexample, if AD^L(ℝ) holds then this version of CHholds for all sets of real numbers inL(ℝ). And if there is aproper class of Woodin cardinals then this version of CH holds for alluniversally Baire sets of reals.

2.1.3 Surjection Version

The third version formulation of CH asserts that there is nosurjection ρ : ℝ → ℵ₂, or,equivalently, that there is no prewellordering of ℝ of lengthℵ₂. For a given pointclass Γ in the hierarchyof definability, the corresponding surjection version of CH assertsthat there is no surjection ρ : ℝ → ℵ₂ such that (the code for) ρ is in Γ.

Here the situation is more interesting. Axioms of definabledeterminacy and large cardinal axioms have bearing on this versionsince they place bounds on how long definable prewellorderings canbe. Let δ̰1nbe the supremum of the lengths of theΣ̰1n-prewellorderingsof reals and let Θ^L(ℝ) be the supremum of thelengths of prewellorderings of reals where the prewellordering isdefinable in the sense of being inL(ℝ). It is a classicalresult that δ̰11 =ℵ₁. Martin showed that δ̰12≤ ℵ₂ and that if there is a measurable cardinalthen δ̰13≤ ℵ₃. Kunen and Martin also showed under PD,δ̰14≤ ℵ₄ and Jackson showed that under PD, for each n< ω, δ̰1n< ℵ_ω. Thus, assuming that there areinfinitely many Woodin cardinals, these bounds hold. Moreover, thebounds continue to hold regardless of the size of2^ℵ₀. Of course, the question is whether thesebounds can be improved to show that the prewellorderings are shorterthan ℵ₂. In 1986 Foreman and Magidor initiated aprogram to establish this. In the most general form they aimed to showthat large cardinal axioms implied that this version of CH held forall universally Baire sets of reals.

2.1.4 Potential Bearing on CH

Notice that in the context of ZFC, these three hierarchies ofversions of CH are all successive approximations of CH and in thelimit case, where Γ is the pointclass of all sets of reals, theyare equivalent to CH. The question is whether these approximations canprovide any insight into CH itself.

There is an asymmetry that was pointed out by Martin, namely, thata definable counterexample to CH is a real counterexample, while nomatter how far one proceeds in verifying definable versions of CH atno stage will one have touched CH itself. In other words, thedefinability approach could refute CH but it could not prove it.

Still, one might argue that although the definability approachcould not prove CH it might provide some evidence for it. In the caseof the first two versions we now know that CH holds for all definablesets. Does this provide evidence of CH? Martin pointed out (before thefull results were known) that this is highly doubtful since in eachcase one is dealing with sets that are atypical. For example, in thefirst version, at each stage one secures the definable version of CHby showing that all sets in the definability class have the perfectset property; yet such sets are atypical in that assuming AC it iseasy to show that there are sets without this property. In the secondversion, at each stage one actually shows not only that eachwell-ordering of reals in the definability class has ordertype lessthan ℵ₂, but also that it has ordertype less thanℵ₁. So neither of these versions really illuminatesCH.

The third version actually has an advantage in this regard sincenot all of the sets it deals with are atypical. For example, while allΣ̰11-setshave length less than ℵ₁, there areΠ̰11-setsof length ℵ₁. Of course, it could turn out thateven if the Foreman-Magidor program were to succeed the sets couldturn out to be atypical in another sense, in which case it would shedlittle light on CH. More interesting, however, is the possibility thatin contrast to the first two versions, it would actually provide anactual counterexample to CH. This, of course, would require thefailure of the Foreman-Magidor program.

2.2 The Foreman-Magidor Program

The goal of the Foreman-Magidor program was to show that largecardinal axioms also implied that the third version of CH held for allsets inL(ℝ) and, more generally, all universally Baire sets. Inother words, the goal was to show that large cardinal axioms impliedthat Θ^L(ℝ) ≤ ℵ₂ and, moregenerally, that Θ^L(A,ℝ)≤ ℵ₂ for each universally BairesetA.

The motivation came from the celebrated results of Foreman,Magidor and Shelah on Martin's Maximum (MM), which showed thatassuming large cardinal axioms one can always force to obtain aprecipitous ideal on ℵ₂ without collapsingℵ₂ (see Foreman, Magidor & Shelah (1988)). The program involved a two-part strategy:

Strengthen this result to show that assuming largecardinal axioms one can always force to obtain asaturatedideal on ℵ₂ without collapsingℵ₂.
Show that the existence ofsuch a saturated ideal implies that Θ^L(ℝ)≤ ℵ₂ and, more generally thatΘ^L(A,ℝ) ≤ ℵ₂ for everyuniversally Baire setA.

This would show that show that Θ^L(ℝ)≤ ℵ₂ and, more generally thatΘ^L(A,ℝ) ≤ ℵ₂ for everyuniversally Baire setA.^[4]

In December 1991, the following result dashed the hopes of thisprogram.

Theorem 2.1(Woodin).: Assume that the non-stationary idealon ℵ₁is saturated and that there is ameasurable cardinal. Then δ̰12 =ℵ₂.

The point is that the hypothesis of this theorem can always beforced assuming large cardinals. Thus, it is possible to haveΘ^L(ℝ) > ℵ₂ (in fact,δ̰13> ℵ₂).

Where did the program go wrong? Foreman and Magidor had anapproximation to (B) and in the end it turned out that (B) istrue.

Theorem 2.2(Woodin).: Assume that there is a properclass of Woodin cardinals and that there is a saturated idealon ℵ₂. Then for every A∈ Γ^∞,Θ^L(A,ℝ)≤ ℵ₂.

So the trouble is with (A).

This illustrates an interesting contrast between our threeversions of the effective continuum hypothesis, namely, that they cancome apart. For while large cardinals rule out definablecounterexamples of the first two kinds, they cannot rule out definablecounterexamples of the third kind. But again we must stress that theycannot prove that thereare such counterexamples.

But there is an important point: Assuming large cardinal axioms(AD^L(ℝ) suffices), although one can produce outermodels in which δ̰13> ℵ₂ it is not currently known how toproduce outer models in which δ̰13> ℵ₃ or even Θ^L(ℝ)> ℵ₃. Thus it is an open possibility thatfrom ZFC +AD^L(ℝ) one can proveΘ^L(ℝ) ≤ ℵ₃. Were this tobe the case, it would follow that although large cardinals cannot ruleout the definable failure of CHthey can rule outthe definable failure of 2^ℵ₀ =ℵ₂. This could provide some insight into the sizeof the continuum, underscoring the centrality ofℵ₂.

Further Reading: For more on the three effective versionsof CH see Martin (1976); for more on the Foreman-Magidor program seeForeman & Magidor (1995) and the introduction to Woodin (1999).

3. The Case for ¬CH

The above results led Woodin to the identification of a“canonical” model in which CH fails and this formed thebasis of his an argument that CH is false. In Section 3.1 we willdescribe the model and in the remainder of the section we will presentthe case for the failure of CH. InSection 3.2 we will introduceΩ-logic and the other notions needed to make the case. InSection 3.3 we will present the case.

3.1 ℙ_max

The goal is to find a model in which CH is false and which iscanonical in the sense that its theory cannot be altered by setforcing in the presence of large cardinals. The background motivationis this: First, we know that in the presence of large cardinal axiomsthe theory of second-order arithmetic and even the entire theory ofL(ℝ) is invariant under set forcing. The importance of this isthat it demonstrates that our main independence techniques cannot beused to establish the independence of questions about second-orderarithmetic (or aboutL(ℝ)) in the presence of largecardinals. Second, experience has shown that the large cardinal axiomsin question seem to answer all of the major known open problems aboutsecond-order arithmetic andL(ℝ) and the set forcing invariancetheorems give precise content to the claim that these axioms are“effectively complete”.^[5]

It follows that if ℙ is any homogeneous partial order inL(ℝ) then the generic extensionL(ℝ)^ℙinherits the generic absoluteness ofL(ℝ). Woodin discoveredthat there is a very special partial orderℙ_max that has this feature. Moreover, themodelL(ℝ)^ℙ_max satisfies ZFC +¬CH. The key feature of this model is that it is“maximal” (or “saturated”) with respect tosentences that are of a certain complexity and which can be shown tobe consistent via set forcing over the model; in other words, if thesesentencescan hold (by set forcing over the model)then theydo hold in the model. To state this more preciselywe are going to have to introduce a few rather technical notions.

There are two ways of stratifying the universe of sets. The firstis in terms of ⟨V_α | α∈ On ⟩, the second is in terms of⟨H(κ) | κ ∈ Card⟩, whereH(κ) is the setof all sets which have cardinality less than κ and whose membershave cardinality less than κ, and whose members of members havecardinality less than κ, and so on. For example,H(ω)=V_ω and the theories of the structuresH(ω₁) andV_ω+1 are mutuallyinterpretable. This latter structure is the structure of second-orderarithmetic and, as mentioned above, large cardinal axioms give us an“effectively complete” understanding of this structure. Weshould like to be in the same position with regard to larger andlarger fragments of the universe and the question is whether we shouldproceed in terms of the first or the second stratification.

The second stratification is potentially morefine-grained. Assuming CH one has that the theories ofH(ω₂) andV_ω+2 are mutuallyinterpretable and assuming larger and larger fragments of GCH thiscorrespondence continues upward. But if CH is false then the structureH(ω₂) is less rich than thestructureV_ω₂. In this event the latterstructure captures full third-order arithmetic, while the formercaptures only a small fragment of third-order arithmetic but isnevertheless rich enough to express CH. Given this, in attempting tounderstand the universe of sets by working up through it level bylevel, it is sensible to use the potentially more fine-grainedstratification.

Our next step is therefore to understandH(ω₂). It actually turns out that we will be able tounderstand slightly more and this is somewhat technical. We will beconcerned with the structure ⟨H(ω₂), ∈,I_NS,A^G⟩⊧ φ, whereI_NS is the non-stationary idealon ω₁ andA^G is theinterpretation of (the canonical representation of) a set ofrealsA inL(ℝ). The details will not be important andthe reader is asked to just think ofH(ω₂) along withsome “extra stuff” and not worry about the detailsconcerning the extra stuff.^[6]

We are now in a position to state the main result:

Theorem 3.1(Woodin 1999).: Assume ZFCand that there isa proper class of Woodin cardinals. Suppose thatA ∈P (ℝ) ∩L(ℝ)and φis aΠ₂-sentence (in the extendedlanguage with two additional predicates )and thereis a set forcing extensionV[G]such that
⟨H(ω₂), ∈,I_NS,A^G⟩⊧ φ; (whereA^Gis theinterpretation ofAinV[G]). Then
L(ℝ)^ℙ_max ⊧ “⟨H(ω₂), ∈,I_NS, A⟩⊧ φ”.

There are two key points: First, the theory ofL(ℝ)^ℙ_max is “effectivelycomplete” in the sense that it is invariant under setforcing. Second, the modelL(ℝ)^ℙ_maxis “maximal” (or “saturated”) in the sensethat it satisfies all Π₂-sentences (about the relevantstructure) that can possibly hold (in the sense that they can be shownto be consistent by set forcing over the model).

One would like to get a handle on the theory of this structure byaxiomatizing it. The relevant axiom is the following:

Definition 3.2 (Woodin 1999).: Axiom(∗): AD^L(ℝ) holds andL(P(ω₁)) is a ℙ_max-genericextension ofL(ℝ).

Finally, this axiom settles CH:

Theorem 3.3(Woodin 1999).: Assume(∗). Then 2^ω =ℵ₂.

3.2 Ω-Logic

We will now recast the above results in terms of a strong logic. Weshall make full use of large cardinal axioms and in this setting weare interested in logics that are “well-behaved” in thesense that the question of what implies what is not radicallyindependent. For example, it is well known that CH is expressible infull second-order logic. It follows that in the presence of largecardinals one can always use set forcing to flip the truth-value of apurported logical validity of full second-order logic. However, thereare strong logics—like ω-logic and β-logic—thatdo not have this feature—they are well-behaved in the sense thatin the presence of large cardinal axioms the question of what implieswhat cannot be altered by set forcing. We shall introduce a verystrong logic that has this feature—Ω-logic. In fact, thelogic we shall introduce can be characterized asthestrongest logic with this feature (seeKoellner (2010) for further discussion of strong logics and for aprecise statement of this result).

3.2.1 Ω-logic

Definition 3.4.: Suppose thatT is acountable theory in the language of set theory and φ is asentence. Then; T ⊧_Ω φ; if for all complete Boolean algebrasB and for all ordinalsα,; ifVBα ⊧ T thenVBα ⊧ φ.

We say that a statement φ is Ω-satisfiable ifthere exists an ordinal α and a complete BooleanalgebraB suchthatVBα ⊧ φ,and we say that φ is Ω-valid if∅ ⊧_Ω φ. So, the above theorem saysthat (under our background assumptions), the statement “φis Ω-satisfiable” is generically invariant and in terms ofΩ-validity this is simply the following:

Theorem 3.5(Woodin 1999).: Assume ZFCand that there isa proper class of Woodin cardinals. Suppose thatTisa countable theory in the language of set theory andφis a sentence. Then for all complete BooleanalgebrasB,; T ⊧_Ω φ iffV^B ⊧ “T ⊧_Ω φ.”

Thus this logic is robust in that the question of what implies whatis invariant under set forcing.

3.2.2 The Ω Conjecture

Corresponding to the semantic relation ⊧_Ωthere is a quasi-syntactic proof relation⊢_Ω. The “proofs” are certain robustsets of reals (universally Baire sets of reals) and the teststructures are models that are “closed” under theseproofs. The precise notions of “closure” and“proof” are somewhat technical and so we will pass overthem in silence.^[7]

Like the semantic relation, this quasi-syntactic proof relation isrobust under large cardinal assumptions:

Theorem 3.6(Woodin 1999).: Assume ZFCand that there isa proper class of Woodin cardinals. SupposeTis acountable theory in the language of set theory, φis asentence, andBis a complete Booleanalgebra. Then; T⊢_Ω φ iffV^B ⊧ ‘T⊢_Ω φ’.

Thus, we have a semantic consequence relation and aquasi-syntactic proof relation, both of which are robust under theassumption of large cardinal axioms. It is natural to ask whether thesoundness and completeness theorems hold for these relations. Thesoundness theorem is known to hold:

Theorem 3.7(Woodin 1999).: AssumeZFC. SupposeTis a countable theory inthe language of set theory and φis a sentence. IfT ⊢_Ω φthenT ⊧_Ω φ.

It is open whether the corresponding completeness theoremholds. The Ω Conjecture is simply the assertion that itdoes:

Conjecture 3.8(ΩConjecture ).: Assume ZFCand that there is a proper class of Woodin cardinals. Then for eachsentence φ,; ∅ ⊧_Ω φ iff ∅ ⊢_Ω φ.

We will need a strong form of this conjecture which we shall callthe Strong Ω Conjecture. It is somewhat technical and so we willpass over it in silence.^[8]

3.2.3 Ω-Complete Theories

Recall that one key virtue of large cardinal axioms is that they“effectively settle” the theory of second-order arithmetic(and, in fact, the theory ofL(ℝ) and more) in the sense that inthe presence of large cardinals one cannot use the method of setforcing to establish independence with respect to statements aboutL(ℝ). This notion of invariance under set forcing played a keyrole inSection 3.1. We can now rephrase this notion in terms ofΩ-logic.

Definition 3.9.: A theoryT isΩ-complete for a collection of sentences Γ if foreach φ ∈ Γ, T ⊧_Ω φ orT ⊧_Ω ¬φ.

The invariance of the theory ofL(ℝ) under set forcing cannow be rephrased as follows:

Theorem 3.10(Woodin 1999).: Assume ZFCand that there isa proper class of Woodin cardinals. Then ZFCisΩ-complete for the collection of sentences of theform “L(ℝ) ⊧ φ”.

Unfortunately, it follows from a series of results originatingwith work of Levy and Solovay that traditional large cardinal axiomsdo not yield Ω-complete theories at the level ofΣ21 sinceone can always use a “small” (and hence large cardinalpreserving) forcing to alter the truth-value of CH.

Theorem 3.11.: AssumeLis astandard large cardinal axiom. Then ZFC + Lis notΩ-complete forΣ21.

3.3 The Case

Nevertheless, if one supplements large cardinal axioms thenΩ-complete theories are forthcoming. This is the centerpiece ofthe case against CH.

Theorem 3.12(Woodin).

Assume that there is a proper class ofWoodin cardinals and that the StrongΩConjecture holds.

There is an axiomAsuchthat
1. ZFC +AisΩ-satisfiable and
2. ZFC +AisΩ-complete for the structureH(ω₂).
Any such axiomAhas the feature that
ZFC +A ⊧_Ω ‘H(ω₂) ⊧ ¬CH’.

Let us rephrase this as follows: For eachA satisfying (1),let

T_A = {φ | ZFC +A ⊧_Ω ‘H(ω₂) ⊧ ¬φ’}.

The theorem says that if there is a proper class of Woodincardinals and the Ω Conjecture holds, then there are(non-trivial) Ω-complete theoriesT_A ofH(ω₂) and all such theories contain ¬CH.

It is natural to ask whether there is greater agreement among theΩ-complete theoriesT_A. Ideally, therewould be just one. A recent result (building on Theorem 5.5) showsthat if there is one such theory then there are many suchtheories.

Theorem 3.13 (Koellner andWoodin 2009).: Assume that there is a proper class ofWoodin cardinals. Suppose thatAis an axiomsuch that; i. ZFC +Ais Ω-satisfiable and
ii. ZFC +AisΩ-complete for the structureH(ω₂).; Then there is an axiomBsuch that; i′. ZFC +Bis Ω-satisfiable and
ii′. ZFC +Bis Ω-complete for the structureH(ω₂); andT_A≠T_B.

How then shall one select from among these theories?Woodin's work in this area goes a good deal beyond Theorem5.1. In addition to isolating an axiom that satisfies (1) of Theorem5.1 (assuming Ω-satisfiability), he isolates a very special suchaxiom, namely, the axiom (∗) (“star”) mentionedearlier.

This axiom can be phrased in terms of (the provability notion of)Ω-logic:

Theorem 3.14(Woodin).

Assume ZFCand that there isa proper class of Woodin cardinals. Then the followingare equivalent:

(∗).
For eachΠ₂-sentence φin the language for thestructure
⟨H(ω₂), ∈,I_NS,A |A ∈ 𝒫(ℝ) ∩L(ℝ)⟩
if
ZFC + “⟨H(ω₂), ∈,I_NS,A |A ∈ 𝒫(ℝ) ∩L(ℝ)⟩ ⊧ φ”
is Ω-consistent, then
⟨H(ω₂), ∈,I_NS,A |A ∈ 𝒫(ℝ) ∩L(ℝ)⟩ ⊧ φ.

It follows that of the varioustheoriesT_A involved in Theorem 5.1, there isone that stands out: The theoryT_(∗) given by(∗). This theory maximizes the Π₂-theory of thestructure ⟨H(ω₂), ∈,I_NS,A |A ∈ 𝒫(ℝ) ∩L(ℝ)⟩.

The continuum hypothesis fails in this theory. Moreover, in themaximal theoryT_(∗) given by (∗) thesize of the continuum is ℵ₂.^[9]

To summarize: Assuming the Strong Ω Conjecture, there is a“good” theory ofH(ω₂) and all suchtheories imply that CH fails. Moreover, (again, assuming the StrongΩ Conjecture) there is a maximal such theory and in that theory2^ℵ₀ = ℵ₂.

Further Reading: For the mathematics concerningℙ_max see Woodin (1999). For an introduction toΩ-logic see Bagaria, Castells & Larson (2006). Formore on incompatible Ω-complete theories see Koellner & Woodin(2009). For more on the case against CH see Woodin (2001a,b, 2005a,b).

4. The Multiverse

The above case for the failure of CH is the strongest known localcase for axioms that settle CH. In this section and the next we willswitch sides and consider the pluralist arguments to the effect thatCH does not have an answer (in this section) and to the effect thatthere is an equally good case for CH (in the next section). In thefinal two section we will investigate optimistic global scenarios thatprovide hope of settling the issue.

The pluralist maintains that the independence results effectivelysettle the undecided questions by showing that they have noanswer. One way of providing a foundational framework for such a viewis in terms of the multiverse. On this view there is not asingleuniverse of set theory but ratheramultiverse of legitimate candidates, some of which may bepreferable to others for certain purposes but none of which can besaid to be the “true” universe. Themultiverseconception of truth is the view that a statement of set theorycan only be said to be true simpliciter if it is true in all universesof the multiverse. For the purposes of this discussion we shall saythat a statement isindeterminate according to the multiverseconception if it is neither true nor false according to themultiverse conception. How radical such a view is depends on thebreadth of the conception of the multiverse.

4.1 Broad Multiverse Views

The pluralist is generally a non-pluralist about certain domains ofmathematics. For example, a strict finitist might be a non-pluralistabout PA but a pluralist about set theory and one might be anon-pluralist about ZFC and a pluralist about large cardinal axiomsand statements like CH.

There is a form of radical pluralism which advocates pluralismconcerning all domains of mathematics. On this view any consistenttheory is a legitimate candidate and the corresponding models of suchtheories are legitimate candidates for the domain ofmathematics. Let us call this thebroadest multiverseview. There is a difficulty in articulating this view, which may bebrought out as follows: To begin with, one must pick a backgroundtheory in which to discuss the various models and this leads to adifficult. For example, according to the broad multiverse conception,since PA cannot prove Con(PA) (by the second incompleteness theorem,assuming that PA is consistent) there are models of PA + ¬Con(PA)and these models are legitimate candidates, that is, they areuniverses within the broad multiverse. Now to arrive at thisconclusion one must (in the background theory) be in a position toprove Con(PA) (since this assumption is required to apply the secondincompleteness theorem in this particular case). Thus, from theperspective of the background theory used to argue that the abovemodels are legitimate candidates, the models in question satisfy afalseΣ01-sentence,namely, ¬Con(PA). In short, there is a lack of harmony betweenwhat is held at the meta-level and what is held at theobject-level.

The only way out of this difficulty would seem to be to regardeach viewpoint—each articulation of the multiverseconception—as provisional and, when pressed, embrace pluralismconcerning the background theory. In other words, one would have toadopt a multiverse conception of the multiverse, a multiverseconception of the multiverse conception of the multiverse, and so on,off to infinity. It follows that such a position can never be fullyarticulated—each time one attempts to articulate the broadmultiverse conception one must employ a background theory but sinceone is a pluralist about that background theory this pass at using thebroad multiverse to articulate the conception does not do theconception full justice. The position is thus difficult toarticulate. One can certainly take the pluraliststance andtry togesture toward orexhibit the view that oneintends by provisionally settling on a particular background theorybut then advocate pluralism regardingthat when pressed. Theview is thus something of a “moving target”. We shall passover this view in silence and concentrate on views that can bearticulated within a foundational framework.

We will accordingly look at views which embrace non-pluralism withregard to a given stretch of mathematics and for reasons of space andbecause this is an entry on set theory we will pass over the longdebates concerning strict finitism, finitism, predicativism, and startwith views that embrace non-pluralism regarding ZFC.

Let thebroad multiverse (based on ZFC) be the collectionof all models of ZFC. The broad multiverse conception of truth (basedon ZFC) is then simply the view that a statement of set theory is truesimpliciter if it is provable in ZFC. On this view the statementCon(ZFC) and other undecidedΠ01-statementsare classified as indeterminate. This view thus faces a difficultyparallel to the one mentioned above concerning radical pluralism.

This motivates the shift to views that narrow the class ofuniverses in the multiverse by employing a strong logic. For example,one can restrict to universes that are ω-models, β-models(i.e., wellfounded), etc. On the view where one takes ω-models,the statement Con(ZFC) is classified as true (though this is sensitiveto the background theory) but the statement PM (all projective setsare Lebesgue measurable) is classified as indeterminate.

For those who are convinced by the arguments (surveyed in theentry “Large Cardinals and Determinacy”)for largecardinal axioms and axioms of definable determinacy, even thesemultiverse conceptions are too weak. We will follow this route. Forthe rest of this entry we will embrace non-pluralism concerning largecardinal axioms and axioms of definable determinacy and focus on thequestion of CH.

4.2 The Generic Multiverse

The motivation behind the generic multiverse is to grant the casefor large cardinal axioms and definable determinacy but deny thatstatements such as CH have a determinate truth value. To be specificabout the background theory let us take ZFC + “There is a properclass of Woodin cardinals” and recall that this large cardinalassumption secures axioms of definable determinacy such as PD andAD^L(ℝ).

Let thegeneric multiverse𝕍 be the result ofclosingV under generic extensions and generic refinements. Oneway to formalize this is by taking an external vantage point and startwith a countable transitive modelM. The generic multiversebased onM is then the smallest set𝕍_Msuch that M ∈𝕍_Mand, for each pair ofcountable transitive models (N,N[G]) such thatN ⊧ ZFC andG⊆ ℙ isN-generic for some partial order in ℙ∈N, if eitherN orN[G] is in𝕍_Mthen bothN andN[G] are in𝕍_M.

Let thegeneric multiverse conception of truth be theview that a statement is true simpliciter iff it is true in alluniverses of the generic multiverse. We will call such a statementageneric multiverse truth. A statement is said tobeindeterminate according to the generic multiverseconception iff it is neither true nor false according to thegeneric multiverse conception. For example, granting our largecardinal assumptions, such a view deems PM (and PD andAD^L(ℝ)) true but deems CH indeterminate.

4.3 The Ω Conjecture and the Generic Multiverse

Is the generic multiverse conception of truth tenable? The answerto this question is closely related to the subject ofΩ-logic. The basic connection between generic multiverse truthand Ω-logic is embodied in the following theorem:

Theorem 4.1(Woodin).

Assume ZFCand that there isa proper class of Woodin cardinals. Then, for eachΠ₂-statement φthefollowing are equivalent:

φis a generic multiverse truth.
φisΩ-valid.

Now, recall that by Theorem 3.5, under our background assumptions,Ω-validity is generically invariant. It follows that given ourbackground theory, the notion of generic multiverse truth is robustwith respect to Π₂-statements. In particular, forΠ₂-statements, the statement “φ isindeterminate” isitself determinate according to thegeneric multiverse conception. In this sense the conception of truthis not “self-undermining” and one is not sent in adownward spiral where one has to countenance multiverses ofmultiverses. So it passes the first test. Whether it passes a morechallenging test depends on the Ω Conjecture.

The Ω Conjecture has profound consequences for the genericmultiverse conception of truth. Let

𝒱_Ω ={φ | ∅ ⊧_Ω φ}

and, for any specifiable cardinal κ, let

𝒱_Ω(H(κ⁺)) ={φ | ZFC ⊧_Ω“H(κ⁺) ⊧ φ”},

where recall thatH(κ⁺) is the collection of setsof hereditary cardinality less than κ⁺. Thus,assuming ZFC and that there is a proper class of Woodin cardinals, theset𝒱_Ω is Turing equivalent to the set ofΠ₂ generic multiverse truths and theset𝒱_Ω(H(κ⁺)) is preciselythe set of generic multiverse truths ofH(κ⁺).

To describe the bearing of the Ω Conjecture on thegeneric-multiverse conception of truth, we introduce two TranscendencePrinciples which serve as constraints on any tenable conception oftruth in set theory—atruth constraint andadefinability constraint.

Definition 4.2 (TruthConstraint).: Any tenable multiverse conception oftruth in set theory must be such that the Π₂-truths(according to that conception) in the universe of sets are notrecursive in the truths aboutH(κ) (according to thatconception), for any specifiable cardinal.

This constraint is in the spirit of those principles of settheory—most notably, reflection principles—which aim tocapture the pretheoretic idea that the universe of sets is so richthat it cannot “be described from below”; more precisely,it asserts that any tenable conception of truth must respect the ideathat the universe of sets is so rich that truth (or even justΠ₂-truth) cannot be described in some specifiablefragment. (Notice that by Tarski's theorem on the undefinabilityof truth, the truth constraint is trivially satisfied by the standardconception of truth in set theory which takes the multiverse tocontain a single element, namely,V.)

There is also a related constraint concerning the definability oftruth. For a specifiable cardinal κ, setY ⊆ ωisdefinable inH(κ⁺)across themultiverse ifY is definable in the structureH(κ⁺) of each universe of the multiverse (possibly byformulas which depend on the parent universe).

Definition 4.3 (DefinabilityConstraint).: Any tenable multiverse conception oftruth in set theory must be such that the Π₂-truths(according to that conception) in the universe of sets are definableinH(κ) across the multiverse universe, for any specifiablecardinal κ.

Notice again that by Tarski's theorem on the undefinabilityof truth, the definability constraint is trivially satisfied by thedegenerate multiverse conception that takes the multiverse to containthe single elementV. (Notice also that if one modifies thedefinability constraint by adding the requirement that the definitionbeuniform across the multiverse, then the constraint wouldautomatically be met.)

The bearing of the Ω Conjecture on the tenability of thegeneric-multiverse conception of truth is contained in the followingtwo theorems:

Theorem 4.4(Woodin).: Assume ZFCand that there isa proper class of Woodin cardinals. Suppose that theΩConjecture holds. Then𝒱_Ωis recursivein𝒱_Ω(H(δ+0)), whereδ₀is the least Woodin cardinal.

Theorem 4.5(Woodin).: Assume ZFCand that there isa proper class of Woodin cardinals. Suppose that theΩConjectureholds. Then𝒱_Ωis definable inH(δ+0), whereδ₀is the least Woodin cardinal.

In other words, if there is a proper class of Woodin cardinals andif the Ω Conjecture holds then the generic multiverse conceptionof truth violates both the Truth Constraint (at δ₀)and the Definability Constraint (at δ₀).

There are actually sharper versions of the above results thatinvolveH(c⁺) in place ofH(δ+0).

Theorem 4.6(Woodin).: Assume ZFCand that there isa proper class of Woodin cardinals. Suppose that theΩConjectureholds. Then𝒱_Ωis recursivein𝒱_Ω(H(c⁺)).

Theorem 4.7(Woodin).: Assume ZFCand that there isa proper class of Woodin cardinals. Suppose that theΩConjecture holds and that theAD⁺Conjectureholds. Then𝒱_Ωis definable inH(c⁺).

In other words, if there is a proper class of Woodin cardinals andif the Ω Conjecture holds then the generic-multiverse conceptionof truth violates the Truth Constraint at the level of third-orderarithmetic, and if, in addition, the AD⁺ Conjectureholds, then the generic-multiverse conception of truth violates theDefinability Constraint at the level of third-order arithmetic.

4.4 Is There a Way Out?

There appear to be four ways that the advocate of the genericmultiverse might resist the above criticism.

First, one could maintain that the Ω Conjecture is just asproblematic as CH and hence like CH it is to be regarded asindeterminate according to the generic-multiverse conception oftruth. The difficulty with this approach is the following:

Theorem 4.8(Woodin).: Assume ZFCand that there isa proper class of Woodin cardinals. Then, for any completeBoolean algebra𝔹,; V ⊧ Ω-conjectureiffV^𝔹 ⊧ Ω-conjecture.

Thus, in contrast to CH, the Ω Conjecture cannot be shown tobe independent of ZFC + “There is a proper class of Woodincardinals” via set forcing. In terms of the generic multiverseconception of truth, we can put the point this way: While thegeneric-multiverse conception of truth deems CH to be indeterminate,it doesnot deem the Ω Conjecture to beindeterminate. So the above response is not available to the advocateof the generic-multiverse conception of truth. The advocate of thatconceptionalready deems the Ω Conjecture to bedeterminate.

Second, one could grant that the Ω Conjecture is determinatebut maintain that it is false. There are ways in which one might dothis but that does not undercut the above argument. The reason is thefollowing: To begin with there is a closely relatedΣ₂-statement that one can substitute for the ΩConjecture in the above arguments. This is the statement that theΩ Conjecture is (non-trivially) Ω-satisfiable, that is,the statement: There exists an ordinal α and a universeV′ ofthe multiverse such that

V′_α ⊧ ZFC + “There is aproper class of Woodin cardinals”

and

V′_α ⊧ “The ΩConjecture”.

This Σ₂-statement is invariant under set forcingand hence is one adherents to the generic multiverse view of truthmust deem determinate. Moreover, the key arguments above go throughwith this Σ₂-statement instead of the ΩConjecture. The person taking this second line of response would thusalso have to maintain that this statement is false. But there issubstantial evidence that this statement istrue. The reasonis that there is no known example of a Σ₂-statementthat is invariant under set forcing relative to large cardinal axiomsand which cannot be settled by large cardinal axioms. (Such astatement would be a candidate for anabsolutely undecidablestatement.) So it is reasonable to expect that this statement isresolved by large cardinal axioms. However, recent advances in innermodel theory—in particular, those in Woodin (2010)—provideevidence that no large cardinal axiom can refute thisstatement. Putting everything together: It is very likely that thisstatement is in facttrue ; so this line of response is notpromising.

Third, one could reject either the Truth Constraint or theDefinability Constraint. The trouble is that if one rejects the TruthConstraint then on this view (assuming the Ω Conjecture)Π₂ truth in set theory is reduciblein the sense ofTuring reducibility to truth inH(δ₀) (or,assuming the Strong Ω Conjecture,H(c⁺)). Andif one rejects the Definability Constraint then on this view (assumingthe Ω Conjecture) Π₂ truth in set theory isreduciblein the sense of definability to truth inH(δ₀) (or, assuming the Strong Ω Conjecture,H(c⁺)). On either view, the reduction is in tensionwith the acceptance of non-pluralism regarding the background theoryZFC + “There is a proper class of Woodin cardinals”.

Fourth, one could embrace the criticism, reject the genericmultiverse conception of truth, and admit that there are somestatements aboutH(δ+0)(orH(c⁺), granting, in addition, theAD⁺ Conjecture) that are true simpliciter but not true inthe sense of the generic-multiverse, and yet nevertheless continue tomaintain that CH is indeterminate. The difficulty is that any suchsentence φ is qualitatively just like CH in that it can be forcedto hold and forced to fail. The challenge for the advocate of thisapproach is to modify the generic-multiverse conception of truth insuch a way that it counts φ as determinate and yet counts CH asindeterminate.

In summary: There is evidence that the only way out is the fourthway out and this places the burden back on the pluralist—thepluralist must come up with a modified version of the genericmultiverse.

Further Reading: For more on the connection betweenΩ-logic and the generic multiverse and the above criticism ofthe generic multiverse see Woodin (2011a). For the bearing of recentresults in inner model theory on the status of the Ω Conjecturesee Woodin (2010).

5. The Local Case Revisited

Let us now turn to a second way in which one might resist the localcase for the failure of CH. This involves a parallel case for CH. InSection 5.1 we will review the main features of the case for ¬CHin order to compare it with the parallel case for CH. InSection 5.2we will present the parallel case for CH. InSection 5.3 we willassess the comparison.

5.1 The Case for ¬CH

Recall that there are two basic steps in the case presented inSection 3.3. The first step involves Ω-completeness (and thisgives ¬CH) and the second step involves maximality (and this givesthe stronger 2^ℵ₀ = ℵ₂). Forease of comparison we shall repeat these features here:

The first step is based on the following result:

Theorem 5.1(Woodin).

Assume that there is a proper class ofWoodin cardinals and that the StrongΩConjecture holds.

There is an axiomAsuchthat
1. ZFC +AisΩ-satisfiable and
2. ZFC +AisΩ-complete for the structureH(ω₂).
Any suchaxiomAhas the feature that
ZFC +A ⊧_Ω“H(ω₂) ⊧ ¬CH”.

Let us rephrase this as follows: For eachA satisfying (1),let

T_A = {φ | ZFC +A ⊧_Ω “H(ω₂) ⊧ ¬φ”}.

The theorem says that if there is a proper class of Woodincardinals and the Strong Ω Conjecture holds, then there are(non-trivial) Ω-complete theoriesT_A ofH(ω₂) and all such theories contain ¬CH. In otherwords, under these assumptions, there is a “good” theoryand all “good” theories imply ¬CH.

The second step begins with the question of whether there isgreater agreement among the Ω-completetheoriesT_A. Ideally, there would be justone. However, this is not the case.

Theorem 5.2 (Koellner andWoodin 1999).: Assume that there is a proper class ofWoodin cardinals. Suppose thatAis an axiomsuch that; i. ZFC +Ais Ω-satisfiable and
ii. ZFC +AisΩ-complete for the structureH(ω₂).
Then there is an axiomBsuch that
i′. ZFC +Bis Ω-satisfiable and
ii′. ZFC +Bis Ω-complete for the structureH(ω₂); andT_A≠T_B.

This raises the issue as to how one is to select from among thesetheories? It turns out that there is a maximal theory amongtheT_A and this is given by the axiom(∗).

Theorem 5.3(Woodin).

Assume ZFCand that there isa proper class of Woodin cardinals. Then the followingare equivalent:

(∗).
For eachΠ₂-sentence φin the language for thestructure
⟨H(ω₂), ∈,I_NS,A |A ∈ 𝒫(ℝ) ∩L(ℝ)⟩
if
ZFC + “⟨H(ω₂), ∈,I_NS,A |A ∈ 𝒫(ℝ) ∩L(ℝ)⟩ ⊧ φ”
is Ω-consistent, then
⟨H(ω₂), ∈,I_NS,A |A ∈ 𝒫(ℝ) ∩L(ℝ)⟩ ⊧ φ.

So, of the various theoriesT_A involvedin Theorem 5.1, there is one that stands out: ThetheoryT_(∗) given by (∗). This theorymaximizes the Π₂-theory of the structure ⟨H(ω₂), ∈,I_NS,A |A ∈ 𝒫(ℝ) ∩L(ℝ)⟩. The fundamental result is that inthis maximal theory

2^ℵ₀ =ℵ₂.

5.2 The Parallel Case for CH

The parallel case for CH also has two steps, the first involvingΩ-completeness and the second involving maximality.

The first result in the first step is the following:

Theorem 5.4 (Woodin1985).: Assume ZFCand that there is aproper class of measurable Woodin cardinals. Then ZFC +CHis Ω-complete forΣ21.

Moreover, up to Ω-equivalence, CH is the uniqueΣ21-statementthat is Ω-complete forΣ21; thatis, lettingT_A be the Ω-complete theorygiven by ZFC +A whereA isΣ21, allsuchT_A are Ω-equivalenttoT_CH and hence (trivially) allsuchT_A contain CH. In other words, there isa “good” theory and all “good” theories implyCH.

To complete the first step we have to determine whether thisresult is robust. For it could be the case that when one considers thenext level,Σ22 (orfurther levels, like third-order arithmetic) CH is no longer part ofthe picture, that is, perhaps large cardinals imply that there is anaxiomA such that ZFC +A is Ω-complete forΣ22 (or,going further, all of third order arithmetic) and yetnot allsuchA have an associatedT_A whichcontains CH. We must rule this out if we are to secure the firststep.

The most optimistic scenario along these lines is this: Thescenario is that there is a large cardinal axiomL andaxiomsA→ such that ZFC+L +A→ isΩ-complete for all of third-order arithmetic and all suchtheories are Ω-equivalent and imply CH. Going further, perhapsfor each specifiable fragmentV_λ of theuniverse of sets there is a large cardinal axiomL andaxiomsA→ such that ZFC+L +A→ isΩ-complete for the entire theory ofV_λand, moreover, that such theories are Ω-equivalent and implyCH. Were this to be the case it would mean that for each such λthere is a unique Ω-complete pictureofV_λ and we would have a uniqueΩ-complete understanding of arbitrarily large fragments of theuniverse of sets. This would make for a strong case for new axiomscompleting the axioms of ZFC and large cardinal axioms.

Unfortunately, this optimistic scenario fails: Assuming theexistence of one such theory one can construct another which differson CH:

Theorem 5.5 (Koellner andWoodin 2009).: Assume ZFCand that there is aproper class of Woodincardinals. SupposeV_λis aspecifiable fragment of the universe (that is sufficientlylarge) and suppose that there is a large cardinalaxiomLand axiomsA→such that; ZFC +L +A→ is Ω-completefor Th(V_λ).; Then there are axiomsB→such that; ZFC +L +B→ is Ω-completefor Th(V_λ); and the first theory Ω-implies CHifand only if the second theory Ω-implies¬CH.

This still leaves us with the question of existence and the answerto this question is sensitive to the Ω Conjecture and theAD⁺ Conjecture:

Theorem 5.6(Woodin).: Assume that there is a properclass of Woodin cardinals and that the ΩConjectureholds. Then there is no recursivetheoryA→suchthat ZFC +A→is Ω-complete for thetheoryofV_δ₀+1, whereδ₀is the least Woodin cardinal.

In fact, under a stronger assumption, the scenario must fail at amuch earlier level.

Theorem 5.7(Woodin).: Assume that there is a properclass of Woodin cardinals and that the ΩConjectureholds. Assume that the AD⁺Conjectureholds. Then there is no recursivetheoryA→suchthat ZFC +A→is Ω-complete for thetheory ofΣ23.

It is open whether there can be such a theory at the level ofΣ22. Itis conjectured that ZFC + ◇ is Ω-complete (assuming largecardinal axioms) forΣ22.

Let us assume that it is answered positively and return to thequestion of uniqueness. For each such axiomA,letT_A be theΣ22theory computed by ZFC +A in Ω-logic. The question ofuniqueness simply asks whetherT_A isunique.

Theorem 5.8 (Koellner andWoodin 2009).: Assume that there is a proper class ofWoodin cardinals. Suppose thatAis an axiomsuch that; i. ZFC +Ais Ω-satisfiable and
ii. ZFC +Ais Ω-complete forΣ22.; Then there is an axiomBsuch that; i′. ZFC +Bis Ω-satisfiable and
ii′. ZFC +Bis Ω-complete forΣ22; andT_A≠T_B.

This is the parallel of Theorem 5.2.

To complete the parallel one would need that CH is among all oftheT_A. This is not known. But it is areasonable conjecture.

Conjecture 5.9.

Assume large cardinalaxioms.

There isan Σ22axiomA such that
1. ZFC +A is Ω-satisfiable and
2. ZFC +A is Ω-complete fortheΣ22.
Any suchΣ22axiomA has the feature that
ZFC +A ⊧_Ω CH.

Should this conjecture hold it would provide a true analogue ofTheorem 5.1. This would complete the parallel with the firststep.

There is also a parallel with the second step. Recall that for thesecond step in the previous subsection we had that although thevariousT_A did not agree, they all contained¬CH and, moreover, from among them there is one that stands out,namely the theory given by (∗), since this theory maximizes theΠ₂-theory of the structure ⟨H(ω₂), ∈,I_NS,A |A ∈ P(ℝ) ∩L(ℝ)⟩. In the present context of CH weagain (assuming the conjecture) have that althoughtheT_A do not agree, they all contain CH. Itturns out that once again, from among them there is one that standsout, namely, the maximum one. For it is known (by a result of Woodinin 1985) that if there is a proper class of measurable Woodincardinals then there is a forcing extension satisfying allΣ22sentences φ such that ZFC + CH + φ isΩ-satisfiable (see Ketchersid, Larson, & Zapletal(2010)). It follows that if the question of existence is answeredpositively with anA that isΣ22thenT_A must be this maximumΣ22theory and, consequently, allT_A agreewhenA isΣ22. So,assuming that there is aT_A whereA isΣ22,then, although not allT_A agree(whenA is arbitrary) there is one that stands out, namely, theone that is maximum forΣ22sentences.

Thus,if the above conjecture holds, then the case of CHparallels that of ¬CH, only nowΣ22 takesthe place of the theory ofH(ω₂).

5.3 Assessment

Assuming that the conjecture holds the case of CH parallels that of¬CH, only nowΣ22 takesthe place of the theory ofH(ω₂): Under thebackground assumptions we have:

1. there areA such that ZFC +A isΩ-complete forH(ω₂)
2. for every suchA theassociatedT_A contains ¬CH, and
3. there isaT_A which is maximal,namely,T_(∗) and this theory contains2^ℵ₀ = ℵ₂.
1. there areΣ22-axiomsAsuch that ZFC +A is Ω-complete forΣ22
2. for every suchA theassociatedT_A contains CH, and
3. there isaT_A which is maximal.

The two situations are parallel with regard to maximality but interms of the level of Ω-completeness the first is stronger. Forin the first case we are not just getting Ω-completeness withregard to the Π₂ theory ofH(ω₂) (withthe additional predicates), rather we are getting Ω-completenesswith regard toall ofH(ω₂). This isarguably an argument in favour of the case for ¬CH, even grantingthe conjecture.

But there is a stronger point. There is evidence coming from innermodel theory (which we shall discuss in the next section) to theeffect that the conjecture is in factfalse. Should thisturn out to be the case it would break the parallel, strengthening thecase for ¬CH.

However, one might counter this as follows: The higher degree ofΩ-completeness in the case for ¬CH is really illusory sinceit is an artifact of the fact that under (∗) the theory ofH(ω₂) is in fact mutually interpretable with that ofH(ω₁) (by a deep result of Woodin). Moreover, thislatter fact is in conflict with the spirit of the TranscendencePrinciples discussed inSection 4.3. Those principles were invoked inan argument to the effect that CH does not have an answer. Thus, whenall the dust settles the real import of Woodin's work on CH (sothe argument goes) isnot that CH is false but rather that CHvery likely has an answer.

It seems fair to say that at this stage the status of the localapproaches to resolving CH is somewhat unsettled. For this reason, inthe remainder of this entry we shall focus on global approaches tosettling CH. We shall very briefly discuss two suchapproaches—the approach via inner model theory and the approachvia quasi-large cardinal axioms.

6. The Ultimate Inner Model

Inner model theory aims to produce “L-like”models that contain large cardinal axioms. For each large cardinalaxiom Φ that has been reached by inner model theory, one has anaxiom of the formV =L^Φ. This axiom has thevirtue that (just as in the simplest case ofV =L) it provides an“effectively complete” solution regarding questionsaboutL^Φ (which, by assumption,isV). Unfortunately, it turns out that the axiomV=L^Φ is incompatible withstrongerlarge cardinal axioms Φ'. For this reason, axioms of this formhave never been considered as plausible candidates for newaxioms.

But recent developments in inner model theory (due to Woodin) showthat everything changes at the level of a supercompact cardinal. Thesedevelopments show that if there is an inner modelN which“inherits” a supercompact cardinal fromV (in themanner in which one would expect, given the trajectory of inner modeltheory), then there are two remarkable consequences: First,Nis close toV (in, for example, the sense that for sufficientlylarge singular cardinals λ,N correctly computesλ⁺). Second,N inherits all known largecardinals that exist inV. Thus, in contrast to the innermodels that have been developed thus far, an inner model at the levelof a supercompact would provide one with an axiom thatcouldnot be refuted by stronger large cardinalassumptions.

The issue, of course, is whether one can have an“L-like” model (one that yields an“effectively complete” axiom) at this level. There isreason to believe that one can. There is now a candidatemodelL^Ω that yields an axiomV=L^Ω with the following features: First,V=L^Ω is “effectively complete.”Second,V =L^Ω is compatible with all largecardinal axioms. Thus, on this scenario, the ultimate theory would bethe (open-ended) theory ZFC +V =L^Ω + LCA,where LCA is a schema standing for “large cardinalaxioms.” The large cardinal axioms will catch instances ofGödelian independence and the axiomV=L^Ω will capture the remaining instances ofindependence. This theory would imply CH and settle the remainingundecided statements. Independence would cease to be an issue.

It turns out, however, that there are other candidate axioms thatshare these features, and so the spectre of pluralism reappears. Forexample, there are axiomsV=LΩSandV=LΩ(∗). Theseaxioms would also be “effectively complete” and compatiblewith all large cardinal axioms. Yet they would resolve variousquestions differently than the axiomV=L^Ω. For example, the axiom,V=LΩ(∗)would imply ¬CH. How, then, is one to adjudicate betweenthem?

Further Reading: For an introduction to inner modeltheory see Mitchell (2010) and Steel (2010). For more on the recentdevelopments at the level of one supercompact and beyond see Woodin(2010).

7. The Structure Theory ofL(V_λ+1)

This brings us to the second global approach, one that promises toselect the correct axiom from amongV =L^Ω,V=LΩS,V=LΩ(∗),and their variants. This approach is based on the remarkable analogybetween the structure theory ofL(ℝ) under the assumption ofAD^L(ℝ) and the structure theory ofL(V_λ+1) under the assumption that there is anelementary embedding fromL(V_λ+1) into itselfwith critical point below λ. Thisembedding assumptionis the strongest large cardinal axiom that appears in theliterature.

The analogy betweenL(ℝ) andL(V_λ+1) is based on the observation thatL(ℝ) is simplyL(V_ω+1). Thus, λis the analogue of ω, λ⁺ is the analogue ofω₁, and so on. As an example of the parallel betweenthe structure theory ofL(ℝ) under AD^L(ℝ) andthe structure theory ofL(V_λ+1) under theembedding axiom, let us mention that in the first case,ω₁ is a measurable cardinal inL(ℝ) and, in thesecond case, the analogue of ω₁—namely,λ⁺—is a measurable cardinal inL(V_λ+1). This result is due to Woodin and isjust one instance from among many examples of the parallel that arecontained in his work.

Now, we have a great deal of information about the structuretheory ofL(ℝ) under AD^L(ℝ). Indeed, as wenoted above, this axiom is “effectively complete” withregard to questions aboutL(ℝ). In contrast, the embedding axiomon its own is not sufficient to imply thatL(V_λ+1) has a structure theory that fullyparallels that ofL(ℝ) under AD^L(ℝ). However,the existence of an already rich parallel is evidence that theparallel extends, and we can supplement the embedding axiom by addingsome key components. When one does so, something remarkable happens:the supplementary axioms becomeforcing fragile. This meansthat they have the potential to erase independence and providenon-trivial information aboutV_λ+1. Forexample, these supplementary axioms might settle CH and muchmore.

The difficulty in investigating the possibilities for thestructure theory ofL(V_λ+1) is that we havenot had the proper lenses through which to view it. The trouble isthat the modelL(V_λ+1) contains a large pieceof the universe—namely,L(V_λ+1)—and the theory of thisstructure is radically underdetermined. The results discussed aboveprovide us with the proper lenses. For one can examine the structuretheory ofL(V_λ+1) in the context of ultimateinner modelslikeL^Ω,LΩS,LΩ(∗),and their variants. The point is that these models can accommodate theembedding axiom and, within each, one will be able to compute thestructure theory ofL(V_λ+1).

This provides a means to select the correct axiom from amongV=L^Ω,V=LΩS,V=LΩ(∗),and their variants. One simply looks at theL(V_λ+1) of each model (where the embeddingaxiom holds) and checks to see which has the true analogue of thestructure theory ofL(ℝ) under the assumption ofAD^L(ℝ). It is already known that certain pieces ofthe structure theorycannot holdinL^Ω. But it is open whether they can holdinLΩS.

Let us consider one such (very optimistic) scenario: The trueanalogue of the structure theory ofL(ℝ) underAD^L(ℝ) holds of theL(V_λ+1)ofLΩSbut not of any of its variants. Moreover, this structure theory is“effectively complete” for the theoryofV_λ+1. Assuming that there is a properclass of λ where the embedding axiom holds, this gives an“effectively complete” theory ofV. And,remarkably, part of that theory is thatVmustbeLΩS. This(admittedly very optimistic) scenario would constitute a very strongcase for axioms that resolve all of the undecided statements.

One should not place too much weight on this particularscenario. It is just one of many. The point is that we are now in aposition to write down a list of definite questions with the followingfeatures: First, the questions on this list will haveanswers—independence is not an issue. Second, if the answersconverge then one will have strong evidence for new axioms settlingthe undecided statements (and hence non-pluralism about the universeof sets); while if the answers oscillate, one will have evidence thatthese statements are “absolutely undecidable” and thiswill strengthen the case for pluralism. In this way the questions of“absolute undecidability” and pluralism are givenmathematical traction.

Further Reading: For more on the structure theory ofL(V_λ+1) and the parallel with determinacy seeWoodin (2011b).

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