## 1. Independence in Cardinal Arithmetic


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


### 1.1 Regular Cardinals


The addition and multiplication of infinite cardinal numbers is
trivial: For infinite cardinals k and l, 



> k + l = k [?] l = max{k,l}.


 The situation becomes interesting when one turns to exponentiation
and the attempt to compute kl for infinite
cardinals. 


 During the dawn of set theory Cantor showed that for every
cardinal k, 



> 2k > k.


 There is no mystery about the size of 2*n* for
finite *n*. The first natural question then is where
20 is located in the aleph-hierarchy: Is it
1, 2, ..., 17
or something much larger? 


 The cardinal 20 is important since
it is the size of the continuum (the set of real numbers). Cantor's
famous *continuum hypothesis* (CH) is the statement that
20 =
1. This is a special case of the *generalized
continuum hypothesis* (GCH) which asserts that for all a,
2a =
a+1. One virtue of GCH is that it gives a
complete solution to the problem of computing
kl for infinite cardinals: Assuming GCH, if
k [?] l then kl =
l+; if cf(k) [?] l [?] k then
kl = k+; and if l <
cf(k) then kl = k. 


 Very little progress was made on CH and GCH. In fact, in the early
era of set theory the only other piece of progress beyond
Cantor's result that 2k > k (and
the trivial result that if k [?] l then
2k [?] 2l) was Konig's
result that cf(2k) > k. The explanation
for the lack of progress was provided by the independence results in
set theory: 



**Theorem 1.1** (Godel
1938a, 1938b). *Assume that* ZFC *is consistent. Then*
ZFC + CH *and* ZFC + GCH *are consistent.* 

To prove this Godel invented the method of *inner
models* --he showed that CH and GCH held in the minimal inner
model *L* of ZFC. Cohen then complemented this result: 



**Theorem 1.2** (Cohen
1963). *Assume that* ZFC *is
consistent. Then* ZFC + !CH *and* ZFC +
!GCH *are consistent.* 
He did this by inventing the method of *outer models* and
showing that CH failed in a *generic extension*
*V**B* of *V*. The combined
results of Godel and Cohen thus demonstrate that assuming the
consistency of ZFC, it is in principle impossible to settle either CH
or GCH in ZFC. 


 In the Fall of 1963 Easton completed the picture by showing that
for infinite regular cardinals k the *only* constraints
on the function k - 2k that are provable in
ZFC are the trivial constraint and the results of Cantor and
Konig: 



**Theorem 1.3** (Easton
1963). *Assume that* ZFC *is
consistent. Suppose* *F* *is a*
(*definable class*) *function defined on infinite
regular cardinals such that*
1. *if* k [?] l *then* *F*(k) [?] *F*(l)*,*
2. *F*(k) > k*, and*
3. cf(*F*(k)) > k*.*

*Then* ZFC + "*For all infinite regular
cardinals* k, 2k = *F*(k)" *is consistent.*

 Thus, set theorists had pushed the cardinal arithmetic of regular
cardinals as far as it could be pushed within the confines of
ZFC. 


### 1.2 Singular Cardinals


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


 It was generally believed that, as in the case for regular
cardinals, the behaviour of the function
k - 2k would be relatively unconstrained
within the setting of ZFC. But then Silver proved the following
remarkable result:[3] 



**Theorem 1.4** (Silver 
1974). *If*
d *is a singular cardinal of uncountable
cofinality, then, if* GCH *holds below*
d*, then* GCH *holds
at* d*.* 
It turns out that (by a deep result of Magidor, published in 1977)
GCH can first fail at o (assuming the
consistency of a supercompact cardinal). Silver's theorem shows
that it cannot first fail at o1
and *this is provable in* ZFC. 


 This raises the question of whether one can "control"
the size of 2d with a weaker assumption
than that d is a singular cardinal of
uncountable cofinality such that GCH holds below
d. The natural hypothesis to consider is
that d is a singular cardinal of uncountable
cofinality which is *a strong limit cardinal*, that is, that
for all a < d,
2a < d. In 1975
Galvin and Hajnal proved (among other things) that under this weaker
assumption there is indeed a bound: 



**Theorem 1.5** (Galvin and Hajnal
1975). *If*
d *is a singular strong limit cardinal of
uncountable cofinality then*

> 2d <
> (|d|cf(d))+.



It is possible that there is a jump--in fact, Woodin showed
(again assuming large cardinals) that it is possible that for all
k, 2k = k++. What the above
theorem shows is that in ZFC there is a provable bound on how big the
jump can be. 


 The next question is whether a similar situation prevails with
singular cardinals of countable cofinality. In 1978 Shelah showed that
this is indeed the case. To fix ideas let us concentrate on
o. 



**Theorem 1.6** (Shelah 1978). *If*
o *is a strong limit cardinal
then*

> 2o <
> (20)+.



One drawback of this result is that the bound is sensitive to the
actual size of 20, which can be anything below
o. Remarkably Shelah was later able to
remedy this with the development of his pcf (possible cofinalities)
theory. One very quotable result from this theory is the
following: 



**Theorem 1.7**
(Shelah 1982). *If*
o *is a strong limit cardinal then*
(*regardless of the size of*
20)


> 2o <
> o4.

 

 In summary, although the continuum function at regular cardinals
is relatively unconstrained in ZFC, the continuum function at singular
cardinals is (provably in ZFC) constrained in significant ways by the
behaviour 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 see
Abraham & 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 and
concentrate on the simplest case, the size of
20. One of Cantor's original approaches
to CH was by investigating "simple" sets of real
numbers (see Hallett (1984), pp. 3-5 and SS2.3(b)). One of the first results in this direction is the
Cantor-Bendixson theorem that every infinite closed set is either
countable or contains a perfect subset, in which case it has the same
cardinality as the set of reals. In other words, CH holds (in this
formulation) when one restricts one's attention to closed sets
of reals. In general, questions about "definable" sets of
reals are more tractable than questions about arbitrary sets of reals
and this suggests looking at definable versions of the continuum
hypothesis. 


### 2.1 Three Versions


There are three different formulations of the continuum
hypothesis--the *interpolant* version,
the *well-ordering* version, and the *surjection*
version. These versions are all equivalent to one another in ZFC but
we shall be imposing a definability constraint and in this case there
can be interesting differences (our discussion follows
Martin (1976)). There is really a hierarchy of notions
of definability--ranging up through the Borel hierarchy, the
projective hierarchy, the hierarchy in *L*(R), and, more
generally, the hierarchy of universally Baire sets--and so each
of these three general versions is really a hierarchy of versions,
each corresponding to a given level of the hierarchy of
definability (for a discussion of the hierarchy of
definability see 
SS2.2.1 and 
SS4.6 of the entry 
 "Large Cardinals and Determinacy").



#### 2.1.1 Interpolant Version


The first formulation of CH is that there is
no *interpolant*, that is, there is no infinite set *A*
of real numbers such that the cardinality of *A* is strictly
between that of the natural numbers and the real numbers. To obtain
definable versions one simply asserts that there is no
"definable" interpolant and this leads to a hierarchy of
definable interpolant versions, depending on which notion of
definability one employs. More precisely, for a given pointclass
G in the hierarchy of definable sets of reals, the corresponding
definable interpolant version of CH asserts that there is no
interpolant in G. 


 The Cantor-Bendixson theorem shows that there is no interpolant in
G in the case where G is the pointclass of closed sets,
thus verifying this version of CH. This was improved by Suslin who
showed that this version of CH holds for G where G is the
class of S11 sets. One cannot go much further within ZFC--to prove stronger
versions one must bring in stronger assumptions. It turns out that
axioms of definable determinacy and large cardinal axioms achieve
this. For example, results of Kechris and Martin show that if
D1*n*-determinacy
holds then this version of CH holds for the pointclass of
S1n+1
sets. Going further, if one assumes AD*L*(R) then
this version of CH holds for all sets of real numbers appearing in
*L*(R). Since these hypotheses follow from large cardinal axioms
one also has that stronger and stronger large cardinal assumptions
secure stronger and stronger versions of this version of the effective
continuum hypothesis. Indeed large cardinal axioms imply that this
version of CH holds for *all* sets of reals in the definability
hierarchy we are considering; more precisely, if there is a proper
class of Woodin cardinals then this version of CH holds for all
universally Baire sets of reals. 


#### 2.1.2 Well-ordering Version


The second formulation of CH asserts that every well-ordering of
the reals has order type less than 2. For a given
pointclass G in the hierarchy, the corresponding definable
well-ordering version of CH asserts that every well-ordering (coded by
a set) in G has order type less than 2. 


 Again, axioms of definable determinacy and large cardinal axioms
imply this version of CH for richer notions of definability. For
example, if AD*L*(R) holds then this version of CH
holds for all sets of real numbers in *L*(R). And if there is a
proper class of Woodin cardinals then this version of CH holds for all
universally Baire sets of reals. 


#### 2.1.3 Surjection Version


The third version formulation of CH asserts that there is no
surjection r : R - 2, or,
equivalently, that there is no prewellordering of R of length
2. For a given pointclass G in the hierarchy
of definability, the corresponding surjection version of CH asserts
that there is no surjection r : R - 2 
such that (the code for) r is in G. 


 Here the situation is more interesting. Axioms of definable
determinacy and large cardinal axioms have bearing on this version
since they place bounds on how long definable prewellorderings can
be. Let 
d1*n*
be the supremum of the lengths of the
S1*n*-prewellorderings
of reals and let Th*L*(R) be the supremum of the
lengths of prewellorderings of reals where the prewellordering is
definable in the sense of being in *L*(R). It is a classical
result that d11 =
1. Martin showed that d12
[?] 2 and that if there is a measurable cardinal
then d13
[?] 3. Kunen and Martin also showed under PD,
d14
[?] 4 and Jackson showed that under PD, for each n
< o, d1*n*
< o. Thus, assuming that there are
infinitely many Woodin cardinals, these bounds hold. Moreover, the
bounds continue to hold regardless of the size of
20. Of course, the question is whether these
bounds can be improved to show that the prewellorderings are shorter
than 2. In 1986 Foreman and Magidor initiated a
program to establish this. In the most general form they aimed to show
that large cardinal axioms implied that this version of CH held for
all universally Baire sets of reals. 


#### 2.1.4 Potential Bearing on CH


Notice that in the context of ZFC, these three hierarchies of
versions of CH are all successive approximations of CH and in the
limit case, where G is the pointclass of all sets of reals, they
are equivalent to CH. The question is whether these approximations can
provide any insight into CH itself. 


 There is an asymmetry that was pointed out by Martin, namely, that
a definable counterexample to CH is a real counterexample, while no
matter how far one proceeds in verifying definable versions of CH at
no stage will one have touched CH itself. In other words, the
definability approach could refute CH but it could not prove it. 


 Still, one might argue that although the definability approach
could not prove CH it might provide some evidence for it. In the case
of the first two versions we now know that CH holds for all definable
sets. Does this provide evidence of CH? Martin pointed out (before the
full results were known) that this is highly doubtful since in each
case one is dealing with sets that are atypical. For example, in the
first version, at each stage one secures the definable version of CH
by showing that all sets in the definability class have the perfect
set property; yet such sets are atypical in that assuming AC it is
easy to show that there are sets without this property. In the second
version, at each stage one actually shows not only that each
well-ordering of reals in the definability class has ordertype less
than 2, but also that it has ordertype less than
1. So neither of these versions really illuminates
CH. 


 The third version actually has an advantage in this regard since
not all of the sets it deals with are atypical. For example, while all
S11-sets
have length less than 1, there are
P11-sets
of length 1. Of course, it could turn out that
even if the Foreman-Magidor program were to succeed the sets could
turn out to be atypical in another sense, in which case it would shed
little light on CH. More interesting, however, is the possibility that
in contrast to the first two versions, it would actually provide an
actual counterexample to CH. This, of course, would require the
failure of the Foreman-Magidor program. 


### 2.2 The Foreman-Magidor Program


The goal of the Foreman-Magidor program was to show that large
cardinal axioms also implied that the third version of CH held for all
sets in *L*(R) and, more generally, all universally Baire sets. In
other words, the goal was to show that large cardinal axioms implied
that Th*L*(R) [?] 2 and, more
generally, that Th*L*(A,R)
[?] 2 for each universally Baire
set *A*. 


 The motivation came from the celebrated results of Foreman,
Magidor and Shelah on Martin's Maximum (MM), which showed that
assuming large cardinal axioms one can always force to obtain a
precipitous ideal on 2 without collapsing
2 (see Foreman, Magidor & Shelah (1988)). The program involved a two-part strategy: 


1. Strengthen this result to show that assuming large
cardinal axioms one can always force to obtain a *saturated*
ideal on 2 without collapsing
2.
2. Show that the existence of
such a saturated ideal implies that Th*L*(R)
[?] 2 and, more generally that
Th*L*(A,R) [?] 2 for every
universally Baire set *A*.


This would show that show that Th*L*(R)
[?] 2 and, more generally that
Th*L*(A,R) [?] 2 for every
universally Baire set *A*.[4]



 In December 1991, the following result dashed the hopes of this
program. 



**Theorem 2.1**
(Woodin). *Assume that the non-stationary ideal
on* 1 *is saturated and that there is a
measurable cardinal. Then* d12 =
2*.* 
The point is that the hypothesis of this theorem can always be
forced assuming large cardinals. Thus, it is possible to have
Th*L*(R) > 2 (in fact,
d13
> 2). 


 Where did the program go wrong? Foreman and Magidor had an
approximation to (B) and in the end it turned out that (B) is
true. 



**Theorem 2.2**
(Woodin). *Assume that there is a proper
class of Woodin cardinals and that there is a saturated ideal
on* 2*. Then for every* A
[?] G[?]*,*
Th*L*(A,R)
[?] 2*.* 
So the trouble is with (A). 


 This illustrates an interesting contrast between our three
versions of the effective continuum hypothesis, namely, that they can
come apart. For while large cardinals rule out definable
counterexamples of the first two kinds, they cannot rule out definable
counterexamples of the third kind. But again we must stress that they
cannot prove that there *are* such counterexamples. 


 But there is an important point: Assuming large cardinal axioms
(AD*L*(R) suffices), although one can produce outer
models in which d13
> 2 it is not currently known how to
produce outer models in which d13
> 3 or even Th*L*(R)
> 3. Thus it is an open possibility that
from ZFC +AD*L*(R) one can prove
Th*L*(R) [?] 3. Were this to
be the case, it would follow that although large cardinals cannot rule
out the definable failure of CH *they can rule out
the definable failure of* 20 =
2. This could provide some insight into the size
of the continuum, underscoring the centrality of
2. 


*Further Reading*: For more on the three effective versions
of CH see Martin (1976); for more on the Foreman-Magidor program see
Foreman & 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 the
basis of his an argument that CH is false. In Section 3.1 we will
describe the model and in the remainder of the section we will present
the case for the failure of CH. In Section 3.2 we will introduce
O-logic and the other notions needed to make the case. In
Section 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 is
canonical in the sense that its theory cannot be altered by set
forcing in the presence of large cardinals. The background motivation
is this: First, we know that in the presence of large cardinal axioms
the theory of second-order arithmetic and even the entire theory of
*L*(R) is invariant under set forcing. The importance of this is
that it demonstrates that our main independence techniques cannot be
used to establish the independence of questions about second-order
arithmetic (or about *L*(R)) in the presence of large
cardinals. Second, experience has shown that the large cardinal axioms
in question seem to answer all of the major known open problems about
second-order arithmetic and *L*(R) and the set forcing invariance
theorems give precise content to the claim that these axioms are
"effectively complete".[5]



 It follows that if P is any homogeneous partial order in
*L*(R) then the generic extension *L*(R)P
inherits the generic absoluteness of *L*(R). Woodin discovered
that there is a very special partial order
P*max* that has this feature. Moreover, the
model *L*(R)P*max* satisfies ZFC +
!CH. The key feature of this model is that it is
"maximal" (or "saturated") with respect to
sentences that are of a certain complexity and which can be shown to
be consistent via set forcing over the model; in other words, if these
sentences *can hold* (by set forcing over the model)
then they *do hold* in the model. To state this more precisely
we are going to have to introduce a few rather technical notions. 


 There are two ways of stratifying the universe of sets. The first
is in terms of <*V*a | a
[?] On > , the second is in terms of
<*H*(k) | k [?] Card> , where *H*(k) is the set
of all sets which have cardinality less than k and whose members
have cardinality less than k, and whose members of members have
cardinality less than k, and so on. For example, *H*(o)
= *V*o and the theories of the structures
*H*(o1) and *V*o+1 are mutually
interpretable. This latter structure is the structure of second-order
arithmetic and, as mentioned above, large cardinal axioms give us an
"effectively complete" understanding of this structure. We
should like to be in the same position with regard to larger and
larger fragments of the universe and the question is whether we should
proceed in terms of the first or the second stratification. 


 The second stratification is potentially more
fine-grained. Assuming CH one has that the theories of
*H*(o2) and *V*o+2 are mutually
interpretable and assuming larger and larger fragments of GCH this
correspondence continues upward. But if CH is false then the structure
*H*(o2) is less rich than the
structure *V*o2. In this event the latter
structure captures full third-order arithmetic, while the former
captures only a small fragment of third-order arithmetic but is
nevertheless rich enough to express CH. Given this, in attempting to
understand the universe of sets by working up through it level by
level, it is sensible to use the potentially more fine-grained
stratification. 


 Our next step is therefore to understand
*H*(o2). It actually turns out that we will be able to
understand slightly more and this is somewhat technical. We will be
concerned with the structure 
<*H*(o2), [?], *I*NS, *A**G*> 
[?] ph, where *I*NS is the non-stationary ideal
on o1 and *A**G* is the
interpretation of (the canonical representation of) a set of
reals *A* in *L*(R). The details will not be important and
the reader is asked to just think of *H*(o2) along with
some "extra stuff" and not worry about the details
concerning the extra stuff.[6]



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



**Theorem 3.1**
(Woodin 1999). *Assume* ZFC *and that there is
a proper class of Woodin cardinals. Suppose that*
*A* [?] *P* (R) [?] *L*(R) *and* ph *is a*
P2*-sentence* (*in the extended
language with two additional predicates* ) *and there
is a set forcing extension* *V*[*G*] *such that*

> <*H*(o2), [?], *I*NS, *A**G*> 
> [?] ph


 (*where* *A**G* *is the
interpretation of* *A* *in* *V*
[*G*])*. Then*

> *L*(R)P*max* [?] 
> "<*H*(o2), [?], *I*NS, A> 
> [?] ph".


There are two key points: First, the theory of
*L*(R)P*max* is "effectively
complete" in the sense that it is invariant under set
forcing. Second, the model *L*(R)P*max*
is "maximal" (or "saturated") in the sense
that it satisfies all P2-sentences (about the relevant
structure) that can possibly hold (in the sense that they can be shown
to be consistent by set forcing over the model). 


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



**Definition 3.2** (Woodin 1999). Axiom
([?]): AD*L*(R) holds and
*L*(P(o1)) is a P*max*-generic
extension of *L*(R). 
 Finally, this axiom settles CH: 



**Theorem 3.3**
(Woodin 1999). *Assume*
([?])*. Then* 2o =
2*.* 
### 3.2 O-Logic


We will now recast the above results in terms of a strong logic. We
shall make full use of large cardinal axioms and in this setting we
are interested in logics that are "well-behaved" in the
sense that the question of what implies what is not radically
independent. For example, it is well known that CH is expressible in
full second-order logic. It follows that in the presence of large
cardinals one can always use set forcing to flip the truth-value of a
purported logical validity of full second-order logic. However, there
are strong logics--like o-logic and b-logic--that
do not have this feature--they are well-behaved in the sense that
in the presence of large cardinal axioms the question of what implies
what cannot be altered by set forcing. We shall introduce a very
strong logic that has this feature--O-logic. In fact, the
logic we shall introduce can be characterized as
the *strongest* logic with this feature (see
Koellner (2010) for further discussion of strong logics and for a
precise statement of this result). 


#### 3.2.1 O-logic



**Definition 3.4.** Suppose that *T* is a
countable theory in the language of set theory and ph is a
sentence. Then

> T [?]O ph


 if for all complete Boolean algebras *B* and for all ordinals
a, 


> if *V**B*a [?] T then *V**B*a [?] ph.


We say that a statement ph is O-*satisfiable* if
there exists an ordinal a and a complete Boolean
algebra *B* such
that *V**B*a [?] ph,
and we say that ph is O-*valid* if
[?] [?]O ph. So, the above theorem says
that (under our background assumptions), the statement "ph
is O-satisfiable" is generically invariant and in terms of
O-validity this is simply the following: 



**Theorem 3.5**
(Woodin 1999). *Assume* ZFC *and that there is
a proper class of Woodin cardinals. Suppose that* *T* *is
a countable theory in the language of set theory and*
ph *is a sentence. Then for all complete Boolean
algebras* *B**,* 


> T [?]O ph iff *V**B* [?] "T [?]O ph."


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


#### 3.2.2 The O Conjecture


Corresponding to the semantic relation [?]O
there is a quasi-syntactic proof relation
[?]O. The "proofs" are certain robust
sets of reals (universally Baire sets of reals) and the test
structures are models that are "closed" under these
proofs. The precise notions of "closure" and
"proof" are somewhat technical and so we will pass over
them in silence.[7]



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



**Theorem 3.6**
(Woodin 1999). *Assume* ZFC *and that there is
a proper class of Woodin cardinals. Suppose* *T* *is a
countable theory in the language of set theory,* ph *is a
sentence, and* *B* *is a complete Boolean
algebra. Then* 


> T
> [?]O ph iff *V**B* [?] 'T
> [?]O ph'.



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



**Theorem 3.7**
(Woodin 1999). *Assume*
ZFC*. Suppose* *T* *is a countable theory in
the language of set theory and* ph *is a sentence. If*
T [?]O ph *then*
T [?]O ph*.* 
It is open whether the corresponding completeness theorem
holds. The O Conjecture is simply the assertion that it
does: 



**Conjecture 3.8**
(O **Conjecture** ). Assume ZFC
and that there is a proper class of Woodin cardinals. Then for each
sentence ph, 


> [?] [?]O ph iff [?] [?]O ph.



 We will need a strong form of this conjecture which we shall call
the Strong O Conjecture. It is somewhat technical and so we will
pass over it in silence.[8]



#### 3.2.3 O-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 of *L*(R) and more) in the sense that in
the presence of large cardinals one cannot use the method of set
forcing to establish independence with respect to statements about
*L*(R). This notion of invariance under set forcing played a key
role in Section 3.1. We can now rephrase this notion in terms of
O-logic. 



**Definition 3.9.** A theory *T* is
O-*complete* for a collection of sentences G if for
each ph [?] G, T [?]O ph or
T [?]O !ph. 
The invariance of the theory of *L*(R) under set forcing can
now be rephrased as follows: 



**Theorem 3.10**
(Woodin 1999). *Assume* ZFC *and that there is
a proper class of Woodin cardinals. Then* ZFC *is*
O*-complete for the collection of sentences of the
form* "*L*(R) [?] ph"*.* 
 Unfortunately, it follows from a series of results originating
with work of Levy and Solovay that traditional large cardinal axioms
do not yield O-complete theories at the level of
S21 since
one can always use a "small" (and hence large cardinal
preserving) forcing to alter the truth-value of CH. 



**Theorem 3.11.**  *Assume* *L* *is a
standard large cardinal axiom. Then* ZFC + L *is not*
O*-complete for*
S21*.* 
### 3.3 The Case


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



**Theorem 3.12**
(Woodin). *Assume that there is a proper class of
Woodin cardinals and that the Strong*
O *Conjecture holds.* 

1. *There is an axiom* *A* *such
that*
	1. ZFC + *A* *is*
	O*-satisfiable and*
	2. ZFC + *A* *is*
	O*-complete for the structure*
	*H*(o2)*.*
2. *Any such axiom* *A* *has the feature that*

> ZFC + *A* [?]O '*H*(o2) [?] !CH'.



 Let us rephrase this as follows: For each *A* satisfying (1),
let 



> *T**A* = {ph | ZFC +
> *A* [?]O '*H*(o2) [?] !ph'}.


 The theorem says that if there is a proper class of Woodin
cardinals and the O Conjecture holds, then there are
(non-trivial) O-complete theories *T**A* of
*H*(o2) and all such theories contain !CH. 


 It is natural to ask whether there is greater agreement among the
O-complete theories *T**A*. Ideally, there
would be just one. A recent result (building on Theorem 5.5) shows
that if there is one such theory then there are many such
theories. 



**Theorem 3.13** (Koellner and
Woodin 2009). *Assume that there is a proper class of
Woodin cardinals. Suppose that* *A* *is an axiom
such that* 


> 
>  i.  ZFC + *A* *is* O*-satisfiable and*
>   
> 
> ii.  ZFC + *A* *is*
> O*-complete for the structure*
> *H*(o2)*.*
> 



*Then there is an axiom* *B* *such that* 


> 
>  i'.  ZFC + *B* *is* O*-satisfiable and*
>   
> 
> ii'.  ZFC + *B* *is* O*-complete for the structure*
> *H*(o2)
> 



*and* *T**A*[?]*T**B**.* 

 How then shall one select from among these theories?
Woodin's work in this area goes a good deal beyond Theorem
5.1. In addition to isolating an axiom that satisfies (1) of Theorem
5.1 (assuming O-satisfiability), he isolates a very special such
axiom, namely, the axiom ([?]) ("star") mentioned
earlier. 


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



**Theorem 3.14**
(Woodin). *Assume* ZFC *and that there is
a proper class of Woodin cardinals. Then the following
are equivalent*: 

1. ([?])*.*
2. *For each*
P2*-sentence* ph *in the language for the
structure*

> <*H*(o2), [?], *I*NS, *A* | *A* [?] P
> (R) [?] *L*(R)> 


*if*

> ZFC + "<*H*(o2), [?], *I*NS, *A* | *A* [?] P
> (R) [?] *L*(R)>  [?] ph"


*is* O*-consistent, then*

> <*H*(o2), [?], *I*NS, *A* | *A* [?] P
> (R) [?] *L*(R)>  [?] ph.

 It follows that of the various
theories *T**A* involved in Theorem 5.1, there is
one that stands out: The theory *T*([?]) given by
([?]). This theory maximizes the P2-theory of the
structure <*H*(o2), [?], *I*NS, *A* | *A* [?] P
(R) [?] *L*(R)> . 


 The continuum hypothesis fails in this theory. Moreover, in the
maximal theory *T*([?]) given by ([?]) the
size of the continuum is 2.[9]



 To summarize: Assuming the Strong O Conjecture, there is a
"good" theory of *H*(o2) and all such
theories imply that CH fails. Moreover, (again, assuming the Strong
O Conjecture) there is a maximal such theory and in that theory
20 = 2. 


*Further Reading*: For the mathematics concerning
P*max* see Woodin (1999). For an introduction to
O-logic see Bagaria, Castells & Larson (2006). For
more on incompatible O-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 local
case for axioms that settle CH. In this section and the next we will
switch sides and consider the pluralist arguments to the effect that
CH does not have an answer (in this section) and to the effect that
there is an equally good case for CH (in the next section). In the
final two section we will investigate optimistic global scenarios that
provide hope of settling the issue. 


 The pluralist maintains that the independence results effectively
settle the undecided questions by showing that they have no
answer. One way of providing a foundational framework for such a view
is in terms of the multiverse. On this view there is not a
single *universe* of set theory but rather
a *multiverse* of legitimate candidates, some of which may be
preferable to others for certain purposes but none of which can be
said to be the "true" universe. The *multiverse
conception of truth* is the view that a statement of set theory
can only be said to be true simpliciter if it is true in all universes
of the multiverse. For the purposes of this discussion we shall say
that a statement is *indeterminate according to the multiverse
conception* if it is neither true nor false according to the
multiverse conception. How radical such a view is depends on the
breadth of the conception of the multiverse. 


### 4.1 Broad Multiverse Views


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


 There is a form of radical pluralism which advocates pluralism
concerning all domains of mathematics. On this view any consistent
theory is a legitimate candidate and the corresponding models of such
theories are legitimate candidates for the domain of
mathematics. Let us call this the *broadest multiverse*
view. There is a difficulty in articulating this view, which may be
brought out as follows: To begin with, one must pick a background
theory in which to discuss the various models and this leads to a
difficult. 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 are
universes within the broad multiverse. Now to arrive at this
conclusion one must (in the background theory) be in a position to
prove Con(PA) (since this assumption is required to apply the second
incompleteness theorem in this particular case). Thus, from the
perspective of the background theory used to argue that the above
models are legitimate candidates, the models in question satisfy a
false
S01-sentence,
namely, !Con(PA). In short, there is a lack of harmony between
what is held at the meta-level and what is held at the
object-level. 


 The only way out of this difficulty would seem to be to regard
each viewpoint--each articulation of the multiverse
conception--as provisional and, when pressed, embrace pluralism
concerning the background theory. In other words, one would have to
adopt a multiverse conception of the multiverse, a multiverse
conception of the multiverse conception of the multiverse, and so on,
off to infinity. It follows that such a position can never be fully
articulated--each time one attempts to articulate the broad
multiverse conception one must employ a background theory but since
one is a pluralist about that background theory this pass at using the
broad multiverse to articulate the conception does not do the
conception full justice. The position is thus difficult to
articulate. One can certainly take the pluralist *stance* and
try to *gesture toward* or *exhibit* the view that one
intends by provisionally settling on a particular background theory
but then advocate pluralism regarding *that* when pressed. The
view is thus something of a "moving target". We shall pass
over this view in silence and concentrate on views that can be
articulated within a foundational framework. 


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


 Let the *broad multiverse* (based on ZFC) be the collection
of all models of ZFC. The broad multiverse conception of truth (based
on ZFC) is then simply the view that a statement of set theory is true
simpliciter if it is provable in ZFC. On this view the statement
Con(ZFC) and other undecided
P01-statements
are classified as indeterminate. This view thus faces a difficulty
parallel to the one mentioned above concerning radical pluralism. 


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


 For those who are convinced by the arguments (surveyed in the
entry 
 "Large Cardinals and Determinacy")
for large
cardinal axioms and axioms of definable determinacy, even these
multiverse conceptions are too weak. We will follow this route. For
the rest of this entry we will embrace non-pluralism concerning large
cardinal axioms and axioms of definable determinacy and focus on the
question of CH. 


### 4.2 The Generic Multiverse


The motivation behind the generic multiverse is to grant the case
for large cardinal axioms and definable determinacy but deny that
statements such as CH have a determinate truth value. To be specific
about the background theory let us take ZFC + "There is a proper
class of Woodin cardinals" and recall that this large cardinal
assumption secures axioms of definable determinacy such as PD and
AD*L*(R). 


 Let the *generic multiverse*
V be the result of
closing *V* under generic extensions and generic refinements. One
way to formalize this is by taking an external vantage point and start
with a countable transitive model *M*. The generic multiverse
based on *M* is then the smallest set 
V*M*
such that 
M [?] V*M*
and, for each pair of
countable transitive models (*N*, *N*[*G*]) such that *N* [?] ZFC and *G*
[?] P is *N*-generic for some partial order in P
[?] *N*, if either *N* or *N*[*G*] is in 
V*M*
then both *N* and *N*[*G*] are in 
V*M*. 


 Let the *generic multiverse conception of truth* be the
view that a statement is true simpliciter iff it is true in all
universes of the generic multiverse. We will call such a statement
a *generic multiverse truth*. A statement is said to
be *indeterminate according to the generic multiverse
conception* iff it is neither true nor false according to the
generic multiverse conception. For example, granting our large
cardinal assumptions, such a view deems PM (and PD and
AD*L*(R)) true but deems CH indeterminate. 


### 4.3 The O Conjecture and the Generic Multiverse


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



**Theorem 4.1**
(Woodin). *Assume* ZFC *and that there is
a proper class of Woodin cardinals. Then, for each*
P2*-statement* ph *the
following are equivalent*: 

1. ph *is a generic multiverse truth.*
2. ph *is*
O*-valid.*

Now, recall that by Theorem 3.5, under our background assumptions,
O-validity is generically invariant. It follows that given our
background theory, the notion of generic multiverse truth is robust
with respect to P2-statements. In particular, for
P2-statements, the statement "ph is
indeterminate" is *itself* determinate according to the
generic multiverse conception. In this sense the conception of truth
is not "self-undermining" and one is not sent in a
downward spiral where one has to countenance multiverses of
multiverses. So it passes the first test. Whether it passes a more
challenging test depends on the O Conjecture. 


 The O Conjecture has profound consequences for the generic
multiverse conception of truth. Let 



> VO =
> {ph | [?] [?]O ph}
> 


 and, for any specifiable cardinal k, let 



> VO(*H*(k+)) =
> {ph | ZFC [?]O"*H*(k+) [?] ph"},


 where recall that *H*(k+) is the collection of sets
of hereditary cardinality less than k+. Thus,
assuming ZFC and that there is a proper class of Woodin cardinals, the
set 
VO is Turing equivalent to the set of
P2 generic multiverse truths and the
set 
VO(*H*(k+)) is precisely
the set of generic multiverse truths of *H*(k+). 


 To describe the bearing of the O Conjecture on the
generic-multiverse conception of truth, we introduce two Transcendence
Principles which serve as constraints on any tenable conception of
truth in set theory--a *truth constraint* and
a *definability constraint*. 



**Definition 4.2** (Truth
Constraint). Any tenable multiverse conception of
truth in set theory must be such that the P2-truths
(according to that conception) in the universe of sets are not
recursive in the truths about *H*(k) (according to that
conception), for any specifiable cardinal. 
This constraint is in the spirit of those principles of set
theory--most notably, reflection principles--which aim to
capture the pretheoretic idea that the universe of sets is so rich
that it cannot "be described from below"; more precisely,
it asserts that any tenable conception of truth must respect the idea
that the universe of sets is so rich that truth (or even just
P2-truth) cannot be described in some specifiable
fragment. (Notice that by Tarski's theorem on the undefinability
of truth, the truth constraint is trivially satisfied by the standard
conception of truth in set theory which takes the multiverse to
contain a single element, namely, *V*.) 


 There is also a related constraint concerning the definability of
truth. For a specifiable cardinal k, set *Y* [?] o
is *definable in* *H*(k+) *across the
multiverse* if *Y* is definable in the structure
*H*(k+) of each universe of the multiverse (possibly by
formulas which depend on the parent universe). 



**Definition 4.3** (Definability
Constraint). Any tenable multiverse conception of
truth in set theory must be such that the P2-truths
(according to that conception) in the universe of sets are definable
in *H*(k) across the multiverse universe, for any specifiable
cardinal k. 
Notice again that by Tarski's theorem on the undefinability
of truth, the definability constraint is trivially satisfied by the
degenerate multiverse conception that takes the multiverse to contain
the single element *V*. (Notice also that if one modifies the
definability constraint by adding the requirement that the definition
be *uniform* across the multiverse, then the constraint would
automatically be met.) 


 The bearing of the O Conjecture on the tenability of the
generic-multiverse conception of truth is contained in the following
two theorems: 



**Theorem 4.4**
(Woodin). *Assume* ZFC *and that there is
a proper class of Woodin cardinals. Suppose that the*
O *Conjecture holds. Then*
VO *is recursive
in*
VO(*H*(d+0))*, where*
d0 *is the least Woodin cardinal.* 

**Theorem 4.5**
(Woodin). *Assume* ZFC *and that there is
a proper class of Woodin cardinals. Suppose that the*
O *Conjecture
holds. Then*
VO *is definable in*
*H*(d+0)*, where*
d0 *is the least Woodin cardinal.* 
In other words, if there is a proper class of Woodin cardinals and
if the O Conjecture holds then the generic multiverse conception
of truth violates both the Truth Constraint (at d0)
and the Definability Constraint (at d0). 


 There are actually sharper versions of the above results that
involve *H*(*c*+) in place of
*H*(d+0). 



**Theorem 4.6**
(Woodin). *Assume* ZFC *and that there is
a proper class of Woodin cardinals. Suppose that the*
O *Conjecture
holds. Then*
VO *is recursive
in* VO(*H*(*c*+))*.* 

**Theorem 4.7**
(Woodin). *Assume* ZFC *and that there is
a proper class of Woodin cardinals. Suppose that the*
O *Conjecture holds and that the*
AD+ *Conjecture
holds. Then*
VO *is definable in*
*H*(*c*+)*.* 
In other words, if there is a proper class of Woodin cardinals and
if the O Conjecture holds then the generic-multiverse conception
of truth violates the Truth Constraint at the level of third-order
arithmetic, and if, in addition, the AD+ Conjecture
holds, then the generic-multiverse conception of truth violates the
Definability 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 generic
multiverse might resist the above criticism. 


 First, one could maintain that the O Conjecture is just as
problematic as CH and hence like CH it is to be regarded as
indeterminate according to the generic-multiverse conception of
truth. The difficulty with this approach is the following: 



**Theorem 4.8**
(Woodin). *Assume* ZFC *and that there is
a proper class of Woodin cardinals. Then, for any complete
Boolean algebra* B*,* 

> 
> *V* [?] O-conjecture
> iff *V*B [?] O-conjecture.



Thus, in contrast to CH, the O Conjecture cannot be shown to
be independent of ZFC + "There is a proper class of Woodin
cardinals" via set forcing. In terms of the generic multiverse
conception of truth, we can put the point this way: While the
generic-multiverse conception of truth deems CH to be indeterminate,
it does *not* deem the O Conjecture to be
indeterminate. So the above response is not available to the advocate
of the generic-multiverse conception of truth. The advocate of that
conception *already* deems the O Conjecture to be
determinate. 


 Second, one could grant that the O Conjecture is determinate
but maintain that it is false. There are ways in which one might do
this but that does not undercut the above argument. The reason is the
following: To begin with there is a closely related
S2-statement that one can substitute for the O
Conjecture in the above arguments. This is the statement that the
O Conjecture is (non-trivially) O-satisfiable, that is,
the statement: There exists an ordinal a and a universe *V'* of
the multiverse such that 



> *V'*a [?] ZFC + "There is a
> proper class of Woodin cardinals"


 and 



> *V'*a [?] "The O
> Conjecture".


 This S2-statement is invariant under set forcing
and hence is one adherents to the generic multiverse view of truth
must deem determinate. Moreover, the key arguments above go through
with this S2-statement instead of the O
Conjecture. The person taking this second line of response would thus
also have to maintain that this statement is false. But there is
substantial evidence that this statement is *true*. The reason
is that there is no known example of a S2-statement
that is invariant under set forcing relative to large cardinal axioms
and which cannot be settled by large cardinal axioms. (Such a
statement would be a candidate for an *absolutely undecidable*
statement.) So it is reasonable to expect that this statement is
resolved by large cardinal axioms. However, recent advances in inner
model theory--in particular, those in Woodin (2010)--provide
evidence that no large cardinal axiom can refute this
statement. Putting everything together: It is very likely that this
statement is in fact *true* ; so this line of response is not
promising. 


 Third, one could reject either the Truth Constraint or the
Definability Constraint. The trouble is that if one rejects the Truth
Constraint then on this view (assuming the O Conjecture)
P2 truth in set theory is reducible *in the sense of
Turing reducibility* to truth in *H*(d0) (or,
assuming the Strong O Conjecture, *H*(*c*+)). And
if one rejects the Definability Constraint then on this view (assuming
the O Conjecture) P2 truth in set theory is
reducible *in the sense of definability* to truth in
*H*(d0) (or, assuming the Strong O Conjecture,
*H*(*c*+)). On either view, the reduction is in tension
with the acceptance of non-pluralism regarding the background theory
ZFC + "There is a proper class of Woodin cardinals". 


 Fourth, one could embrace the criticism, reject the generic
multiverse conception of truth, and admit that there are some
statements about
*H*(d+0)
(or *H*(*c*+), granting, in addition, the
AD+ Conjecture) that are true simpliciter but not true in
the sense of the generic-multiverse, and yet nevertheless continue to
maintain that CH is indeterminate. The difficulty is that any such
sentence ph is qualitatively just like CH in that it can be forced
to hold and forced to fail. The challenge for the advocate of this
approach is to modify the generic-multiverse conception of truth in
such a way that it counts ph as determinate and yet counts CH as
indeterminate. 


 In summary: There is evidence that the only way out is the fourth
way out and this places the burden back on the pluralist--the
pluralist must come up with a modified version of the generic
multiverse. 


*Further Reading*: For more on the connection between
O-logic and the generic multiverse and the above criticism of
the generic multiverse see Woodin (2011a). For the bearing of recent
results in inner model theory on the status of the O Conjecture
see Woodin (2010). 


## 5. The Local Case Revisited


Let us now turn to a second way in which one might resist the local
case for the failure of CH. This involves a parallel case for CH. In
Section 5.1 we will review the main features of the case for !CH
in order to compare it with the parallel case for CH. In Section 5.2
we will present the parallel case for CH. In Section 5.3 we will
assess the comparison. 


### 5.1 The Case for !CH


Recall that there are two basic steps in the case presented in
Section 3.3. The first step involves O-completeness (and this
gives !CH) and the second step involves maximality (and this gives
the stronger 20 = 2). For
ease 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 of
Woodin cardinals and that the Strong*
O *Conjecture holds.* 

1. *There is an axiom* *A* *such
that*
	1. ZFC + *A* *is*
	O*-satisfiable and*
	2. ZFC + *A* *is*
	O*-complete for the structure*
	*H*(o2)*.*
2. *Any such
axiom* *A* *has the feature that* 
> ZFC +
> *A* [?] O"*H*(o2) [?] !CH".

Let us rephrase this as follows: For each *A* satisfying (1),
let 



> *T**A* = {ph | ZFC +
> *A* [?]O "*H*(o2) [?] !ph"}.


 The theorem says that if there is a proper class of Woodin
cardinals and the Strong O Conjecture holds, then there are
(non-trivial) O-complete theories *T**A* of
*H*(o2) and all such theories contain !CH. In other
words, under these assumptions, there is a "good" theory
and all "good" theories imply !CH. 


 The second step begins with the question of whether there is
greater agreement among the O-complete
theories *T**A*. Ideally, there would be just
one. However, this is not the case. 



**Theorem 5.2** (Koellner and
Woodin 1999). *Assume that there is a proper class of
Woodin cardinals. Suppose that* *A* *is an axiom
such that* 


> 
>  i.  ZFC +
> *A* *is* O*-satisfiable and*
>   
> 
> ii.  ZFC + *A* *is*
> O*-complete for the structure*
> *H*(o2)*.*
> 


*Then there is an axiom* *B* *such that* 



> 
>  i'.  ZFC + *B* *is* O*-satisfiable and*
>   
> 
> ii'.  ZFC + *B* *is* O*-complete for the structure*
> *H*(o2)
> 



*and* *T**A*[?]*T**B**.* 

 This raises the issue as to how one is to select from among these
theories? It turns out that there is a maximal theory among
the *T**A* and this is given by the axiom
([?]). 



**Theorem 5.3**
(Woodin). *Assume* ZFC *and that there is
a proper class of Woodin cardinals. Then the following
are equivalent*: 

1. ([?])*.*
2. *For each*
P2*-sentence* ph *in the language for the
structure* 
> <*H*(o2), [?], *I*NS,
> *A* | *A* [?] P
> (R) [?] *L*(R)> 


 *if* 



> ZFC + "<*H*(o2), [?], *I*NS, *A* | *A* [?] P
> (R) [?] *L*(R)>  [?] ph"


 *is* O*-consistent, then* 



> <*H*(o2), [?], *I*NS, *A* | *A* [?] P
> (R) [?] *L*(R)>  [?] ph.



 So, of the various theories *T**A* involved
in Theorem 5.1, there is one that stands out: The
theory *T*([?]) given by ([?]). This theory
maximizes the P2-theory of the structure <*H*(o2), [?], *I*NS,
*A* | *A* [?] P
(R) [?] *L*(R)> . The fundamental result is that in
this maximal theory 



> 20 =
> 2.


### 5.2 The Parallel Case for CH


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


 The first result in the first step is the following: 



**Theorem 5.4** (Woodin
1985). *Assume* ZFC *and that there is a
proper class of measurable Woodin cardinals. Then* ZFC +
CH *is* O*-complete for*
S21*.* 
Moreover, up to O-equivalence, CH is the unique
S21-statement
that is O-complete for
S21; that
is, letting *T**A* be the O-complete theory
given by ZFC + *A* where *A* is
S21, all
such *T**A* are O-equivalent
to *T*CH and hence (trivially) all
such *T**A* contain CH. In other words, there is
a "good" theory and all "good" theories imply
CH. 


 To complete the first step we have to determine whether this
result is robust. For it could be the case that when one considers the
next level,
S22 (or
further levels, like third-order arithmetic) CH is no longer part of
the picture, that is, perhaps large cardinals imply that there is an
axiom *A* such that ZFC + *A* is O-complete for
S22 (or,
going further, all of third order arithmetic) and yet *not* all
such *A* have an associated *T**A* which
contains CH. We must rule this out if we are to secure the first
step. 


 The most optimistic scenario along these lines is this: The
scenario is that there is a large cardinal axiom *L* and
axioms *A*- such that ZFC
+ *L* + *A*- is
O-complete for all of third-order arithmetic and all such
theories are O-equivalent and imply CH. Going further, perhaps
for each specifiable fragment *V*l of the
universe of sets there is a large cardinal axiom *L* and
axioms *A*- such that ZFC
+ *L* + *A*- is
O-complete for the entire theory of *V*l
and, moreover, that such theories are O-equivalent and imply
CH. Were this to be the case it would mean that for each such l
there is a unique O-complete picture
of *V*l and we would have a unique
O-complete understanding of arbitrarily large fragments of the
universe of sets. This would make for a strong case for new axioms
completing the axioms of ZFC and large cardinal axioms. 


 Unfortunately, this optimistic scenario fails: Assuming the
existence of one such theory one can construct another which differs
on CH: 



**Theorem 5.5** (Koellner and
Woodin 2009). *Assume* ZFC *and that there is a
proper class of Woodin
cardinals. Suppose* *V*l *is a
specifiable fragment of the universe (that is sufficiently
large) and suppose that there is a large cardinal
axiom* *L* *and axioms*
*A*- *such that* 


> ZFC + *L* + *A*- is O-complete
> for Th(*V*l).


 *Then there are axioms* *B*- *such that* 


> ZFC + *L* +
> *B*- is O-complete
> for Th(*V*l)


 *and the first theory* O*-implies* CH *if
and only if the second theory* O*-implies*
!CH. 
 This still leaves us with the question of existence and the answer
to this question is sensitive to the O Conjecture and the
AD+ Conjecture: 



**Theorem 5.6**
(Woodin). *Assume that there is a proper
class of Woodin cardinals and that the* O *Conjecture
holds. Then there is no recursive
theory* *A*- *such
that* ZFC +
*A*- *is* O*-complete for the
theory
of* *V*d0+1*, where*
d0 *is the least Woodin cardinal.* 
In fact, under a stronger assumption, the scenario must fail at a
much earlier level. 



**Theorem 5.7**
(Woodin). *Assume that there is a proper
class of Woodin cardinals and that the* O *Conjecture
holds. Assume that the* AD+ *Conjecture
holds. Then there is no recursive
theory* *A*- *such
that* ZFC +
*A*- *is* O*-complete for the
theory of*
S23*.* 
It is open whether there can be such a theory at the level of
S22. It
is conjectured that ZFC + * is O-complete (assuming large
cardinal axioms) for
S22. 


 Let us assume that it is answered positively and return to the
question of uniqueness. For each such axiom *A*,
let *T**A* be the
S22
theory computed by ZFC + *A* in O-logic. The question of
uniqueness simply asks whether *T**A* is
unique. 



**Theorem 5.8** (Koellner and
Woodin 2009). *Assume that there is a proper class of
Woodin cardinals. Suppose that* *A* *is an axiom
such that* 


> 
>  i. ZFC + *A* *is* O*-satisfiable and*
>   
> 
> ii. ZFC + *A* *is* O*-complete for*
> S22.
> 



*Then there is an axiom* *B* *such that* 


> 
>  i'. ZFC + *B* *is* O*-satisfiable and*
>   
> 
> ii'. ZFC + *B* *is* O*-complete for*
> S22
> 



*and* *T**A*[?]*T**B**.* 
This is the parallel of Theorem 5.2. 


 To complete the parallel one would need that CH is among all of
the *T**A*. This is not known. But it is a
reasonable conjecture. 



**Conjecture 5.9.** Assume large cardinal
axioms. 

1. There is
an S22
axiom *A* such that
	1. ZFC + *A* is O-satisfiable and
	2. ZFC + *A* is O-complete for
	the
	S22.
2. Any such
S22
axiom *A* has the feature that 
> ZFC +
> *A* [?]O CH.

Should this conjecture hold it would provide a true analogue of
Theorem 5.1. This would complete the parallel with the first
step. 


 There is also a parallel with the second step. Recall that for the
second step in the previous subsection we had that although the
various *T**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 the
P2-theory of the structure <*H*(o2), [?], *I*NS, 
*A* | *A* [?] P
(R) [?] *L*(R)> . In the present context of CH we
again (assuming the conjecture) have that although
the *T**A* do not agree, they all contain CH. It
turns out that once again, from among them there is one that stands
out, namely, the maximum one. For it is known (by a result of Woodin
in 1985) that if there is a proper class of measurable Woodin
cardinals then there is a forcing extension satisfying all
S22
sentences ph such that ZFC + CH + ph is
O-satisfiable (see Ketchersid, Larson, & Zapletal
(2010)). It follows that if the question of existence is answered
positively with an *A* that is
S22
then *T**A* must be this maximum
S22
theory and, consequently, all *T**A* agree
when *A* is
S22. So,
assuming that there is a *T**A* where *A* is
S22,
then, although not all *T**A* agree
(when *A* is arbitrary) there is one that stands out, namely, the
one that is maximum for
S22
sentences. 


 Thus, *if* the above conjecture holds, then the case of CH
parallels that of !CH, only now
S22 takes
the place of the theory of *H*(o2). 


### 5.3 Assessment


Assuming that the conjecture holds the case of CH parallels that of
!CH, only now
S22 takes
the place of the theory of *H*(o2): Under the
background assumptions we have: 


1. 1. there are *A* such that ZFC + *A* is
	O-complete for *H*(o2)
	2. for every such *A* the
	associated *T**A* contains !CH, and
	3. there is
	a *T**A* which is maximal,
	namely, *T*([?]) and this theory contains
	20 = 2.
2. 1. there are
	S22-axioms *A*
	such that ZFC + *A* is O-complete for
	S22
	2. for every such *A* the
	associated *T**A* contains CH, and
	3. there is
	a *T**A* which is maximal.


 The two situations are parallel with regard to maximality but in
terms of the level of O-completeness the first is stronger. For
in the first case we are not just getting O-completeness with
regard to the P2 theory of *H*(o2) (with
the additional predicates), rather we are getting O-completeness
with regard to *all* of *H*(o2). This is
arguably an argument in favour of the case for !CH, even granting
the conjecture. 


 But there is a stronger point. There is evidence coming from inner
model theory (which we shall discuss in the next section) to the
effect that the conjecture is in fact *false*. Should this
turn out to be the case it would break the parallel, strengthening the
case for !CH. 


 However, one might counter this as follows: The higher degree of
O-completeness in the case for !CH is really illusory since
it is an artifact of the fact that under ([?]) the theory of
*H*(o2) is in fact mutually interpretable with that of
*H*(o1) (by a deep result of Woodin). Moreover, this
latter fact is in conflict with the spirit of the Transcendence
Principles discussed in Section 4.3. Those principles were invoked in
an argument to the effect that CH does not have an answer. Thus, when
all the dust settles the real import of Woodin's work on CH (so
the argument goes) is *not* that CH is false but rather that CH
very likely has an answer. 


 It seems fair to say that at this stage the status of the local
approaches to resolving CH is somewhat unsettled. For this reason, in
the remainder of this entry we shall focus on global approaches to
settling CH. We shall very briefly discuss two such
approaches--the approach via inner model theory and the approach
via 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 cardinal
axiom Ph that has been reached by inner model theory, one has an
axiom of the form *V* = *L*Ph. This axiom has the
virtue that (just as in the simplest case of *V* = *L*) it provides an
"effectively complete" solution regarding questions
about *L*Ph (which, by assumption,
is *V*). Unfortunately, it turns out that the axiom *V*
= *L*Ph is incompatible with *stronger*
large cardinal axioms Ph'. For this reason, axioms of this form
have never been considered as plausible candidates for new
axioms. 


 But recent developments in inner model theory (due to Woodin) show
that everything changes at the level of a supercompact cardinal. These
developments show that if there is an inner model *N* which
"inherits" a supercompact cardinal from *V* (in the
manner in which one would expect, given the trajectory of inner model
theory), then there are two remarkable consequences: First, *N*
is close to *V* (in, for example, the sense that for sufficiently
large singular cardinals l, *N* correctly computes
l+). Second, *N* inherits all known large
cardinals that exist in *V*. Thus, in contrast to the inner
models that have been developed thus far, an inner model at the level
of a supercompact would provide one with an axiom that
could *not* be refuted by stronger large cardinal
assumptions. 


 The issue, of course, is whether one can have an
"*L*-like" model (one that yields an
"effectively complete" axiom) at this level. There is
reason to believe that one can. There is now a candidate
model *L*O that yields an axiom *V*
= *L*O with the following features: First, *V*
= *L*O is "effectively complete."
Second, *V* = *L*O is compatible with all large
cardinal axioms. Thus, on this scenario, the ultimate theory would be
the (open-ended) theory ZFC + *V* = *L*O + LCA,
where LCA is a schema standing for "large cardinal
axioms." The large cardinal axioms will catch instances of
Godelian independence and the axiom *V*
= *L*O will capture the remaining instances of
independence. This theory would imply CH and settle the remaining
undecided statements. Independence would cease to be an issue. 


 It turns out, however, that there are other candidate axioms that
share these features, and so the spectre of pluralism reappears. For
example, there are axioms *V*
= *L*O*S*
and *V*
= *L*O([?]). These
axioms would also be "effectively complete" and compatible
with all large cardinal axioms. Yet they would resolve various
questions differently than the axiom *V*
= *L*O. For example, the axiom, *V*
= *L*O([?])
would imply !CH. How, then, is one to adjudicate between
them? 


*Further Reading*: For an introduction to inner model
theory see Mitchell (2010) and Steel (2010). For more on the recent
developments at the level of one supercompact and beyond see Woodin
(2010). 


## 7. The Structure Theory of *L*(*V*l+1)


This brings us to the second global approach, one that promises to
select the correct axiom from among *V* = *L*O, *V*
= *L*O*S*,
*V*
= *L*O([?]),
and their variants. This approach is based on the remarkable analogy
between the structure theory of *L*(R) under the assumption of
AD*L*(R) and the structure theory of
*L*(*V*l+1) under the assumption that there is an
elementary embedding from *L*(*V*l+1) into itself
with critical point below l. This *embedding assumption*
is the strongest large cardinal axiom that appears in the
literature. 


 The analogy between *L*(R) and
*L*(*V*l+1) is based on the observation that
*L*(R) is simply *L*(*V*o+1). Thus, l
is the analogue of o, l+ is the analogue of
o1, and so on. As an example of the parallel between
the structure theory of *L*(R) under AD*L*(R) and
the structure theory of *L*(*V*l+1) under the
embedding axiom, let us mention that in the first case,
o1 is a measurable cardinal in *L*(R) and, in the
second case, the analogue of o1--namely,
l+--is a measurable cardinal in
*L*(*V*l+1). This result is due to Woodin and is
just one instance from among many examples of the parallel that are
contained in his work. 


 Now, we have a great deal of information about the structure
theory of *L*(R) under AD*L*(R). Indeed, as we
noted above, this axiom is "effectively complete" with
regard to questions about *L*(R). In contrast, the embedding axiom
on its own is not sufficient to imply that
*L*(*V*l+1) has a structure theory that fully
parallels that of *L*(R) under AD*L*(R). However,
the existence of an already rich parallel is evidence that the
parallel extends, and we can supplement the embedding axiom by adding
some key components. When one does so, something remarkable happens:
the supplementary axioms become *forcing fragile*. This means
that they have the potential to erase independence and provide
non-trivial information about *V*l+1. For
example, these supplementary axioms might settle CH and much
more. 


 The difficulty in investigating the possibilities for the
structure theory of *L*(*V*l+1) is that we have
not had the proper lenses through which to view it. The trouble is
that the model *L*(*V*l+1) contains a large piece
of the universe--namely,
*L*(*V*l+1)--and the theory of this
structure is radically underdetermined. The results discussed above
provide us with the proper lenses. For one can examine the structure
theory of *L*(*V*l+1) in the context of ultimate
inner models
like *L*O, *L*O*S*, *L*O([?]),
and their variants. The point is that these models can accommodate the
embedding axiom and, within each, one will be able to compute the
structure theory of *L*(*V*l+1). 


 This provides a means to select the correct axiom from among *V*
= *L*O, *V*
= *L*O*S*,
*V*
= *L*O([?]),
and their variants. One simply looks at the
*L*(*V*l+1) of each model (where the embedding
axiom holds) and checks to see which has the true analogue of the
structure theory of *L*(R) under the assumption of
AD*L*(R). It is already known that certain pieces of
the structure theory *cannot* hold
in *L*O. But it is open whether they can hold
in *L*O*S*. 


 Let us consider one such (very optimistic) scenario: The true
analogue of the structure theory of *L*(R) under
AD*L*(R) holds of the
*L*(*V*l+1)
of *L*O*S*
but not of any of its variants. Moreover, this structure theory is
"effectively complete" for the theory
of *V*l+1. Assuming that there is a proper
class of l where the embedding axiom holds, this gives an
"effectively complete" theory of *V*. And,
remarkably, part of that theory is that *V* *must*
be *L*O*S*. This
(admittedly very optimistic) scenario would constitute a very strong
case for axioms that resolve all of the undecided statements. 


 One should not place too much weight on this particular
scenario. It is just one of many. The point is that we are now in a
position to write down a list of definite questions with the following
features: First, the questions on this list will have
answers--independence is not an issue. Second, if the answers
converge then one will have strong evidence for new axioms settling
the undecided statements (and hence non-pluralism about the universe
of sets); while if the answers oscillate, one will have evidence that
these statements are "absolutely undecidable" and this
will strengthen the case for pluralism. In this way the questions of
"absolute undecidability" and pluralism are given
mathematical traction. 


*Further Reading*: For more on the structure theory of
*L*(*V*l+1) and the parallel with determinacy see
Woodin (2011b).