a tutorial in set-theoretic geology · a tutorial in set-theoretic geology joel david hamkins new...

Introduction The Ground Axiom Upward glance Set-theoretic Geology Controlling the Mantles Multiverse + Extras

A tutorial in set-theoretic geology

Joel David Hamkins

New York University, Philosophy

The City University of New York, MathematicsCollege of Staten Island of CUNY

The CUNY Graduate Center

London, August 1–6, 2011Summer school in Set Theory and Higher-Order Logic:Foundational Issues and Mathematical Developments

Set-theoretic geology, London 2011 Joel David Hamkins, New York


Part of the work I shall discuss is collaborative work with GunterFuchs, College of Staten Island of CUNY, and Jonas Reitz,New York City Tech, CUNY.

A preprint of our joint paper, "Set-theoretic geology," whichintroduces the topic, is available athttp://arxiv.org/abs/1107.4776.



Philosophical motivations and set-theoretic geology

Philosophy of set theory

A fascinating debate is currently underway in the emerging areaknown as the philosophy of set theory:

Universe view. The philosophical position that there is anabsolute set-theoretic background, in which set-theoreticquestions will have their final answers. The task of settheory is to discover those answers.

multiverse view. The philosophical position that there arenumerous distinct but often closely-related concepts of set,giving rise to distinct set-theoretic universes. The task ofset theory is to investigate these universes and theirinterconnections.




Set-theoretic geologyPart of the evidence for the multiverse position is whatset-theorists have discovered is the enormous diversity ofset-theoretic possibility.

On the multiverse perspective, a major part of the goal of settheory is to understand the structural relations between allthese various set-theoretic worlds.

Set-theoretic geology aims to fulfill the part of this goal as itarises in consideration of forcing, one of the principalset-theoretic methods for building alternative set-theoreticworlds.

In set-theoretic geology, we seek to study the set-theoreticuniverse in the context of all its grounds and forcing extensions.




ForcingForcing is a set-theoretic method (Cohen, 1962) forconstructing a larger model of set theory from a given model.

Begin with a ground model V |= ZFC and poset P ∈ V . Adjoinan ideal “generic” element G, a V -generic filter G ⊆ P, and withit construct the forcing extension V [G], akin to a field extension.

V ⊆ V [G]

Objects of V [G] are constructible algebraically from objects inV and the new object G. The ground model V has a surprisingdegree of access to the objects and truths of V [G].

Forcing has been used to construct diverse models of settheory.




A new perspective

Forcing is naturally viewed as a method of building outer asopposed to inner models of set theory.

Nevertheless, a simple switch in perspective enables forcing asa method of producing inner models as well. Namely, we lookfor how the universe V itself might have arisen via forcing.

A transitive class W ⊆ V is a ground if the universe wasobtained by set forcing over W . That is, if V = W [G] for someW -generic filter G ⊆ P ∈W .

This change in viewpoint leads to set-theoretic geology.




A new fundamental question

Set-theoretic geology begins with a theorem of Laver (2004),which answers a fundamental question about forcing that couldhave been asked decades earlier, but which was not.

Question

Is the ground model V definable in its forcing extensions V [G]?

It turns out that it is.




Geology begins by recognizing the ground

Theorem (Laver 2007, independently Woodin)

The universe V is a definable class in every set-forcingextension V [G].

There is a first-order formula ϕ in the language of set theoryand parameter r ∈ V such that

x ∈ V ⇐⇒ V [G] |= ϕ(x , r).

The proof uses my methods on approximation and covering,and if there is time later I shall explain the definition in detail.



The ground axiom and consequences

The Ground AxiomUpon learning of Laver’s theorem, Jonas Reitz and I formulatedthe Ground Axiom. The idea was to try to undo forcing, to do it‘backwards’.

Definition (Hamkins,Reitz)

The Ground Axiom is the assertion that the universe V has nonontrivial grounds.

That is, V satisfies the ground axiom GA if there is no transitiveinner model W |= ZFC such that V = W [G] for some nontrivialW -generic filter G ⊆ P ∈W .

In other words, the ground axiom asserts that the universe Vwas not obtained by forcing.




GA is first orderThe ground axiom asserts that the universe is not a nontrivialforcing extension of any transitive inner model.

In other words, GA asserts that whenever W is a transitiveproper class model of ZFC and G ⊆ P is W -generic fornontrivial forcing P, then V 6= W [G].

Although this formulation of the ground axiom is prima faciesecond-order, since it quantifies over inner models,nevertheless Reitz proved that the ground axiom is first-orderaxiomatizable.

Theorem (Reitz)

The ground axiom is first-order expressible in set theory.




GroundsDefinition

A transitive class W is a ground of V if W |= ZFC andV = W [G] for some W -generic G ⊆ P ∈W .

Laver’s theorem shows that each ground W is definable in Vfrom parameter r :

x ∈W if and only if V |= ϕ(x , r).

So by varying the parameter r , we reach all possible grounds.

Possibly some parameters r do not succeed in defining aground. Reitz observed that whether ϕ(x , r) defines a ground isa first-order property of r .




The grounds form a parameterized family

Theorem

There is a parameterized family Wr | r ∈ V such that1 Every Wr is a ground of V and r ∈Wr .2 Every ground of V is Wr for some r.3 The relation “x ∈Wr ” is first order.

This reduces second-order statements about grounds tofirst-order statements about parameters.

For example, the relation “V = Wr [G] by Wr -generic filterG ⊆ P ∈Wr ” is first order in arguments (r ,G,P).




Reducing Second to First order

Thus, we may quantify over grounds by quantifying over theparameters used to define them.

Conclusion

The ground axiom is equivalent to the first-order assertion∀r V = Wr .

Second-order questions about grounds Wr become first-orderquestions about the index r .




Natural models of GA

The ground axiom holds in many canonical models of settheory:

The constructible universe L.Extensions L[0]], L[µ], L[~E ].Many other canonical inner models of large cardinals,including many instances of the core model.

These models are among the most highly regular models of settheory that are known. They exhibit numerous highly attractivestructural features, such as the GCH, diamond, V = HOD,condensation and so on.




Consequences of GA

Since all the known models of GA were highly structuredmodels, satisfying GCH diamond, and so on, it was natural toinquire to what extent these regularity features wereconsequences of GA.

Test Question

Does the ground axiom imply CH?

After all, the only way we know how to violate CH is by forcing,and under GA the universe is not a forcing extension, so anaffirmative answer seems reasonable.



Building models of the ground axiom

Obtaining GA in an extensionNevertheless, GA does not settle CH.

Theorem (Reitz)

Every model of ZFC has an extension, preserving any desiredVα, which is a model of GA.

It follows that the ground axiom is compatible with anyset-theoretic behavior that can occur inside any Vα of anymodel of set theory.

This includes CH, ¬CH, ♦, ¬♦ and so on. Every Σ2 statementthat is consistent with ZFC is consistent with ZFC + GA.

Thus, the ground axiom has essentially NO regularityconsequences.




Forcing GA

Theorem (Reitz)

Every model of ZFC has an extension, preserving any desiredVα, which is a model of GA.

The paradoxical situation is that although GA asserts that theuniverse is not obtained by forcing, Reitz obtains GA in aforcing extension, but by using class forcing.

The point is that GA is concerned only with set forcingextensions, for which P is a set, and we can perform properclass forcing whose resulting extension is not obtainable by setforcing over any model.

Reitz’s method is to force a very strong version of V = HOD.




Making sets definable

Suppose x is an arbitrary set of natural numbers. Perhaps x isnot definable. Can we make it definable in a forcing extension?

Yes. Easton’s theorem gives us complete control over the GCHpattern on regular cardinals. So we may find a forcingextension V [G] in which

n ∈ x ⇐⇒ 2ℵn = ℵn+1.

Thus, the set x becomes definable in the forcing extension.

We may now iterate this idea to make every set definable fromordinal parameters.




Continuum coding axiom

Definition

The continuum coding axiom CCA is the assertion that everyset of ordinals is coded into the GCH pattern on a block ofregular cardinals.

That is, whenever x ⊆ γ, then there is an ordinal λ such thatα ∈ x ⇐⇒ 2ℵλ+α+1 = ℵλ+α+2 for all α < γ.

Theorem (Folklore)

There is a class forcing extension V [G] in which the CCA holds.




Forcing the CCATheorem (Folklore)

There is a class forcing extension V [G] in which the CCA holds.

Proof.

Traditional method: bookkeeping iteration.

Simpler method: generic coding. Let P be the Easton-supportforcing iteration, whose conditions may decide generically atstage α, using the lottery sum, either to force the GCH at ℵα+1or to force ¬GCH at ℵα+1. The generic filter decides which.

It is dense that any set x ⊆ ORD that is added is subsequentlycoded into the GCH pattern "generic bookkeeping".

Consequently, V [G] |= CCA.




Theorem (Reitz)

The continuum coding axiom implies the ground axiom.

Proof.

Suppose V |= CCA and V = W [g] for some g ⊆ P ∈W . Sincethe GCH pattern is not affected above |P|, it follows that everyset in V is coded into W , and so V ⊆W and so V = W .

By starting the forcing above α, Reitz obtains:

Corollary

Every model V of set theory has a class forcing extensionV [G], preserving any desired Vα, which is a model of GA.

Consequently, GA has no regularity consequences such as CHor diamond.




An amusing application: MA+GA is consistentMartin’s axiom MA is the assertion that every c.c.c. partialorder P and small family D of dense subsets of P has a filterF ⊆ P meeting every D ∈ D.

One usually achieves MA via long iterated forcing, adding allthe various witnessing filters.

Furthermore, MA is customarily conceived of as assertingprecisely that a lot of c.c.c. forcing has already been performed.

Nevertheless, Reitz’s argument shows one can also have GA:

Theorem (Reitz)

Every model of ZFC has an extension that is a model ofZFC + GA + MA.




GA and V=HOD

Thus, we can obtain GA by forcing strong version of V = HOD.

Question

Is GA consistent with V 6= HOD?

The answer is yes.

Theorem (Hamkins,Reitz,Woodin)

Every model of set theory has an extension which is a model ofGA plus V 6= HOD.

Will give details later. Sketch: first extend to V ⊆ V |= CCA; then addV [G] a Cohen subset to every regular cardinal. By homogeneity, thisachieves V 6= HOD. Use details of approximation and cover for GA.



Amalgamation of forcing extensions

Upward amalgamation

Let us turn now to an upward-oriented question.

To what extent may we amalgamate forcing extensions? Arethe forcing extensions of a given model upward directed?

If V [G] and V [H] are two forcing extensions of V , must there bea common extension V [K ]?

There are meta-mathematical issues with formulating thequestion. After all, if our only method of referring to V [G] andV [H] together is by means of a common extension V [K ] inwhich they both already exist, then the issue becomes moot.

The toy model perspective can be illuminating.




Non-amalgamationTheorem (Woodin)

If W is a countable transitive model of ZFC, then there areW-generic Cohen reals c and d such that W [c] and W [d ] haveno common extension to a model W |= ZFC with the sameordinals.

Proof.

W is countable. Build c and d in stages. Fix a "bad" real z,such as a real coding all of W . Fix Dn dense sets in W . Let c0meet D0, and d0 = 00 · · · 0 up to same length of c0, followed by1, followed by z(0), and then extended to meet D0. Extend c0 toc1 by adding 0s to length of d0, then 1, then meet D1, etc. Thiscodes z into c ⊕ d .




Extending to large finite familiesThe argument generalizes to add three reals c, d and e, anytwo of which can be amalgamated, but not all three. And so on.

Theorem

If W |= ZFC is countable, then for every n there are W-genericCohen reals c1, . . . , cn such that any proper subfamily of theextensions W [c1], . . . ,W [cn] is amalgamable, but the wholefamily is not.

Proof.

The proof is similar. Enumerate the dense sets Dn. Fixforbidden real z. Build ci in steps, extending all but one eachtime, adding 0s to the other, and coding z(k) there.




Infinite version

Does it generalize to infinitely many extensions? If a family offorcing extensions is finitely amalgamable, can one find acommon forcing extension of them all?

One shouldn’t ask for too much.

If W is countable, we may build extensions W [Gn] that collapsemore and more cardinals of W , so that

⋃n W [Gn] collapses all

the cardinals of W . Thus, the W [Gn] have no commonextension M[H].

So one wants to consider only forcing extensions of uniformlybounded size in W .




No extension with the sequence

Another way to ask for too much:

If W is countable, let z be any real that cannot be added byforcing over W , such as a real coding all of W . Let W [dn] bemutually generic Cohen reals. Modify dn on the first bit only toagree with z(n), producing cn. Thus, each cn is W -genericCohen real, and any finitely many are mutually generic. But noextension M[G] has 〈cn | n < ω〉, since from this sequence wecould build z.

So we should not expect the sequence of generic objects in thecommon extension.




Finding a common extensionWith the right goal it turns out that one CAN find a commonextension:

Theorem

If W is a countable model of ZFC and

W ⊆W [G0] ⊆W [G1] ⊆W [G2] ⊆ · · ·

is a countable tower of forcing extensions, with forcing ofbounded size in W, then there is a common forcing extensionW [H] above them all.

Thus, any finitely amalgamable family of forcing extensions isfully amalgamable.




Proof. Suppose (special case) increasing tower of extensions

W ⊆W [g0] ⊆W [g0 × g1] ⊆W [g0 × g1 × g2] ⊆ · · · ,

with gn ⊆ Qn mutually W -generic. We seek a single commonextension W [H] containing them all.

The proof method is "hiding the generics." Pick δ > |Qn|, let θ = 2δ.Let P =

∏α<θ Rα be the finite support product of all size < δ posets,

with unbounded repetition. P is δ-c.c. Since W is countable, there isW -generic H ⊆ P. In fact,

⋃n<ω W [g0 × g1 × · · · × gn]-generic.

Key step. Externally, let 〈θn | n < ω〉 cofinal in θ, with Rθn = Qn.Modify H by surgery to H∗ with gn at θn. H∗ is still a filter on P. It isalso W -generic, since every antichain A ⊆ P in W is bounded in θ,hence involves only finitely many θn.

But gn ∈W [H∗], so W [g0 × · · · × gn] ⊆W [H∗] for every n < ω, asdesired.



Grounds, bedrock, solid bedrock, directedness

Geology

Let us turn now to the development of set-theoretic geology.

We shall dig a bit deeper underground, under the grounds,hitting bedrock or solid bedrock, eventually uncovering themantle, the inner mantles, the generic mantle, the inner genericmantles and ultimately the outer core.

Let’s explore the underground...




Bedrock models

Suppose that the universe is a forcing extension of a ground.This ground may be a forcing extension of a deeper ground.And that ground may be a forcing extension of a still-deeperground.

Do we eventually hit bedrock?

To hit bedrock would mean to have a ground model W thatcannot be further reduced to a deeper ground. In other words,a minimal ground.

A bedrock is a ground that is minimal among all grounds.Equivalently, a bedrock of V is a ground of V that satisfies theground axiom.




A bottomless modelTheorem (Reitz)

Every model of set theory V has a class-forcing extension V [G]having no bedrock.

Proof.

By forcing if necessary, assume GCH in V . Let P be the class productforcing ΠαQα, where Qα generically chooses to force the GCH or itsnegation at ℵα+1, and consider the extension V [G]. Every set ofordinals x ∈ V will be coded into the GCH pattern of V [G].Meanwhile, the tail extensions V [Gα], where Gα ⊆ P [α,∞) areground models of V [G]. If V [G] = W [h] where h ⊆ Q ∈W isW -generic, then V ⊆W by the coding, and above |Q|, W will containall the tails of G, and so V [Gα] ⊆W ⊆ V [G] for large enough α. SoV [Gα+1] is a still-deeper ground.




Unique bedrock

Suppose that the universe V has a bedrock, that is, a groundW that cannot be further decomposed as a forcing extension.

Open Question

Is the bedrock unique when it exists?

We don’t know.

For all the models in which we are able to calculate the answer,the answer is yes.




Solid bedrock

Consider the collection of all grounds of V under the inclusionrelation.

A bedrock is a minimal ground.A solid bedrock is smallest ground, that is, a groundcontained in all other grounds.

Note that the solid bedrock is unique when it exists.




CCA implies solid bedrock

Observation

If the continuum coding axioms CCA holds, then V is a solidbedrock in all its forcing extensions.

Proof.

Suppose that the CCA holds in V . Then every set in V is codedinto the GCH pattern of V . Set forcing V [G] preserves thiscoding above the size of the forcing. Similarly, the coding ispreserved to any ground W ⊆ V [G]. Thus, every set in V iscoded in W , and consequently V ⊆W . So V is a solid bedrockin V [G].




How many grounds?

Theorem

If there is a solid bedrock, then there are only set manygrounds. That is, there is a set I such that every ground is Wrfor some r ∈ I.

Proof.

If M is a solid bedrock, then M is a ground of V = M[G] andevery other ground W is trapped M ⊆W ⊆ M[G] = V . Fromthis it follows that W = M[G ∩ B0] for some completesubalgebra B0 ⊆ B. There are only a set of possible B0.




Common ground

Open Question

Do any two grounds contain a common ground?

That is, for any r and s, is there t with Wt ⊆Wr ∩Ws?

In other words, are the grounds downward directed?

This question could have been asked forty years ago.

I place it at the foundation of any serious investigation offorcing. If we are to claim any serious understanding of forcing,we must know the answer to this fundamental question.




Set-directednessBecause we are also able to quantify over any set of indices rfor grounds Wr , it is also sensible and natural to seek acommon ground below any set of grounds.

Question

Are the grounds downward set-directed? That is, for any set A,is there t with

Wt ⊆⋂r∈A

Wr ?

This is true, of course, in any model with a solid bedrock. But itis also true in Reitz’s bottomless model, where there is nobedrock.




Downward directedness hypothesesDefinition

1 The Downward Directed Grounds Hypothesis DDG assertsthat the grounds are downward directed.

For every r and s there is t such that Wt ⊆Wr ∩Ws.

2 The Strong DDG asserts that they are downwardset-directed.

For every A there is t with Wt ⊆⋂

r∈A Wr .

The strong DDG holds in every model for which we are able todetermine the answer. Also, the strong DDG holds if V = L[A].

Meanwhile, Woodin has a candidate counterexample model.




The Mantle

Burrowing deeper underground, the principal new concept ofset-theoretic geology is the mantle.

Definition

The mantle M is the intersection of all grounds.

Thus, the mantle removes whatever forcing might have beenperformed when forming the universe.




Mantle is definable

Theorem

The mantle is a first-order definable transitive class, containingall ordinals.

Proof.

We have the first-order definable parameterization of groundsWr . So we may define the mantle by x ∈ M if and only if∀r x ∈Wr .

And the mantle is easily seen to be transitive and contain allordinals.




Ancient ParadiseThe analysis of the mantle engages with an interestingphilosophical view:

Ancient Paradise. This is the philosophical view that there is ahighly regular core underlying the universe of set theory, aninner model obscured over the eons by the accumulating layersof debris heaped up by innumerable forcing constructions sincethe beginning of time. If we could sweep the accumulatedmaterial away, we should find an ancient paradise.

The mantle, of course, wipes away an entire strata of forcing.

So the ancient paradise view suggests that the mantle may behighly regular.




Every model is a mantleUnfortunately, our initial main theorem tends to refute thisperspective:

Theorem (Fuchs, Hamkins, Reitz)

Every model of ZFC is the mantle of another model of ZFC.

By sweeping away the accumulated sands of forcing, what wefind is not a highly regular ancient core, but rather: an arbitrarymodel of set theory.

Conclusion: we will not be able to prove any highly regularstructural features of the mantle.

The proof will come later.




Reducing Second to First orderAs mentioned earlier, the parameterized family Wr | r ∈ V ofgrounds reduces 2nd order properties about grounds to 1storder properties about parameters.

The Ground Axiom holds if and only if ∀r Wr = V .Wr is a bedrock if and only if ∀s (Ws ⊆Wr =⇒ Ws = Wr ).Wr is a solid bedrock if and only if ∀s (Wr ⊆Ws).The mantle is defined by M = x | ∀r (x ∈Wr ) .The DDG asserts ∀r , s ∃t Wt ⊆Wr ∩Ws.The strong DDG asserts ∀A existst Wt ⊆

⋂r∈A Wr .

So all of our questions about the nature and structure of thegrounds and of the mantle are first-order questions in thelanguage of set theory, expressible in ZFC.



The mantle under directedness

The Mantle under Directedness

Under the downward directed grounds hypothesis, the mantle iswell behaved.

Theorem

1 If the DDG holds, then the mantle is constant across thegrounds, and M |= ZF.

2 If the Strong DDG holds, then M |= ZFC.

The hypothesis in (2) can be weakened.

Let’s describe the proof.




DDG implies mantle is invariant to grounds

Theorem

If the grounds are downward directed, that is, if the DDG holds,then the mantle is constant across grounds.

Proof.

Suppose the grounds of V are downward directed. Fix aground W . Any ground of W is a ground of V ; so the mantle ofV is contained in the mantle of W . Conversely, if a is not in themantle of V , then a /∈W ′ some ground W ′, and so a /∈W ∩W ′.By directedness, there is a ground W ⊆W ∩W ′. But W is aground of W and a /∈ W , and so a is not in the mantle of W . Sothe mantle is constant among the ground models of V .




ZF in the mantle

Theorem

If the mantle is constant across the grounds, then it is a modelof ZF.

Proof.

If the mantle is constant across the grounds, then it is definablein every ground, so it satisfies ZF by the intersection theorembelow.

Intersection Theorem

IfW is family of ZFC models, all with same ordinals and ∩W isa class in every W ∈ W, then ∩W |= ZF.




Intersection Theorem

IfW is a family of ZFC models with same ordinals and ∩W is aclass in every W ∈ W, then

1 ∩W |= ZF.2 IfW is locally realized, meaning every y ∈ ∩W has

W ∈ W with P(y)∩W = P(y)W , then ∩W |= ZFC.

Proof.⋂W is transitive, contains ORD and closed under Gödel

operations. Remains only to show almost universal: everyA ⊆

⋂W has A ⊆ B ∈

⋂W. Use B = Vα ∩ (

⋂W) for large α.

For ZFC, consider any set y in⋂W, realize P(y × y)∩W in W .

So y has well-orders in W that survive to⋂W.




Intersecting models of ZFC

There are some interesting boundary cases:

Descending ORD-length sequencesM0 ⊇ M1 ⊇ · · · ⊇ Mα ⊇ · · · have ∩αMα |= ZFC.It is not true for set-length sequences: the intersection offirst ω1 iterates of a normal ultrapower does not model ZFC.Similarly, HODω = ∩nHODn may not satisfy ZFC.The intersection of two ZFC models, even two grounds,need not satisfy ZFC.

There are some very interesting meta-mathematical issues withHODn sequence.




Intersection of two grounds

Observation

The intersection of two grounds may not model ZFC.

Proof.

Suppose G ⊆ P is L-generic for the forcing to make 2ℵn = ℵn+2for all n < ω. Use P = ΠnQn, where Qn = Add(ℵn,ℵn+2). Let cbe a Cohen real over L[G], and let V = L[G][c]. Considergrounds L[G] and L[G c]. Same cardinals. Note thatL[G] ∩ L[G c] has every Gn for n ∈ c, and so GCH holds at ℵnif and only if n ∈ c. But c does not exist in L[G] ∩ L[G c].

In fact, there is no largest ZF model inside L[G] ∩ L[G c].


Laura

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ZFC in the mantle

Theorem

If the grounds are downward set-directed, that is, if the strongDDG holds, then the mantle is a model of ZFC.

Proof.

Under finite directedness, we’ve already established that themantle M |= ZF. To get choice, use locally downwardset-directed, which follows from strong DDG.

The point is that any y ∈ M has P(y × y)M realized in someground Wr , and so all the well-orderings of y in Wr survive tothe mantle M.




Set many groundsObservation

The mantle is a ground ⇐⇒ the universe has a solid bedrock.

Earlier we saw this implies there are only set many grounds.

Question

Is the solid bedrock axiom equivalent to the assertion that thereare only set many grounds?

A strong counterexample to this would be a model V havingonly set many grounds, but no minimal ground. Any such modelwould of course also be a counterexample to downwardset-directedness and the generic strong DDG hypothesis.




The generic mantle

Earlier, we defined the mantle to be the intersection of allgrounds of V .

Define now that the generic mantle gM is the intersection of allgrounds of all forcing extensions of V .

This is the intersection of a larger collection of grounds, and so

gM ⊆ M.




Theorem

The generic mantle is a definable transitive class, contains allordinals, is invariant under forcing, and is a model of ZF.

Proof.

Clearly transitive and contains all ordinals.

Invariant under forcing: easy to see gMV ⊆ gMV [G]. If x /∈ gMV , thenx /∈W some ground W ⊆ V [H]. So there is a condition q forcingx /∈ Wr . Now we may assume H is V [G]-generic, so V [G][H] has aground omitting x . So x /∈ gMV [G]. Thus, gMV = gMV [G].

Now argue gM |= ZF by the intersection theorem, since it isintersection of ZFC models in each of which it is a class.




The generic multiverse

The generic mantle of a model is naturally considered in acontext that includes all its forcing extensions, their grounds,subsequent forcing extensions, and so on.

The generic multiverse is the family of universes obtained byclosing under forcing extensions and grounds.

There are various philosophical motivations to study thegeneric multiverse.




The generic multiverse

Woodin introduced the generic multiverse essentially to rejectit, to defeat a certain multiverse view of truth: the idea that to beTrue means to be true in the generic multiverse.

We don’t hold that view of truth, but nevertheless find thegeneric multiverse to be a natural context for set-theoreticinvestigation. Indeed, it is a principal focus for geology.

The generic multiverse naturally partitions the larger multiverseof models of set theory into equivalence (meta)classes,consisting of models reachable from one another by passing toforcing extensions and ground models.




Formalization of generic multiverse

Because the generic multiverse concept is clearly second-orderor higher-order, however, there are certain difficulties offormalization and meta-mathematical issues that need to beaddressed. This is particularly true when one wants to considerthe generic multiverse of the full set-theoretic universe V , ratherthan merely the generic multiverse of a toy countable model.

The standard approaches to second-order set theory, after all,such as Gödel-Bernays set theory or Kelly-Morse set theory, donot seem to provide a direct account of the generic multiverseof V , whose forcing extensions are of course not directlyavailable, even as GBC or KM classes.




But many multiverse questions are first order

Nevertheless, many generic multiverse questions arefirst-order.

We already treated grounds, bedrocks, solid bedrocks, theground axiom, the mantle, the generic mantle, and so on infirst-order set theory.

We have a first-order manner of treating truth in the forcingextensions of V . It is first-order to state that ϕ holds in someforcing extension of V or all forcing extensions of V .

Thus, also first-order to assert that ϕ holds in some forcingextension of a ground of some forcing extension.




The toy model formalizationSometimes we seek the entire generic multiverse.

Toy model perpsective. Analogous tocountable-transitive-model approach to forcing. Use acountable W |= ZFC; consider all forcing extensions, grounds ofW , as constructed in V .

The toy model generic multiverse of W depends on thebackground in which it is computed.

The toy model approach is used in:Countable-transitive-model approach to forcing.Our previous non-amalgamation result W [c], W [d ].Formalization of IMH.




Generic multiverse possibility

We may view the generic multiverse as a Kripke model ofpossibility. Thus, ϕ is multiverse possible, written ♦m ϕ, if itholds somewhere in the generic multiverse.

A multiverse path is 〈U0, . . . ,Un〉, where each Ui+1 is either aground or forcing extension of Ui .

Woodin has argued that ♦m ϕ if and only if ϕ holds in a modelreachable by a multiverse path of length three, specifically, in aforcing extension of a ground of a forcing extension.

This implies that ♦m ϕ is first-order expressible.

Directedness implies that two steps suffice.



Generic mantle in the generic multiverse

Generic mantle is forcing invariant

Corollary

The generic mantle is constant across the generic multiverse.Indeed, it is the intersection of the generic multiverse.

Proof.

Since the generic mantle is invariant by set forcing, it ispreserved from ground to extension and vice versa. Thus, allmodels in the generic multiverse have the same genericmantle. Hence, the generic mantle is contained within theintersection of the generic multiverse. Conversely, it is theintersection of part of the generic multiverse. So the genericmantle is the intersection of the generic multiverse.




Generic mantle = largest forcing-invariant classCorollary

The generic mantle is the largest forcing-invariant class.

Proof.

We showed it is forcing-invariant above. Any other class that isdefinable and invariant by forcing will be preserved to forcingextensions and grounds, and will therefore be contained withinthe generic mantle. So the generic mantle is the largest suchforcing-invariant class.

The generic mantle is thus a highly canonical object, the largestforcing-invariant definable class. It should become a centralfocus of attention for those set-theorists interested in forcingand models of set theory.




Inclusion of modelsLemma

Any generic ground of V contained in V is a ground of V .

Proof.

If W is a ground of V [G] and W ⊆ V ⊆ V [G], then by generalforcing facts, W is a ground of V .

Question

If W is in the generic multiverse of V and W ⊆ V , must W be aground of V?

Is the ⊆ relation the same as the “is a ground model of”relation? Yes, if grounds are dense in generic multiverse.




DDG in all grounds

Theorem. The following are equivalent

1 The DDG holds: the grounds of V are downward directed.2 The DDG holds in some forcing extension of V .3 The DDG holds in every ground of V .

Proof.

3 =⇒ 1 =⇒ 2 are easy. For 2 =⇒ 3, suppose grounds of V [G] aredownward directed and W and W ′ are grounds of U, a ground of V .So W ,W ′ are grounds of V [G], and so by 2 there is ground W ofV [G] with W ⊆W ∩W ′. Since W ⊆ U ⊆ V [G], it follows that W is aground of U, and so 3 holds.





1 The generic DDG holds: in every forcing extension, the groundsare downward directed.

2 The grounds of V are downward directed and dense belowgeneric grounds.

3 The grounds of V are downward directed and dense belowgrounds of ground extensions.

Proof.

(1 =⇒ 3) If W is a ground of Wr [G], then W = W Wr [G]t some

t ∈Wr [G], having name t . If G is V -generic, form V [G] and apply (1)to get W V [G]

s below V and W . So there is condition p forcing this.Argue that W Wr

s ⊆W even when G is not V -generic. (3 =⇒ 2)immediate. (2 =⇒ 1) Use grounds of V .




The generic DDG

Corollary

If the generic DDG holds, then:1 the mantle is the same as the generic mantle;2 the class of ground extensions is closed under forcing

extensions and grounds;

Proof.

If the grounds are dense below the generic grounds, then themantle is the same as the generic mantle. Also, the class ofground extensions is closed under forcing extensions andgrounds.




Generic multiverse

So if the DDG holds in all extensions, then the collection ofground extensions Wr [G] is closed under forcing extensionsand grounds.

Theorem

If DDG holds in all extensions, then the generic multiverse of Vconsists precisely of the ground extensions.

In other words, every model in the generic multiverse can bereached in two steps: first go down to a ground Wr , and then goup to a forcing extension of that ground Wr [G].




Up-down may not suffice

Consider the dual two-step, first go to a forcing extension V [G],and then to a ground W V [G]

r . That is, go "up-and-then-down."

Observation

If ZFC is consistent, there is a toy-model generic multiverse notexhausted by the generic grounds.

Proof.

If V [c] and V [d ] are not amalgamable, then V [d ] is not ageneric ground of V [c], but they have the same multiverse.





1 The generic strong DDG holds. That is, in every forcingextension of V , the grounds are downward set-directed.

2 The grounds of V are downward set-directed and dense belowthe generic grounds.

3 The grounds of V are downward set-directed and dense belowthe grounds of every ground extension.

Proof.

1 =⇒ 3 not hard. 3 =⇒ 2 immediate. For 2 =⇒ 1, basically getbelow the grounds of any extension and apply strong DDG in V (butthere are subtleties).




Generic mantle has ZFC

Theorem

If the generic strong local DDG holds, then the generic mantleis a model of ZFC.

Proof.

We already know gM |= ZF. Use strong local DDG to argue forAC, but there are subtleties.




Solid generic bedrockQuestion

When does the universe have a generic solid bedrock? That is,when is the generic mantle also a ground?

This phenomenon is not universal, in light of the following.

Corollary

There is a class extension V [G] with the generic strong localDDG, but no bedrock and no generic bedrock.

Proof.

Use Reitz’s bottomless model. It continues to be bottomless inevery set forcing extension.




If V is constructible from a set

Theorem

If V is constructible from a set, then this is true throughout thegeneric multiverse.

Proof.

Being L[x ] is clearly preserved by forcing extensions. SupposeW is a ground of V = L[x ] = W [G], where G ⊆ P ∈W . Pickname x = (x)G. Let a ∈W be set of ordinals coding transitiveclosure t = TC(P, x); consider L[a]. Note L[a] ⊆W . Also, Gis L[a]-generic, and so L[a] ⊆W ⊆ L[a][G]. This implies that Wis a forcing extension of L[a], and hence constructible from aset.




Bedrocks in L[a]

Theorem

If V = L[a] and there is no bedrock, then this is true throughoutthe generic multiverse.

Proof.

Suppose V is constructible from a set and there is no bedrock.Thus, there are class many grounds. This remains true in anyset-forcing extension, and so there is no bedrock in anyextension. Also, it is preserved to grounds. So it is truethroughout the generic multiverse.




The Generic HOD

HOD is the class of hereditarily ordinal definable sets.

HOD |= ZFC

The generic HOD, introduced by Fuchs, is the intersection of allHODs of all forcing extensions.

gHOD =⋂G

HODV [G]

The original motivation was to identify a very large canonicalforcing invariant class.




The Generic HOD

Facts

1 gHOD is constant across the generic multiverse.2 The HODs of all forcing extensions are downward

set-directed.3 Consequently, gHOD is locally realized and gHOD |= ZFC.4 The following inclusions hold.

HOD

∪

gHOD ⊆ gM ⊆ M




V = L[a] implies generic strong DDG

Theorem

If V = L[a], then the generic strong DDG holds.

Proof.

Consider HODV Coll(ω,α). By homogeneity, these are contained in

HODV . They are downward set-directed. Vopenka’s theorem,that every set is generic over HOD, implies that these modelsare grounds of V , since we need only add a. They are densebelow the grounds, since if V = W [g] via g ⊆ P ∈W , thenabsorb P into Coll(ω, α) for large enough α, and observe thatHODV [G] ⊆ HODW ⊆W , as desired. Similarly dense in genericgrounds.




Theorem

If V = L[a], then the following are equivalent:1 There are only set many grounds.2 The bedrock axiom.3 The solid bedrock axiom.

Proof.

(1 =⇒ 2) Apply strong DDG. (2 =⇒ 3) Suppose that W is abedrock. It must be W = HODV Coll(ω,α)

some large α. ButHODV Coll(ω,β)

are dense below grounds, so W is a solid bedrock.(3 =⇒ 1) proved generally.




The structure of all groundsTheorem

1 The collection of grounds between a fixed ground W andthe universe V is an upper semi-lattice.

2 If the grounds of V are downward directed, then thegrounds of V are an upper semi-lattice.

3 The grounds of V need not be a complete uppersemi-lattice, even if V = L[a].

4 The grounds of V need not be a lattice, even when thegrounds are downward set-directed, and even if theuniverse is constructible from a set.

(1) and (2) are soft. (3) and (4) follow from fact that two groundsmay have no largest common ground.



Separating the notions

HOD

∪

gHOD ⊆ gM ⊆ M

To what extent can we control and separate these classes?

We answer with our main theorems.



Realizing V as the Mantle


If V |= ZFC, then there is a class extension V in which

V = MV = gMV = gHODV = HODV

In particular, as mentioned earlier, every model of ZFC is themantle and generic mantle of another model of ZFC.

It follows that we cannot expect to prove ANY regularityfeatures about the mantle or the generic mantle.



Forcing definability

The initial idea goes back to McAloon (1971), to make setsdefinable by forcing.

For an easy case, consider an arbitrary real x ⊆ ω. It may nothappen to be definable in V .

With an infinite product, we can force the GCH to hold at ℵnexactly when x(n) = 1.

In the resulting forcing extension V [G], the original real x isdefinable, without parameters.



Proof sketch of first separation theoremWe start in V |= ZFC and want V [G] withV = MV [G] = gMV [G] = gHODV [G] = HODV [G].

Let Qα generically decide whether to force GCH or ¬GCH at ℵα (*).Let P = ΠαQα, with set support. Consider V [G] for generic G ⊆ P.

Every set in V becomes coded unboundedly into the continuumfunction of V [G]. Hence, definable in V [G] and all extensions.

So V ⊆ gHOD. Consequently V ⊆ gHOD ⊆ gM ⊆ M and V ⊆ HOD.

Every tail segment V [Gα] is a ground of V [G]. Also, ∩αV [Gα] = V .Thus, M ⊆ V . Consequently, V = gHOD = gM = M.

HODV [G] ⊆ HODV [Gα], since P α is densely almost homogeneous.

So HODV [G] ⊆ V .

In summary, V = MV [G] = gMV [G] = gHODV [G] = HODV [G], as desired.




Other combinations are also possible.1 Every model of set theory V has an extension V with

V = MV = gMV = gHODV = HODV

2 Every model of set theory V has an extension W with

V = MW = gMW = gHODW but HODW = W

3 Every model of set theory V has an extension U with

V = HODU = gHODU but MU = U

4 Lastly, every V has an extension Y with

Y = HODY = gHODY = MY = gMYSet-theoretic geology, London 2011 Joel David Hamkins, New York


Second separation theorem: mantles low, HOD highFor the 2nd separation theorem, we want V [G] with

V = MV [G] = gMV [G] = gHODV [G] but HODV [G] = V [G]

Balance the forces on M, gM, gHOD and HOD.Force to V [G] where every set in V is coded unboundedlyin the GCH pattern.Also ensure that G is definable, but not robustly.The proof uses self-encoding forcing:

Add a subset A ⊆ κ. Then code this set A into theGCH pattern above κ. Then code THOSE setsinto the GCH pattern, etc. Get extension V [G(κ)]in which G(κ) is definable.



Keeping HODs low, Mantles highFor the 3rd separation, we keep the HODs low and the Mantlehigh, V = HODV [G] = gHODV [G] but MV [G] = V [G].

Such a model V [G] will of course be a model of the GroundAxiom plus V 6= HOD. Recall


Every V |= ZFC has a class forcing extensionV [G] |= GA + V 6= HOD.

We modified the argument to obtain:

Theorem

If V |= ZFC, then there is a class extension V [G] in which

V = HODV [G] = gHODV [G] but MV [G] = V [G]



Mantles high, HODs high

Lastly,

Theorem

If V |= ZFC, then there is V [G] in which

V [G] = HODV [G] = gHODV [G] = MV [G] = gMV [G]

This is possible by forcing the Continuum Coding Axiom CCA.



The Inner Mantles

When the mantle M is a model of ZFC, we may consider themantle of the mantle, iterating to reveal the inner mantles:

M1 = M Mα+1 = MMαMλ =

⋂α<λ

Mα

Continue as long as the model satisfies ZFC.

The Outer Core is reached if Mα has no grounds,Mα |= ZFC + GA.

Conjecture. Every model of ZFC is the αth inner mantle ofanother model, for arbitrary α ≤ ORD.

Philosophical view: ancient paradise?


Laura

Sticky Note

Harrington theorem:

Laura

Sticky Note

Extnt to which it can be revived?


Large cardinal indestructibility across the multiverse

Large cardinals across the multiverse

We now turn to the question of set-theoretic features that mayhold throughout the generic multiverse.

Large cardinal indestructibility, the question of whether a largecardinal property is preserved to a forcing extension, has beena focus of study for decades.

Let us enlarge the problem to the question of whether a largecardinal exhibits its large cardinal property throughout asubstantial portion of the generic multiverse.




Supercompactness

Theorem

If κ is supercompact, then there is a forcing extension V [G] inwhich κ remains supercompact, becomes indestructible by<κ-directed closed forcing, and the Ground Axiom holds.

Thus, in this model, the supercompactness of κ is indestructibleboth upward by <κ-directed closed forcing and (vacuously)downward to ground models.

Question

Can we arrange that κ is supercompact throughout the<κ-directed closed multiverse?




No for supercompact cardinals (and measurables)

Theorem

No supercompact cardinal κ is indestructible throughout the<κ-directed closed multiverse. For every cardinal κ, there is a<κ-directed closed generic ground in which κ is notmeasurable.

Proof.

The forcing P to add a slim κ-Kurepa tree always destroys themeasurability of κ. But P can be absorbed into the collapseforcing P ∗ Coll(κ,2κ) ∼= Coll(κ,2κ). So the extension by P is a<κ-directed closed generic ground of V in which κ is notmeasurable.




But Yes for weakly compact cardinals

Theorem

If κ is supercompact, then there is a class extension V [G] suchthat κ is weakly compact throughout the <κ-closed multiverseof V [G].

The proof uses:

Lemma

If κ is weakly compact and this is indestructible by <κ-closedforcing, then κ retains this property throughout the <κ-closedgeneric multiverse.

There are a large number of similar such questions in this area.



Defining the ground

Approximation and cover properties

The definition of V inside V [G] makes use of the following keyideas:

Definition (Hamkins)

1 W ⊆ V has δ cover property if every A ⊆W with A ∈ V ,|A|V < δ is covered A ⊆ B by some B ∈W with |B|W < δ.

2 W ⊆ V has δ approximation property if every A ⊆W withA ∈ V and all small approximations A ∩ B in W , whenever|B|W < δ, is already in the ground model A ∈W .



Defining the ground

Uniqueness of groundsLemma (Hamkins)

If V ⊆ V [G] and G ⊆ P ∗ Q with P nontrivial and Q is≤ |P|-strategically closed, then V [G] has the δ cover andapproximation properties for δ = |P|+.

Lemma (Laver,Hamkins)

If W ,W ′ ⊆ V have δ cover and approximation properties,P(δ)W = P(δ)W ′

, (δ+)W = (δ+)W ′= (δ+)V , then W = W ′.

Laver had first proved the lemma for small forcing, and Iextended it to cover and approximation property.

This lemma provides the definition of W inside the forcingextension W [G], using parameter P(δ)W .



Defining the ground

GA + V 6= HOD


Every model of set theory has an extension which is a model ofGA plus V 6= HOD.

First extend to V ⊆ V |= CCA; then add V [G] a Cohen subsetto every regular cardinal. By homogeneity, this achievesV 6= HOD. Details of approximation and cover establish GA.



Defining the ground

Questions

Set-theoretic geology is a young area, and there are a largenumber of open questions.

To what extent does the mantle satisfy ZF or ZFC?Are the grounds or generic grounds downward directed?downward set-directed? locally?Is the bedrock unique when it exists?Is gM = M? Is gM = gHOD?Does the generic mantle satisfy ZFC?Does inclusion = ‘ground model of’ in the genericmultiverse?



Case study: Multiverse view of the Continuum Hypothesis CH

Time permitting, let’s briefly discuss some issues in thephilosophy of set theory regarding the question of thecontinuum hypothesis, and how it is treated by the universeperspective in comparison with the multiverse view.




Case study: the Continuum Hypothesis

The continuum hypothesis (CH) is the assertion that every setof reals is either countable or equinumerous with R.

This was a major open question from the time of Cantor, andappeared at the top of Hilbert’s famous list of open problems atthe dawn of the 20th century.

The continuum hypothesis is now known to be neither provablenor refutable from the usual ZFC axioms of set theory.

Gödel proved that CH holds in the constructible universe L.

Cohen proved that L has a forcing extension L[G] with ¬CH.




CH in the MultiverseMore important than mere independence, both CH and ¬CHare forceable over any model of set theory. Every V has:

V [~c], collapsing no cardinals, such that V [~c] |= ¬CH.V [G], adding no new reals, such that V [G] |= CH.

That is, both CH and ¬CH are easily forceable. We can turn CHon and off like a lightswitch.

We have a deep understanding of how CH can hold and fail,densely in the multiverse, and we have a rich experience in theresulting models. We know, in a highly detailed manner,whether one can obtain CH or ¬CH over any model of settheory, while preserving any number of other features of themodel.




The CH is settled

The multiverse perspective is that the CH question is settled.

The answer consists of our detailed understanding of how theCH both holds and fails throughout the multiverse, of how thesemodels are connected and how one may reach them from eachother while preserving or omitting certain features.

Fascinating open questions about CH remain, of course, but themost important essential facts are known.

In particular, I shall argue that the CH can no longer be settledin the manner that set theorists formerly hoped it might be.




Traditional Dream solution for settling CH

Set theorists traditionally hoped to settle CH this way:

Step 1. Produce a set-theoretic assertion Φ expressing anatural ‘obviously true’ set-theoretic principle. (e.g. AC)

Step 2. Prove that Φ determines CH.That is, prove that Φ =⇒ CH,or prove that Φ =⇒ ¬CH.

And so, CH would be settled, since everyone would accept Φand its consequences.




Dream solution will never be realized

I argue that this template is now unworkable.

The reason is that because of our rich experience andfamiliarity with models having CH and ¬CH, the mere fact of Φdeciding CH immediately casts doubt on its naturality. So wecannot accept such a Φ as obviously true.

In other words, success in the second step exactly underminesthe first step.

Let me present two examples illustrating how this plays out.




Freiling: “a simple philosophical ‘proof’ of ¬CH”

The Axiom of Symmetry (Freiling JSL, 1986)

Asserts that for any function f mapping reals to countable setsof reals, there are x , y with y /∈ f (x) and x /∈ f (y).

Freiling justifies the axiom on pre-theoretic grounds, withthought experiments throwing darts. The first lands at x , soalmost all y have y /∈ f (x). By symmetry, x /∈ f (y).

“Actually [the axiom], being weaker than our intuition,does not say that the two darts have to do anything.All it claims is that what heuristically will happen everytime, can happen.”

Thus, Freiling carries out step 1 in the template.




Freiling carries out step 2

Theorem (Freiling)

The axiom of symmetry is equivalent to ¬CH.

Proof.

If CH, let f (r) be the set of predecessors of r under a fixedwell-ordering of type ω1. So x ∈ f (y) or y ∈ f (x) by linearity.If ¬CH, then for any ω1 many xα, there must be y /∈

⋃α f (xα),

but f (y) contains at most countably many xα.

Thus, Freiling exactly carries out the template.

Was his proposal received as a solution of CH? No.




Objections to Symmetry

Many mathematicians, ignoring Freiling’s pre-reflective appeal,objected from a perspective of deep experience withnon-measurable sets and functions, including extremeviolations of Fubini. For them, the pre-reflective argumentssimply fell flat.

We have become skeptical of naive uses of measure preciselybecause we know the pitfalls; we know how badly behaved setsand functions can be with respect to measure concepts.

Because of our detailed experience, we are not convinced thatAS is intuitively true. Thus, the reception follows my prediction.

And similarly for other dream solutions of CH.




Another example using the dream template

Consider the following set-theoretic principle:

The powerset size axiom PSA

PSA asserts that whenever a set is strictly larger than anotherin cardinality, then it also has strictly more subsets:

∀x , y |x | < |y | ⇒ |P(x)| < |P(y)|.

Set-theorists understand this axiom very well.




Powerset size axiom: |x | < |y | ⇒ |P(x)| < |P(y)|How is this axiom received in non-logic mathematical circles?

Extremely well!

To many mathematicians, this principle is Obvious, as naturaland appealing as AC. Many are surprised to learn it is not atheorem. (Ask your colleagues!)

Meanwhile, set theorists do not agree. Why not? In part,because we know how to achieve all kinds of crazy patternsκ 7→ 2κ via Easton’s theorem. Cohen’s ¬CH model violates it;Martin’s axiom violates it; Luzin’s hypothesis violates it. PSAfails under many of the axioms, such as PFA, MM that are oftenfavored particularly by set-theorists with the universe view.




Powerset size axiomSo we have a set-theoretic principle

which many mathematicians find to be obviously true;which expresses an intuitively clear pre-reflective principleabout the concept of size;which set-theorists know is safe and (relatively) consistent;

is almost universally rejected by set-theorists when proposedas a fundamental axiom.

We are too familiar with the ways that PSA can fail, and havetoo much experience working in models where it fails.

But imagine an alternative history, in which PSA is used tosettle a prominent early question and is subsequently adoptedas a fundamental axiom.




Similarly, with CH

I claim that the dream template will not settle CH, because assoon as we know that a proposed principle Φ implies CH orimplies ¬CH, we cannot accept Φ as obvious.

Conclusion. The dream template is unworkable.

We simply have too much experience in and familiarity with theCH and ¬CH worlds. We therefore understand deeply how Φcan fail in worlds that seem perfectly acceptableset-theoretically.




Other attempts to settle CHMore sophisticated attempts to settle CH do not rely on thistraditional template.

Woodin has advanced arguments to settle CH based onΩ-logic, and based on Ultimate-L.

To the extent that an argument aims to settle CH, what theMultiversist desires is an explicit explanation of how ourexperience in the CH or in the ¬CH worlds was somehowillusory, as it seems it must be for the argument to succeed.

Since we have an informed, deep understanding of how it couldbe that CH holds or fails, even in worlds close to any givenworld, it is difficult to regard these worlds as imaginary.




Thank youJoel David HamkinsNew York University, PhilosophyThe City University of New York, Mathematics

http://jdh.hamkins.org

Grateful acknowledgement to

National Science Foundation (USA), for support 2008-2011.

CUNY Research Foundation

Simons Foundation

A preprint of G. Fuchs, J. D. Hamkins and J. Reitz, "Set-theoreticgeology," is available at http://arxiv.org/abs/1107.4776.


a tutorial in set-theoretic geology · a tutorial in set-theoretic geology joel david hamkins new...

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