pluralism-inspired mathematics, including a recent

Set-theoretic geology Generic multiverse Isomorphism Absolute truth Inside models Universal algorithm

Pluralism-inspired mathematics, including arecent breakthrough in set-theoretic geology

Joel David Hamkins

City University of New York

CUNY Graduate CenterMathematics, Philosophy, Computer Science

College of Staten IslandMathematics

MathOverflow

Set-theoretic Pluralism 2016Symposium I, University of Aberdeen

Pluralism-inspired Mathematics, STP 2016 Joel David Hamkins, New York


Downward-directed grounds

Set-theoretic geology

In set theory, forcing is naturally viewed as a method of buildingouter as opposed to inner models of set theory.

Set-theoretic geology inverts this perspective by studying howthe set-theoretic universe V was obtained by forcing, bystudying the fundamental structure of the grounds of V .




A fundamental question

Question (Reitz 2006)

Suppose that the set-theoretic universe V was obtained byforcing over two different grounds V = W0[G0] = W1[G1]. Mustthere be a common deeper ground?

V

W0 W1

W?

In other words, are the grounds downward directed?




Downward directedness

Downward-directedness expresses a fundamental property ofthe structure of grounds.

The question could have been asked 50 years ago.

My view is that it is difficult for us to claim a deep undertandingof forcing, if we lack the knowledge whether or not theforcing-theoretic ground models exhibit this basic, elementaryfeature.




Some previous knowledgeWe knew many instances where it was true.

True in every model where we were able to determineanswer.True in the bottomless models of Reitz—no bedrock.(Fuchs,JDH,Reitz) True in any model of the form L[A],where A is a set.

Meanwhile, there were attempts to find a counterexample.

Woodin proposed a candidate counterexample model,based on inner model considerations.S. Friedman proposed a candidate counterexample, builtby forcing.

It turns out, however, that there is no counterexample.




Usuba’s breakthrough

This question is now answered by Toshimichi Usuba.

Theorem (Usuba)

The grounds of V are downward-directed. Any two groundshave a common deeper ground.

Indeed, any set-indexed family of grounds have a commondeeper ground.

I should like to explain the proof.

To my way of thinking, this is the most important advance in settheory in recent years.




Ground model enumeration theorem

Theorem (Fuchs,JDH,Reitz)

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 ground axiom (JDH,Reitz) asserts ∀r V = Wr .

The mantle is M =⋂

r Wr .




Downward directedness hypotheses

Definition

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 set I there is t with Wt ⊆⋂

r∈I Wr .

By the ground-model enumeration theorem, these areexpressible in the first-order language of set theory.




Downward-directed Grounds hypothesis

Theorem (Usuba)

The strong DDG is true.

In other words, for any set I, there is a ground W that iscontained in every ground Wr with r ∈ I.

Let me present a detailed proof.




Review of some tree combinatorics

Konig’s Lemma

Every infinite finitely-branching tree has a cofinal branch.

In other words, every tree of height ω with all levels finite has abranch.

How does this generalize to higher cardinals?

Aronszajn trees, of course, block one approach.

But consider a tree of height ω2 with all levels countable.

More generally, consider a tall narrow tree...




Every tall narrow tree has a branch, but not too manyLemma (Kurepa)

If T is a tree, height λ, all levels size < δ, with δ regular,δ < cof(λ), then T has a cofinal branch, but fewer than δ many.

Proof.

For levels β with cofinality δ, find νβ < β such that level β nodesseparate already by level νβ. By Fodor, there is stationaryS ⊆ λ with constant value ν. So all nodes on level β ∈ Sseparate by level ν. By pigeon-hole, one node on level ν hassuccessors on cofinally many levels, and these must cohere toform a branch.

Meanwhile, any two cofinal branches have separated by level ν,so there are fewer than δ many branches.




Tall narrow trees gain no new branches

Lemma

If W ⊆ V is an inner model of ZFC and T is a tall narrow tree inW, then all cofinal branches of T are already in W.

Proof.

We know T has fewer than δ many branches in W . If b is a newbranch through T in V , then there must be a level by which b isdistinguished from those branches. So there is a node p on b,but not on any branch in W . But Tp is a tall narrow tree in W ,and so has a branch in W , contradiction.




Approximation and cover properties

Suppose that W is a transitive class and that δ is a cardinal.

Definition

The extension W ⊆ V satisfies the δ-approximationproperty, if whenever A ⊆W , A ∈ V and A ∩ a ∈W for alla ∈W of size less than δ, then A ∈W .The extension W ⊆ V satisfies the δ-cover property, if forevery A ⊆W with A ∈ V and |A| < δ, there is B ∈W withA ⊆ B and |B| < δ.

These concepts are central in set-theoretic geology, used in myproof of the ground-model definability theorem.




Uniform cover propertyConsider a uniform version of covering: every sequence ofsmall sets is uniformly covered.

The uniform δ-cover property for W ⊆ V

If Ai ⊆W with |Ai | < δ every i ∈ I, with I ∈W , then there is acovering sequence in W :

〈Bi | i ∈ I〉 ∈W .Ai ⊆ Bi .|Bi | < δ.

Using the axiom of choice, it suffices to consider 〈Aα | α < λ〉for λ ordinal and Aα ⊆ λ.




Uniform covering, alternative formulations

If W ⊆ V and δ regular, δ ≤ λ, then the following are equivalent:

1 For every 〈Aα | α < λ〉 with Aα ⊆ λ size < δ, there is〈Bα | α < λ〉 ∈W with Aα ⊆ Bα and |Bα| < δ.

2 For every A ⊆ λ× λ, with all vertical sections size < δ,there is B ∈W with A ⊆ B and all sections size < δ.

3 For every function f : λ→ λ there is B ∈W with B ⊆ λ× λ,all vertical sections size < δ and f ⊆ B.

Let’s call this: λ-uniform δ-covering.




Uniform covering implies approximationLemma

If W ⊆ V has λ-uniform δ-cover property, with δ regular and λ stronglimit, then it has δ+-approximation property for subsets of λ.

Proof.

Assume s ∈ 2λ has all δ+-small approximations in W ; aim to shows ∈W . Assume inductively that s � α ∈W all α < η. By uniformcovering, can find a tree T ∈W height η, levels size < δ, such thats � α ∈ T for α < η. If δ < cof(η), this is tall narrow tree, and sos ∈W .

Otherwise, cof(η) ≤ δ. By a simple closure argument, find cofinalJ ⊆ η size δ, such that distinct nodes p,q on level β ∈ J havep(α) 6= q(α) some α ∈ J. By approximation assumption, s � J ∈W ,and this determines s.




Bukovsky ground model characterization

Theorem (Bukovsky, 1973)

Suppose that W ⊆ V is an inner model of ZFC. Then thefollowing are equivalent:

1 W is a ground of V .2 For some cardinal δ, the extension W ⊆ V exhibits the

uniform δ-cover property.

The direction 1→ 2 is immediate, using the δ-chain condition.

The direction 2→ 1 is reminiscent of Vopenka’s proof thatevery set is HOD-generic, uses an infinitary logic.




Downward-directed Grounds hypothesis

Let us now put it together to prove the main result.

Theorem (Usuba)

The downward-directed grounds hypothesis is true.

Fix any set I and consider the grounds Wr for r ∈ I. We seek tofind a ground W with W ⊆Wr for all r ∈ I.

For each ground Wr , we may realize V as a forcing extensionV = Wr [Gr ], where Gr ⊆ Qr ∈Wr is Wr -generic.

Let κ be regular, larger than every |Qr |Wr and larger than |I|.




Consider large θ = iθ > κ.

Lemma

There is A ⊆ θ with L[A] ⊆Wr all r ∈ I, such that L[A] ⊆ V has<θ-uniform κ+-covering property.

Proof.

Let h : θ → θ be universal, in that every t : λ→ λ for λ < θ occurs asa block in h. Since Wr ⊆ V has uniform κ-covering, there isH0,r ⊆ θ × θ with all vertical sections size < κ and h ⊆ H0,r ∈Wr .Continuing, for η < κ find Hη,r ∈Wr with all sections size < κ andHν,s ⊆ Hη,r for ν < η, s ∈ I. Let H be the union of all Hη,r , soH ⊆ θ × θ, with all sections size ≤ κ. Note that H ∈Wr all r ∈ I byapproximation property, since H ∩ a = Hη,r ∩ a for κ-small a ∈W . LetA ⊆ θ code H ⊆ θ × θ. So L[A] ⊆Wr all r ∈ I. Also, L[A] ⊆ V has<θ-uniform κ+-covering, since h ⊆ H.




So for each θ = iθ above κ, we have L[Aθ] ⊆⋂

r Wr and L[Aθ] ⊆ Vhas <θ-uniform κ+-covering, hence κ++-approximation for boundedsubsets of θ.

It follows by the ground-model definability theorem that V L[A]θ is

definable in Vθ from parameter p = (2<κ++

)L[Aθ ].

Some such p must be used for unboundedly many θ, and the V L[Aθ ]θ

cohere by the ground-model definability theorem. Let W =⋃θ V L[Aθ ]

θ

be the union of these.

Note that W is closed under Godel operations, is almost universaland has well-orders. So W |= ZFC.

Also, W ⊆Wr for all r ∈ I, since Wθ ⊆ L[Aθ] ⊆Wr .

Finally, W ⊆ V has uniform κ++-cover property, since V L[Aθ ]θ ⊆ Vθ

had <θ-uniform κ++-covering. So by Bukovsky, W is a ground of V ,contained in every Wr for r ∈ I, as desired. QED




Conclusion: in any model of ZFC, any set-indexed family ofgrounds have a common deeper ground.

W ⊆⋂r

Wr ⊆ V

Jonas mentioned several consequences of the DDG in histutorial, so let us briefly mention some of them.



Consequences of DDG

Consequences

Usuba has proved that the strong DDG holds. This settlesmany prominent open questions of set-theoretic geology.

Corollaries

1 Bedrock models are unique when they exist.2 The mantle is absolute by forcing.3 The mantle is a model of ZFC.4 The mantle is the same as the generic mantle.5 The mantle is the largest forcing-invariant class, and equal

to the intersection of the generic multiverse.



Consequences of DDG

Inclusion in the generic multiverseThe generic multiverse of M consists of all models obtainableby successively passing to a forcing extension or a ground.

Question

If M,N are in the same generic multiverse and M ⊆ N, must Mbe a ground of N?

Theorem

Yes. The inclusion relation agrees with the ground-of relation inthe generic multiverse.

The point is that the DDG implies that the generic multiverse ofM consists of all the forcing extensions of grounds of M, sincethis collection is closed under forcing extensions and grounds.



Consequences of DDG

Modal logic of forcingI had introduced the forcing modalities.

�ϕ if ϕ holds in all forcing extensions.♦ϕ if ϕ holds in some forcing extension.

This modal language can express diverse forcing principles.

Maximality principle (Stavi+Vaananen, indep. JDH) asserts

♦�ϕ→ ϕ

Question (JDH)

What exactly are the ZFC-provable forcing validities?

Theorem (JDH,Lowe)

If ZFC is consistent, then the ZFC-provably valid principles offorcing are precisely those in the modal theory S4.2.



Consequences of DDG

Up and down forcing modalitiesIn set-theoretic geology, we naturally have two pairs of forcingmodalities:

−�ϕ if ϕ holds in all forcing extensions.∧♦ϕ if ϕ holds in some forcing extension.

−�ϕ if ϕ holds in all grounds.∨♦ϕ if ϕ holds in some grounds.

Question (JDH,Lowe)

What exactly are the mixed-modality principles of forcing?

For example, these temporal-logic-like principles are valid:

ϕ→ −� ∨♦ϕ ϕ→ −� ∧♦ϕ



Consequences of DDG

Modal logic of grounds

The DDG implies (and is nearly equivalent to) the validity ofaxiom .2 for the downward-logic.

∨♦ −�ϕ→ −� ∨♦ϕ

Corollary

If ZFC is consistent, then the ZFC-provably valid downwardprinciples of forcing are exactly S4.2.

The point is that Benedikt Lowe and I had already proven S4.2as an upper bound, but our previously best lower bound wasS4. With the DDG, we get S4.2, so this theory hits it exactly.



Consequences of DDG

Large cardinal→ few grounds

Theorem (Usuba)

If there is a hyper-huge cardinal, then the universe has abedrock.

In other words, the mantle is a ground.

This is an amazing connection between large cardinalexistence and the structure of grounds!

A cardinal κ is hyper-huge, if for every ordinal λ there isj : V → M with critical point κ, j(κ) > λ and M j(λ) ⊆ M.

supercompact < super-huge < hyper-huge < super almost2-huge



Consequences of DDG

Lemma

If κ is hyper-huge and W is a ground of V , then V is a forcingextension of W by forcing of size < κ.

Proof.

V = W [G] for some G ⊆ P ∈W . Pick inaccessible λ > |P| and letj : V → M witness hyper-hugeness. Fix stationary partition〈Sα | α < j(λ)〉 of Cofω ∩λ in W , and argue thatβ ∈ j " j(λ) ⇐⇒ j(~S)β is stationary in sup j " j(λ) in j(W ).Conclusion: j " j(λ) ∈ j(W ). It follows that

Wj(λ) ⊆ j(W )j(λ) ⊆ Mj(λ) ⊆ Vj(λ) = W [G]j(λ).

So Mj(λ) is extension of Wj(λ) by forcing size < j(κ). By elementarity,Vλ is a forcing extension of Wλ by forcing of size < κ.

Thus, V has a bedrock, by the strong DDG.



The generic multiverse

Let us turn now to other topics.

The generic multiverse of a model of set theory M is thecollection of models arising successively as forcing extensionsor grounds of models already in the class.

We’d like to understand basic structural features of the genericmultiverse.



Amalgamation

Question

If M is a model of set theory with forcing extensions M[G] andM[H], is there a common further extension?

M

M[G] M[H]

M[K ]?

In other words, are forcing extensions upward directed?



Non-amalgamation

Theorem (Woodin)

Every countable transitive model of set theory W hasnon-amalgamable forcing extensions W [c] and W [d ], meaningthat there is no common extension W |= ZFC with the sameordinals.

Build M-generic Cohen reals c and d in stages, so as to meetall the dense sets of M, while also coding some “bad” real z.Individually, M[c] and M[d ] are fine, but any model with both cand d will be able to decode z.



Extending to large finite families

Similarly, we can have three extensions M[c], M[d ] and M[e],such that any two of them can be amalgamated, as in M[c,d ],but there is no common extension of all three.

And similarly for any finite number.



Upper bound in the generic multiverse?

Question

If we successively add Cohen reals

M[c0] ⊆ M[c1] ⊆ M[c2] ⊆ · · ·

Is there an upper bound, an extension M[d ] with

M[cn] ⊆ M[d ]

for all n?



Countable closure

Theorem (JDH,Venturi)

If M is a countable model of ZFCand

M[c0] ⊆ M[c0][c1] ⊆ M[c0][c1][c2] · · ·

are generic extensions by adding Cohen reals, then there is anM-generic Cohen d with M[d ] an upper bound of the chain.

Proof.

The idea is to construct an M-generic d ⊆ N× N, whose nth

slice dn agrees with cn except on a finite set.



Upward closure in the generic multiverse

Theorem (Fuchs,JDH,Reitz)

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.



Isomorphic to a forcing extension?

Question

Can a model of set theory be isomorphic to one of its nontrivialforcing extensions?

M ∼= M[G]?



A model of set theory V can easily be elementarily equivalentto a forcing extension V [G], that is, have the same theory

V ≡ V [G].

For example, if you add a Cohen real c, and then add anotherd , then

V [c] ≡ V [c][d ],

because we may merge the two Cohen reals into one, and soV [c] and V [c ∗ d ] are two extensions by adding a Cohen real,which is homogeneous and so all extensions have the sametheory.

But of course, V [c] will not be isomorphic to V [c ∗ d ] in V [c ∗ d ].



Isomorphic to forcing extension?

No transitive ∈-standard model is isomorphic to a nontrivialforcing extension.

M 6∼= M[G]

This is because distinct transitive sets are never isomorphic: ifπ : M ∼= N is an ∈-isomorphism, then π(x) = x by ∈-induction,and so M = N.



Nevertheless

Theorem

There is a model M of set theory (if consistent) isomorphic toall its forcing extensions by a Cohen real M ∼= M[c].

Lemma (Smorynski)

If M,N are countable, computably saturated, same theory andsame standard system, then M ∼= N.

Lemma (JDH)

If M is countable computably saturated, then so also are allforcing extensions M[G].



Theorem

There is a model M of set theory (if consistent) that isisomorphic to all its forcing extensions M[c] by a Cohen real.

Proof.

Let N be any countable computably saturated model. Let M = N[d ]be a forcing extension by a Cohen real. Consider any M[c].

M[c] = N[d ][c] = N[d ∗ c].

Consequently, M ≡ M[c], since the forcing is homogeneous.

M and M[c] have the same arithmetic, hence same standardsystem.

M and M[c] are computably saturated.

Hence, M ∼= M[c].



My favorite situation

A philosophical concern...

leads to interesting mathematical questions...

whose answers illuminate the philosophical issue.



The philosophical question

Question

To what extent does definiteness of mathematical objects leadto definiteness of our theory of mathematical truth about thoseobjects?

Many mathematicians express a commitment to thedefiniteness of the natural numbers 0,1,2, . . . and the structure〈N,+, ·,0,1, <〉.

Does this also commit us to the definiteness of arithmetic truth?



Definiteness of truth

For example, Feferman and others have defended a viewwhereby arithmetic truth has a definite character, whilehigher-order truth, such as set-theoretic assertions at the levelof P(N) and above, are less definite.

For example, on such a view you might view the continuumhypothesis as a vague mathematical assertion, not capable ofgenuine resolution.



From structure to truth

Solomon Feferman (EFI 2013):In my view, the conception [of the bare structure of thenatural numbers] is completely clear, and thence allarithmetical statements are definite.

It is Feferman’s ‘thence’ to which I call attention.

Donald Martin (EFI 2012):What I am suggesting is that the real reason for confidencein first-order completeness is our confidence in the fulldeterminateness of the concept of the natural numbers.



Structure to truth

So we are interested in the philosophical question concerningthe extent one gets definiteness of the theory of truth for astructure merely from the definiteness of the objects andrelations of the structure itself.

Or is the definiteness of the theory of truth for a structure a kindof higher-order ontological commitment requiring its ownjustification?



The mathematical question

Question (Yang)

Can a mathematical structure exist inside two different modelsof set theory, which disagree on the theory of that structure?

The answer is yes, and indeed, this is pervasive.



Satisfaction is not absoluteTheorem (JDH, Yang)

If ZFC is consistent, then there are M1,M2 |= ZFC which have thesame natural numbers and arithmetic structure

〈N,+, ·,0,1, <〉M1 = 〈N,+, ·,0,1, <〉M2 ,

but which disagree on arithmetic truth.

•N

M1 M2

There is a sentence σ in M1 and M2

M1 believes N |= σ

M2 believes N |= ¬σ



ProofFix any countable M1 |= ZFC with 〈N,TA〉M1 computablysaturated.

Claim there are sentences σ, τ with same 1-type inNM1 = 〈N,+, ·,0,1, <〉M1 , but M1 thinks σ is true and τ false inNM1 . (Proof: consider the type p(s, t) containing ϕ(s) ⇐⇒ ϕ(t)and s ∈ TA and t /∈ TA; this is finitely realized, since TA is notdefinable.)

By back-and-forth, there is automorphism π : NM1 → NM1 withπ(τ) = σ.

Build a copy M2 of M1 so that π extends to an isomorphismπ∗ : M1 → M2. So NM1 = NM2 . But M1 thinks σ is true, yet M2thinks σ = π∗(τ) is false.



A generalization

Theorem

For any countable M |= ZFC, any structure N ∈ M finitelanguage, any S ⊆ N in M not definable in N . Then there areM ≺ M1 and M ≺ M2 with NM1 = NM2 , yet SM1 6= SM2 .

Note SM1 and SM2 share all properties of S in M.

Proof.

Fix M ≺ M1 countable computably saturated. So 〈N ,S〉M1 iscomputably saturated. Since S not definable, there area,b ∈ NM1 with same 1-type in NM1 , but a ∈ S,b /∈ S. So∃π : NM1 ∼= NM1 with π(b) = a. Extend π to π∗ : M1 ∼= M2. Soa ∈ SM1 but a = π(b) /∈ SM2 .



Corollary

Every countable model of set theory M has elementaryextensions M ≺ M1 and M ≺ M2, which agree on their naturalnumbers NM1 = NM2 , their reals RM1 = RM2 and theirhereditarily countable sets 〈HC,∈〉M1 = 〈HC,∈〉M2 , but whichdisagree on their theories of projective truth.

•HC

ω

M1 M2

M1,M2 |= ZFC

NM1 = NM2 RM1 = RM2

〈HC,∈〉M1 = 〈HC,∈〉M2

M1 believes HC |= σ

M2 believes HC |= ¬σ



Corollary

Every countable model of set theory M has elementaryextensions M ≺ M1 and M ≺ M2, which have a transitiverank-initial segment 〈Vδ,∈〉M1 = 〈Vδ,∈〉M2 in common, butwhich disagree on truth in this structure.

•Vδ

M1 M2

M1,M2 |= ZFC

VM1δ

= VM2δ|= ZFC

M1 believes Vδ |= σ

M2 believes Vδ |= ¬σ



Iterated truth predicates

Begin with the standard model 〈N,+, ·,0,1, <〉.

Add a truth predicate 〈N,+, ·,0,1, <,Tr0〉, where Tr0 is a truthpredicate for arithmetic assertions.

Add a truth predicate for that structure, 〈N,+, ·,0,1, <,Tr0,Tr1〉,where Tr1 is a truth predicate for assertions in the languagewith Tr0.

And so on 〈N,+, ·,0,1, <,Tr0, . . . ,Trn〉...



Truth about truth is not absolute

Corollary

For every countable model of set theory M and any naturalnumber n, there are M ≺ M1 and M ≺ M2 with NM1 = NM2 andsame iterated truths up to n

〈N,+, ·,0,1, <,Tr0, . . . ,Trn〉M1 = 〈N,+, ·,0,1, <,Tr0, . . . ,Trn〉M2

but which disagree on the next order of truth Trn+1.

The point: Trn+1 is not definable in 〈N,+, ·,0,1,Tr0, . . . ,Trn〉.



Disagreement on the Church-Kleene ordinal

Corollary

Every countable model of set theory M has elementaryextensions M ≺ M1 and M ≺ M2, which agree on their standardmodel of arithmetic NM1 = NM2 and have a computable linearorder C on N in common, yet M1 thinks 〈N,C〉 is a well-orderand M2 does not.

Proof.

Being the computable index of a well-order is Π11-complete and

hence not definable in 〈N,+, ·,0,1, <〉.



Disagreement on definabilityTheorem

Every countable model of set theory M has M ≺ M1 andM ≺ M2, which agree on

〈N,+, ·,0,1, <〉M1 = 〈N,+, ·,0,1, <〉M2

and which have a set A ⊆ N in common, yet M1 thinks A isfirst-order definable in N and M2 thinks it is not.

The proof relies on the non-absoluteness theorem, plus:

Lemma (Andrew Marks)

There is B ⊆ N× N, such that {n ∈ N | Bn is arithmetic } is notdefinable in the structure 〈N,+, ·,0,1, <,B〉.



Precise violation of ZFC

Theorem

Every countable model of set theory M |= ZFC has elementaryextensions M1 and M2, with a transitive rank-initial segment〈Vδ,∈〉M1 = 〈Vδ,∈〉M2 in common, such that M1 thinks that theleast natural number n for which Vδ violates Σn-collection iseven, but M2 thinks it is odd.

•Vδ

M1 M2M1,M2 |= ZFC

VM1δ

= VM2δ

n is least with ¬Σn-collection in Vδ

M1 believes n is even

M2 believes n is odd



ProofSuppose that M |= ZFC, and let

T1 = ∆(M) + Vδ ≺ V + {m ∈ Vδ | m ∈ M }+ the least n such that Vδ 2 Σn-collection is even,

Consistent via the reflection theorem. Similar with theory T2, wherewe assert n is odd.

Let 〈M1,M2〉 be a computably saturated model pair, with M1 |= T1 andM2 |= T2. It follows that 〈V M1

δ ,V M2δ 〉 is a computably saturated model

pair of elementary extensions of 〈M,∈M〉, which are thereforeelementarily equivalent in the language of set theory with constantsfor elements of M, and hence isomorphic by an isomorphismrespecting those constants. So without loss〈M,∈M〉 ≺ 〈Vδ,∈〉M1 = 〈Vδ,∈〉M2 . Meanwhile, M1 thinks that this Vδviolates Σn-collection first at an even n and M2 thinks it does so firstfor an odd n, as desired.



Disagreement about whether Vδ |= ZFCTheorem

If M is a countable model of set theory in which the worldly cardinalsform a stationary proper class, then there are M ≺ M1 and M ≺ M2with 〈Vδ,∈〉M1 = 〈Vδ,∈〉M2 , but M1 thinks Vδ |= ZFC and M2 thinksVδ 6|= ZFC.

•Vδ

M1 M2

M1,M2 |= ZFC

〈Vδ,∈〉M1 = 〈Vδ,∈〉M2

M1 believes Vδ |= ZFC

M2 believes Vδ 6|= ZFC



Models inside models

Question

If M is a model of set theory ZFC, when do we expect to find astructure inside M that is, externally, a model of ZFC?

If 〈M,∈M〉 thinks m is a set with binary relation E , we canextract this to an actual structure 〈m, E〉

Domain of m consists of a for which M |= a ∈ m.a E b iff M |= a E b.



From Brice Halimi:

Theorem

Every model of ZFC has an element that is a model of ZFC.Specifically, if 〈M,∈M〉 |= ZFC, then there is 〈m,E〉 in M, whichwhen extracted as an actual structure, satisfies ZFC.

Obviously wrong? After all, perhaps M satisfies ¬Con(ZFC).

No, that objection is wrong, since it conflates the object theoryZFC with the actual ZFC. So let’s give a proof.



Every model has a model inside

Theorem

Every model of ZFC has an element that is, externally, a modelof ZFC. Specifically, if 〈M,∈M〉 |= ZFC, then there is 〈m,E〉 inM, which when extracted as an actual structure, satisfies ZFC.

Proof.

If M is ω-nonstandard, then by the reflection theorem plusoverspill, there is some 〈V M

δ ,∈〉M satisfying a nonstandardfragment of ZFC, and hence satisfies the actual ZFC externally.

If M is ω-standard, then it must satisfy Con(ZFC) and so onehas the Henkin model inside M.



Universal algorithm

There is a universal algorithm, capable of computing anyfunction, in the right universe.

Theorem

There is a Turing machine program p, such that for any functionf : N→ N, there is a model M |= ZFC (assuming consistency),such that program p inside M computes f on standard input.

Inside M, on standard input n, the program p with output f (n).

Related to old theorem of Mostowsk and Kripke: independentΠ0

1 assertions.



Rosser treeTheorem

There is a universal program p, which can compute anyf : N→ N on standard input, in the right universe.

Proof 1. Use the Rosser tree: place desired theory at root, andassign to successor nodes either T + ρ or T + ¬ρ, usingRosser sentence over previous theory T .

Every branch through the theory is consistent.

If initial theory proves ¬Con(T ), then the difference between ρand ¬ρ is which proof comes first. Computable!

So the universal algorithm is: search for proofs of the nextRosser assertion. You can thereby compute the path the theoryfollowed.



Theorem

There is a universal program p, which can compute anyf : N→ N on standard input, in the right universe.

Proof 2 (Vadim Kosoy)

There is a program p that searches for a proof that program pdisagrees with a certain finite list of function input/outputvalues. For the first such proof that is found, output thecorrespond output.

The theory cannot prove that the function disagrees with anyparticular value, since then it wouldn’t. So for any function f , itis consistent with the theory that p computes exactly f (n) oninput n.



Thanks to Toshimichi Usuba.

Various other parts of this work were joint variously with GiorgioVenturi, Gunter Fuchs, Jonas Reitz, Ruizhi Yang.

Thank you.

Slides and articles available on http://jdh.hamkins.org.

Joel David HamkinsCity University of New York


pluralism-inspired mathematics, including a recent

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