University of Oxford University College

Categoricity Categorical cardinals Gaps Number of categorical cardinals Implications Philosophical discussion

Categorical set theories

Joel David HamkinsProfessor of Logic

Sir Peter Strawson Fellow

University of OxfordUniversity College

Munich Center for Mathematical Philosophy, 24 June 2021

Categorical set theories Joel David Hamkins


This talk includes joint work in progress with:

Hans Robin Solberg, Oxford University



Categoricity

A theory is categorical if it identifies a unique mathematicalstructure up to isomorphism: any two models of the theory areisomorphic.



Categoricity and structuralism

According to the philosophy of structuralism, we care about ourmathematical structures only up to isomorphism.

Categoricity is thus central to structuralism, because it enablesus easily to refer to mathematical structures.

In mathematical practice, rather than providing a specific copyof a structure, taken out of storage like the platinum rod inParis, and announcing that we only care about it up toisomorphism, we instead simply state that we are thinking of amodel of the categorical theory T .

So we pick out the structure by describing a feature thatdetermines that structure up to isomorphism.
























Categoricity in second-order logic

Infinite structures can never be categorically characterized by afirst-order theory, in light of the Löwenheim-Skolem theorem.

Meanwhile, many of our fundamental structures in mathematicsadmit categorical descriptions in second-order logic.



Categoricity in second-order logic

Infinite structures can never be categorically characterized by afirst-order theory, in light of the Löwenheim-Skolem theorem.

Meanwhile, many of our fundamental structures in mathematicsadmit categorical descriptions in second-order logic.



Categoricity in Mathematics

Natural numbers structure (Dedekind)

〈N,0,S〉 is the unique model of Dedekind Arithmetic.

Real field (Huntington 1903)

〈R,+, ·,0,1〉 is the unique complete ordered field.

Complex field

〈C,+, ·〉 is the unique algebraically closed field of characteristic0 and size continuum.








Complex field









Complex field




Quasi-categoricity in set theory

Consider the second-order Zermelo-Fraenkel set theory ZF2.

second-order separation/replacement axioms.

Zermelo observed that if M |= ZF2, thenM is well-founded. So we might as well use ∈.M is correct about power sets. So M is some Vα.OrdM must be regular.

So M must be Vκ for some inaccessible cardinal κ.

And conversely all such Vκ are models of ZF2.




Consider the second-order Zermelo-Fraenkel set theory ZF2.second-order separation/replacement axioms.








Zermelo observed that if M |= ZF2, then

M is well-founded. So we might as well use ∈.M is correct about power sets. So M is some Vα.OrdM must be regular.







Zermelo observed that if M |= ZF2, thenM is well-founded.

So we might as well use ∈.M is correct about power sets. So M is some Vα.OrdM must be regular.







Zermelo observed that if M |= ZF2, thenM is well-founded. So we might as well use ∈.

M is correct about power sets. So M is some Vα.OrdM must be regular.







Zermelo observed that if M |= ZF2, thenM is well-founded. So we might as well use ∈.M is correct about power sets.

So M is some Vα.OrdM must be regular.







Zermelo observed that if M |= ZF2, thenM is well-founded. So we might as well use ∈.M is correct about power sets. So M is some Vα.

OrdM must be regular.


























Zermelo’s theorem

Theorem (Zermelo 1930)

The models of ZF2 are exactly the models 〈Vκ,∈〉, where κ isan inaccessible cardinal.

These are now known as the Grothendieck-Zermelo universes.



Zermelo’s theorem

Theorem (Zermelo 1930)

The models of ZF2 are exactly the models 〈Vκ,∈〉, where κ isan inaccessible cardinal.

These are now known as the Grothendieck-Zermelo universes.



Zermelo quasi-categoricity

Corollary (Quasi-categoricity, Zermelo 1930)

For any two models of ZF2, one of them is isomorphic to a rankinitial segment of the other.

It follows that any two models of ZF2 agree on ordinarymathematical statements, which are absolute between all thevarious inaccessible Vκ.

For example, they all agree on arithmetic statements,statements about the reals, as well as many set-theoreticstatements, such as the continuum hypothesis (a point made byKreisel).

















Our ProjectTo investigate when quasi-categoricity rises to full categoricity.

Question

Which models of ZF2 satisfy fully categorical theories?

In many instances we can form categorical theories byaugmenting ZF2 with a first-order sentence, forming a theory

ZF2 + σ

that is true in exactly one Vκ.

In other cases, we form a categorical theory with asecond-order sentence or with a theory.




Question



ZF2 + σ






Question



ZF2 + σ






Question



ZF2 + σ





Easy Examples

Suppose that κ is the least inaccessible cardinal.

Then Vκ is characterized by the theory

ZF2 + “there are no inaccessible cardinals.”

The least inaccessible cardinal is therefore first-ordersententially categorical.

Similar ideas apply to the next inaccessible cardinal, and thenext and so on quite a long way.



Easy Examples








Easy Examples








Easy Examples








Main Definitions

1 κ is first-order sententially categorical, if there is afirst-order sentence σ in the language of set theory, suchthat Vκ is categorically characterized by ZF2 + σ.

2 κ is first-order theory categorical, if there is a first-ordertheory T in the language of set theory, such that Vκ iscategorically characterized by ZF2 + T . (Leibnizian)

3 κ is second-order sententially categorical, if there is asecond-order sentence σ in the language of set theory,such that Vκ is categorically characterized by ZF2 + σ.

4 κ is second-order theory categorical, if there is asecond-order theory T in the language of set theory, suchthat Vκ is categorically characterized by ZF2 + T .

Generalize to Σmn -categoricity or even Σα

n -categoricity.



Main Definitions






n -categoricity.



Main Definitions






n -categoricity.



Main Definitions






n -categoricity.



Main Definitions






n -categoricity.



Equivalently

Since Zermelo characterized the inaccessible cardinals κ asthose for which Vκ |= ZFC2, we can say that κ is first-ordersententially categorical if there is a first-order sentence σ suchthat κ is the only inaccessible cardinal for which Vκ |= σ.

And similarly with the other notions. This is about categoricalcharacterizations of Vκ for inaccessible κ.



Equivalently

Since Zermelo characterized the inaccessible cardinals κ asthose for which Vκ |= ZFC2, we can say that κ is first-ordersententially categorical if there is a first-order sentence σ suchthat κ is the only inaccessible cardinal for which Vκ |= σ.

And similarly with the other notions. This is about categoricalcharacterizations of Vκ for inaccessible κ.



Abundance of easy examples

The least inaccessible κ is characterized by “there are noinaccessible cardinals.”

The next one is characterized by “there is exactly oneinaccessible cardinal.”

The αth inaccessible cardinal (start with 0) is characterized by“there are exactly α inaccessible cardinals.” (Need α to beabsolutely expressible.)

So quite a few inaccessible cardinals at the bottom aresententially categorical, up to the ωCK

1 th inaccessible andbeyond.














The αth inaccessible cardinal (start with 0) is characterized by“there are exactly α inaccessible cardinals.”

(Need α to beabsolutely expressible.)





















Beyond countably many inaccessible cardinals

But similarly, the ω1th inaccessible cardinal is sententiallycategorical, and the ω2nd, and more.

The ωαth inaccessible cardinal is sententially categorical, if α issufficiently describable.

We seem thus to open the door to the possibility of gaps in thecategorical cardinals, since there can’t be so many sentences.















Categoricity is a smallness notion

Notice that categoricity is a kind of anti-large-cardinal notion.

It is the smallest of large cardinals that seem to be categorical.



Categoricity is a smallness notion

Notice that categoricity is a kind of anti-large-cardinal notion.

It is the smallest of large cardinals that seem to be categorical.



Absoluteness

Observation

Categoricity is downward absolute from V to any Vθ. If κ iscategorical and θ > κ, then Vθ knows that κ is categorical.

Proof.

Vθ can verify that Vκ has the theory that it has.

And there are fewer challenges to categoricity in Vθ than in V .

So κ is categorical inside Vθ.



Absoluteness

Observation


Proof.






Absoluteness

Observation


Proof.






Absoluteness

Observation


Proof.






Absoluteness

Theorem

Sentential categoricity (first and secod order) is absolutebetween V and any Vθ, both upward and downward.

Proof.Suppose κ is categorical in Vθ via sentence σ. So Vκ is the firstinaccessible model of σ, and this is a characterization in V .

Conversely, if κ is categorical in V , then same sentence works insideany Vλ.



Absoluteness

Theorem






Absoluteness

Theorem






Eventual non-categoricity

Theorem

If κ is not sententially categorical, then it is eventually notcategorical in all sufficiently large Vθ.

Proof.

Assume κ is not sententially categorical.

For each sentence σ true in Vκ, find κσ 6= κ such that σ alsotrue in Vκσ .

If θ > κ and above all κσ, then Vθ sees κ not categorical.




Theorem


Proof.







Theorem


Proof.







Theorem


Proof.






Every sentence true anywhere can be categorical

Theorem

Any sentence or theory true in some universe Vκ can be categorical.

Specifically, if σ is true in some Vκ |= ZFC2, then either this is acategorical characterization of Vκ, or it is consistent with ZFC2 that itis such a categorical characterization of such a universe.

Proof.

If Vκ is the only model of ZFC2 + σ, then this is a categoricalcharacterization and we are done.

Otherwise, there are two such models. If κ < δ are the first twoinstances, then inside Vδ, the theory ZFC2 + σ is a categoricalcharacterization of Vκ.




Theorem



Proof.






Theorem



Proof.






Theorem



Proof.





Categoricity depends on what there is

Conclusion

Every theory that can be true in a set-theoretic universe canalso be a categorical characterization of such a universe.

So the answer to the question “Which extensions of ZFC2 canbe categorical?” is that these are exactly the same extensionsthat can be true at all.

The categoricity of a sentence or theory depends on what elsethere is. Seen in this light, categoricity assertions often amountto anti-large-cardinal assertions.




Conclusion







Conclusion






The categoricity paradoxIt is mildly paradoxical when Vδ satisfies a theory T and also theassertion that T is categorical.

Vδ thinks that T characterizes some strictly smaller Vκ, even thoughVδ thinks T is true.

I find the situation similar to models of PA + ¬Con(PA), which aremistaken about consistency, since they think both that PA is true, yetalso inconsistent.

The phenomenon occurs already with ZFC2 itself. If κ0 < κ1 are thefirst two inaccessible cardinals, then Vκ1 thinks that ZFC2 is true andalso that ZFC2 is a categorical characterization of Vκ0 .

Perhaps ultimately, we don’t really want to know which extensions ofZFC2 can be categorical, but rather which extensions are categorical,in the fully complete set-theoretic universe V , not artificially truncatedat some inaccessible cardinal δ.































Successor inaccessibles

Let κ be the next inaccessible cardinal above κ.

Theorem

If κ is second-order sententially categorical, then κ isfirst-order sententially categorical.

Proof.

If ψ is the second-order sentence, then the next inaccessiblecardinal can see that Vκ satisfies ψ, and this will characterizeκ in a first-order manner.





Theorem


Proof.






Theorem


Proof.




Limits

Theorem

If κ is inaccessible and the sententially categorical cardinals areunbounded in the inaccessible cardinals below κ, then κ isfirst-order theory categorical.

Proof.Note that κ might not be a limit of inaccessibles.

Vκ can see characterizing assertions about the smaller inaccessibles.

So no inaccessible δ < κ can have Vδ with same theory as Vκ.

And no larger θ > κ can have same theory, since in Vθ either thereare new sententially categorical cardinals, or else the sententiallycategorical cardinals will not be unbounded in the inaccessibles.



Limits

Theorem








Limits

Theorem








Limits

Theorem








Limits

Theorem








Mahlo cardinals not first-order categorical

Theorem

No Mahlo cardinal is first-order theory categorical.

Proof.

If κ is Mahlo, then Vδ ≺ Vκ for a stationary set of δ, whichtherefore includes many inaccessible cardinals. So Vκ is notcharacterized by any first-order sentence or theory.



Mahlo cardinals not first-order categorical

Theorem

No Mahlo cardinal is first-order theory categorical.

Proof.

If κ is Mahlo, then Vδ ≺ Vκ for a stationary set of δ, whichtherefore includes many inaccessible cardinals. So Vκ is notcharacterized by any first-order sentence or theory.



Mahlo cardinals can be second-order categorical

Theorem

The least Mahlo cardinal is second-order sententiallycategorical, but not first-order theory categorical.

Proof.

Being Mahlo is a Π11 property: every club C ⊆ κ has a regular

cardinal. So the least one is second-order sententiallycategorical.

But no Mahlo cardinal is first-order categorical by previous.




Theorem


Proof.







Theorem


Proof.






Weakening Mahloness

Can weaken the hypotheses in these observations.

Inaccessible κ is uplifting ([HJ14]), if arbitrarily largeinaccessible λ with Vκ ≺ Vλ.

Weaker than Mahlo. Actually only need a single nontrivialinstance Vκ ≺ Vλ.

And actually only need Vκ ≡ Vλ for non-categoricity.



Weakening Mahloness







Weakening Mahloness







Weakening Mahloness







The rank elementary forest

Consider the relation κ � λ if and only if Vκ ≺ Vλ, forinaccessible cardinals κ and λ.

This is a forest order, since predecessors of any node arelinearly ordered.

Observation

Every first-order theory categorical cardinal is a stump in therank elementary forest, a disconnected root node with nothingabove it.

Converse is not true, since we can have Vκ ≡ Vλ withoutVκ ≺ Vλ.






Observation








Observation








Observation





Gaps in the sententially categorical cardinalsTheorem

If there are uncountably many inaccessible cardinals, then there aregaps in the first-order sententially categorical cardinals.

Proof.

Assume uncountably many inaccessible cardinals.

So there is a non sententially categorical cardinal.

Fix κ inaccessible, not first-order sententially categorical.

Any sufficiently large Vθ can see this.

Let θ be smallest inaccessible that thinks that there is an inaccessiblecardinal that is not first-order sententially categorical.

This is a sententially categorical characterization.

So there are gaps.





Proof.







So there are gaps.





Proof.







So there are gaps.





Proof.







So there are gaps.





Proof.







So there are gaps.





Proof.







So there are gaps.





Proof.







So there are gaps.



More gapsSame analysis works with second-order sentential categoricity.

But also with theory categoricity:

Theorem

If enough inaccessibles, then there is first-order sententiallycategorical cardinal larger than some inaccessible cardinal notcategorical by sentences or theories, first or second order.

Proof.

Assume at least c+ many inaccessible cardinals.

So there is an inaccessible not second-order theory categorical.

So there is some inaccessible θ that can see this.

Let θ be least inaccessible that can see this.

This property charactizes θ.





Theorem


Proof.










Theorem


Proof.










Theorem


Proof.










Theorem


Proof.










Theorem


Proof.










Theorem


Proof.










Theorem


Proof.








The number of categorical cardinals

At most countably many sententially categorical cardinals.

And if there are infinitely many inaccessibles, then there will beinfinitely many sententially categorical cardinals.

The first ω many inaccessible cardinals are sententiallycategorical.

At most c many theory categorical cardinals.
























How many categorical cardinals

Question

How many theory categorical cardinals must there be?

If continuum many inaccessibles, must there be this manytheory categorical cardinals?If uncountably many inaccessibles, must there beuncountably many theory categorical cardinals?Must the first ω1 inaccessible cardinals be theorycategorical?




Question

How many theory categorical cardinals must there be?If continuum many inaccessibles, must there be this manytheory categorical cardinals?

If uncountably many inaccessibles, must there beuncountably many theory categorical cardinals?Must the first ω1 inaccessible cardinals be theorycategorical?




Question

How many theory categorical cardinals must there be?If continuum many inaccessibles, must there be this manytheory categorical cardinals?If uncountably many inaccessibles, must there beuncountably many theory categorical cardinals?

Must the first ω1 inaccessible cardinals be theorycategorical?




Question

How many theory categorical cardinals must there be?If continuum many inaccessibles, must there be this manytheory categorical cardinals?If uncountably many inaccessibles, must there beuncountably many theory categorical cardinals?Must the first ω1 inaccessible cardinals be theorycategorical?



Theorem

It is relatively consistent with ZFC that there are a proper class ofinaccessible cardinals, but only countably many theory categoricalcardinals.

Proof.

Assume proper class of inaccessibles.

Force to V [G] collapsing cV to ω.

Forcing is small, and hence neither creates nor destroysinaccessibles.

Forcing is homogeneous and definable. So creates no categoricalcardinals.

So at most cV (now countable) many theory categorical cardinals inV [G].



Theorem


Proof.








Theorem


Proof.








Theorem


Proof.








Theorem


Proof.








Theorem

If at least c many inaccessibles, then in some forcing extension,preserving continuum and having exactly same inaccessiblecardinals, the first continuum many are all first-order theorycategorical.

Proof.

Let κα be αth inaccessible cardinal.

Enumerate 〈Aα | α < c〉 subsets Aα ⊆ ω. Force to code Aα abovesupremum of inaccessibles below κα.

So Aα becomes definable in the theory of Vκα .

Each κα becomes theory categorical in V [G].

If GCH in ground model, this forcing preserves all cardinals andcofinalities.



Theorem


Proof.








Theorem


Proof.








Theorem


Proof.








Theorem


Proof.








Forcing categoricityCan also arrange intermediate number:

Theorem

It is relatively consistent that the number of first-order theorycategorical cardinals is ω1, with large continuum and a proper class ofinaccessible cardinals.

Proof.

Start with a model having CH and first ω1 many inaccessibles alltheory categorical by coding reals into GCH pattern.

Force with Add(ω, θ) to make continuum large.

This preserves the GCH coding and all inaccessibles, so still have ω1many theory categorical cardinals.

But forcing is definable and homogeneous, so no new categoricalcardinals.




Theorem


Proof.








Theorem


Proof.








Theorem


Proof.








Theorem


Proof.








Theorem


Proof.







Variations

The argument is extremely flexible.

We could have arranged to have exactly ℵ17 many first-ordertheory categorical cardinals, while the continuum is ℵω2+5, orwhatever, in diverse other possible combinations.



Forcing a given cardinal to become categorical

Theorem

For any inaccessible κ, there is a forcing extension V [G] with sameinaccessible cardinals, in which κ is first-order sententiallycategorical.

Indeed, any countable collection of inaccessible cardinals can bemade first-order sententially categorical in a forcing extension withexactly the same inaccessible cardinals.

Proof idea.

Shoot a club C through the singulars in κ. This makes κ inaccessible,but not Mahlo. Now force to destroy GCH only at successors of limitpoints of C. So κ is first inaccessible that is a limit of failures ofGCH.

An instance of the killing-them-softly phenomenon (Erin Carmody).



Forcing a given cardinal to become categorical

Theorem

For any inaccessible κ, there is a forcing extension V [G] with sameinaccessible cardinals, in which κ is first-order sententiallycategorical.

Indeed, any countable collection of inaccessible cardinals can bemade first-order sententially categorical in a forcing extension withexactly the same inaccessible cardinals.

Proof idea.

Shoot a club C through the singulars in κ. This makes κ inaccessible,but not Mahlo. Now force to destroy GCH only at successors of limitpoints of C. So κ is first inaccessible that is a limit of failures ofGCH.

An instance of the killing-them-softly phenomenon (Erin Carmody).



Theorem

Complete implication diagram for categoricity:

κ is first-ordersententially categorical

κ is first-ordertheory categorical

κ is second-ordersententially categorical

κ is second-ordertheory categorical

None of these implications are reversible and no other implicationsare provable.

The positive implication are easy.



Theorem

Complete implication diagram for categoricity:





None of these implications are reversible and no other implicationsare provable.

The positive implication are easy.







To prove no implications reverse, need:

A second-order sententially categorical cardinal that is notfirst-order theory categorical.

A first-order theory categorical cardinal that is not second-ordersententially categorical.






























Categoricity Venn diagramTheorem

Implications between the various categoricity notions are thoseshown in the following Venn diagram, and if there are at least c+ manyinaccessible cardinals, then every cell of the diagram is inhabited.

second-ordertheory

categorical

second-ordersententiallycategorical

first-ordertheory

categorical

first-ordersententiallycategorical



Metatheory or object theory?

Question

Is categoricity a metatheoretic or object-theoretic matter?

The notion of categorical cardinal was offered in the objecttheory, as part of the ZFC large cardinal development.

But categoricity is often thought to be a metatheoretic matter.




Question







Question






Zermelo’s quasi-categoricity theorem

How are we to take Zermelo’s quasi-categoricity theorem?

Is it part of the object theory or the meta-theory?

Related to two interpretations of second-order logic.Most set theorists understand Zermelo’s theorem as atheorem of ZFC, part of the object theory.Many philosophers seem to take it as a metatheoreticanalysis of set theory in second-order logic.Related to Zermelo’s views on set-theoretic potentialism.












Related to two interpretations of second-order logic.

Most set theorists understand Zermelo’s theorem as atheorem of ZFC, part of the object theory.Many philosophers seem to take it as a metatheoreticanalysis of set theory in second-order logic.Related to Zermelo’s views on set-theoretic potentialism.






Related to two interpretations of second-order logic.Most set theorists understand Zermelo’s theorem as atheorem of ZFC, part of the object theory.

Many philosophers seem to take it as a metatheoreticanalysis of set theory in second-order logic.Related to Zermelo’s views on set-theoretic potentialism.






Related to two interpretations of second-order logic.Most set theorists understand Zermelo’s theorem as atheorem of ZFC, part of the object theory.Many philosophers seem to take it as a metatheoreticanalysis of set theory in second-order logic.

Related to Zermelo’s views on set-theoretic potentialism.









Unifying the approaches

Suppose one wants to understand categoricity issues in the contextof second-order logic, taken as independently meaningful.

To clarify the second-order context, however, one will want to adoptprinciples governing the content of second-order logic. One will adoptvarious comprehension principles and so on, governing whichsecond-order objects there are.

But this amounts to undertaking first-order set theory, but in themeta-theory.

Ultimately, therefore, the two approaches merge. To interpretsecond-order logic as independently meaningful is precisely toundertake first-order set theory in the metatheory.
























Internalized categoricity

Adopting the object-theoretic account in effect internalizescategoricity notions relative to a fixed model of ZFC.

But this is different than true externalized categoricity.

Imagine a model of ZFC with nonstandard finite n manyinaccessible cardinals.

Inside the model, these are all sententially categorical, the k thinaccessible. But for nonstandard k , this is not actuallyexpressible in the metatheory.
























Tension between categoricity and reflection

Mathematicians often point to categoricity as a positive featureof our accounts of the natural numbers, the real numbers, thecomplex numbers.

We know these structures, because we have categoricalaccounts of them.

And yet, set theorists point to reflection principles as afundamental expectation of the set-theoretic universe.

But reflection principles are at heart anti-categorical!

Set theorists would not like V to have a categorical description.

This seems to be a philosophical puzzle to be sorted.
















































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

Joel David HamkinsOxford


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