Categorical cardinals Joel David Hamkins Professor of Logic Sir - - PowerPoint PPT Presentation

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Categorical cardinals

Joel David Hamkins

Professor of Logic Sir Peter Strawson Fellow

University of Oxford University College

CUNY Set Theory Seminar, 26 June 2020

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

This talk includes joint work in progress with: Hans Robin Solberg, Oxford University

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Categoricity

A theory is categorical if it identifies a unique mathematical structure up to isomorphism: any two models of the theory are isomorphic.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Categoricity

A theory is categorical if it identifies a unique mathematical structure up to isomorphism: any two models of the theory are isomorphic. For infinite structures, this is of course impossible in first-order logic, because of the Löwenheim-Skolem theorem.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Categoricity

A theory is categorical if it identifies a unique mathematical structure up to isomorphism: any two models of the theory are isomorphic. For infinite structures, this is of course impossible in first-order logic, because of the Löwenheim-Skolem theorem. Meanwhile, many of our fundamental structures in mathematics admit categorical descriptions in second-order logic.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Categoricity in Mathematics

Natural numbers structure (Dedekind) N, 0, S is the unique model of Dedekind Arithmetic.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

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.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

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 characteristic 0 and size continuum.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Quasi-categoricity in set theory

Consider the second-order set theory ZF2.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Quasi-categoricity in set theory

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

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Quasi-categoricity in set theory

Consider the second-order set theory ZF2. second-order separation/replacement axioms. Zermelo observed that if M | = ZF2, then

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Quasi-categoricity in set theory

Consider the second-order set theory ZF2. second-order separation/replacement axioms. Zermelo observed that if M | = ZF2, then M is well-founded.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Quasi-categoricity in set theory

Consider the second-order 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 ∈.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Quasi-categoricity in set theory

Consider the second-order 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.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Quasi-categoricity in set theory

Consider the second-order 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.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Quasi-categoricity in set theory

Consider the second-order 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.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Quasi-categoricity in set theory

Consider the second-order 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. So M must be Vκ for some inaccessible cardinal κ.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Quasi-categoricity in set theory

Consider the second-order 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. So M must be Vκ for some inaccessible cardinal κ. And conversely all such Vκ are models of ZF2.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Zermelo quasi-categoricity

Theorem (Zermelo 1930) The models of ZF2 are exactly the models Vκ, ∈, where κ is an inaccessible cardinal.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Zermelo quasi-categoricity

Theorem (Zermelo 1930) The models of ZF2 are exactly the models Vκ, ∈, where κ is an inaccessible cardinal. These are now known as the Grothendieck-Zermelo universes.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Zermelo quasi-categoricity

Theorem (Zermelo 1930) The models of ZF2 are exactly the models Vκ, ∈, where κ is an inaccessible cardinal. These are now known as the Grothendieck-Zermelo universes. Corollary (Quasi-categoricity, Zermelo 1930) For any two models of ZF2, one of them is isomorphic to a rank initial segment of the other.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Our Project

To investigate when quasi-categoricity rises to full categoricity.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Our Project

To investigate when quasi-categoricity rises to full categoricity. Question Which models of ZF2 satisfy fully categorical theories?

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Our Project

To 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 by augmenting ZF2 with a first-order sentence, forming a theory ZF2 + σ that is true in exactly one Vκ.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Our Project

To 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 by augmenting 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 a second-order sentence or with a theory.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Easy Examples

Suppose that κ is the least inaccessible cardinal.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Easy Examples

Suppose that κ is the least inaccessible cardinal. Then Vκ is characterized by the theory ZF2 + “there are no inaccessible cardinals.”

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

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-order sententially categorical.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

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-order sententially categorical. Similar ideas apply to the next inaccessible cardinal, and the next and so on quite a long way.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Main Definitions

1 κ is first-order sententially categorical, if there is a

first-order sentence σ in the language of set theory, such that Vκ is categorically characterized by ZF2 + σ.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Main Definitions

1 κ is first-order sententially categorical, if there is a

first-order sentence σ in the language of set theory, such that Vκ is categorically characterized by ZF2 + σ.

2 κ is first-order theory categorical, if there is a first-order

theory T in the language of set theory, such that Vκ is categorically characterized by ZF2 + T. (Leibnizian)

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Main Definitions

1 κ is first-order sententially categorical, if there is a

first-order sentence σ in the language of set theory, such that Vκ is categorically characterized by ZF2 + σ.

2 κ is first-order theory categorical, if there is a first-order

theory T in the language of set theory, such that Vκ is categorically characterized by ZF2 + T. (Leibnizian)

3 κ is second-order sententially categorical, if there is a

second-order sentence σ in the language of set theory, such that Vκ is categorically characterized by ZF2 + σ.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Main Definitions

1 κ is first-order sententially categorical, if there is a

first-order sentence σ in the language of set theory, such that Vκ is categorically characterized by ZF2 + σ.

2 κ is first-order theory categorical, if there is a first-order

theory T in the language of set theory, such that Vκ is categorically characterized by ZF2 + T. (Leibnizian)

3 κ is second-order sententially categorical, if there is a

second-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 a

second-order theory T in the language of set theory, such that Vκ is categorically characterized by ZF2 + T.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Main Definitions

1 κ is first-order sententially categorical, if there is a

first-order sentence σ in the language of set theory, such that Vκ is categorically characterized by ZF2 + σ.

2 κ is first-order theory categorical, if there is a first-order

theory T in the language of set theory, such that Vκ is categorically characterized by ZF2 + T. (Leibnizian)

3 κ is second-order sententially categorical, if there is a

second-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 a

second-order theory T in the language of set theory, such that Vκ is categorically characterized by ZF2 + T. Generalize to Σm

n -categoricity or even Σα n-categoricity.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Equivalently

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

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Equivalently

Since Zermelo characterized the inaccessible cardinals κ as those for which Vκ | = ZFC2, we can say that κ is first-order sententially categorical if there is a first-order sentence σ such that κ is the only inaccessible cardinal for which Vκ | = σ. And similarly with the other notions. This is about categorical characterizations of Vκ for inaccessible κ.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Abundance of easy examples

The least inaccessible κ is characterized by “there are no inaccessible cardinals.”

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Abundance of easy examples

The least inaccessible κ is characterized by “there are no inaccessible cardinals.” The next one is characterized by “there is exactly one inaccessible cardinal.”

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Abundance of easy examples

The least inaccessible κ is characterized by “there are no inaccessible cardinals.” The next one is characterized by “there is exactly one inaccessible cardinal.” The αth inaccessible cardinal (start with 0) is characterized by “there are exactly α inaccessible cardinals.”

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Abundance of easy examples

The least inaccessible κ is characterized by “there are no inaccessible cardinals.” The next one is characterized by “there is exactly one inaccessible cardinal.” The αth inaccessible cardinal (start with 0) is characterized by “there are exactly α inaccessible cardinals.” (Need α to be absolutely expressible.)

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Abundance of easy examples

The least inaccessible κ is characterized by “there are no inaccessible cardinals.” The next one is characterized by “there is exactly one inaccessible cardinal.” The αth inaccessible cardinal (start with 0) is characterized by “there are exactly α inaccessible cardinals.” (Need α to be absolutely expressible.) So quite a few inaccessible cardinals at the bottom are sententially categorical, up to the ωCK

1 th inaccessible and

beyond.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Beyond countably many inaccessible cardinals

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

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Beyond countably many inaccessible cardinals

But similarly, the ω1th inaccessible cardinal is sententially categorical, and the ω2nd, and more. The ωαth inaccessible cardinal is sententially categorical, if α is sufficiently describable.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Beyond countably many inaccessible cardinals

But similarly, the ω1th inaccessible cardinal is sententially categorical, and the ω2nd, and more. The ωαth inaccessible cardinal is sententially categorical, if α is sufficiently describable. We seem thus to open the door to the possibility of gaps in the categorical cardinals, since there can’t be so many sentences.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Categoricity is a smallness notion

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

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

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.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Absoluteness

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

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Absoluteness

Observation Categoricity is downward absolute from V to any Vθ. If κ is categorical and θ > κ, then Vθ knows that κ is categorical. Proof. Vθ can verify that Vκ has the theory that it has.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Absoluteness

Observation Categoricity is downward absolute from V to any Vθ. If κ is categorical 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.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Absoluteness

Observation Categoricity is downward absolute from V to any Vθ. If κ is categorical 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.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Successor inaccessibles

Let κ be the next inaccessible cardinal above κ.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Successor inaccessibles

Let κ be the next inaccessible cardinal above κ. Theorem If κ is second-order sententially categorical, then κ is first-order sententially categorical.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Successor inaccessibles

Let κ be the next inaccessible cardinal above κ. Theorem If κ is second-order sententially categorical, then κ is first-order sententially categorical. Proof. If ψ is the second-order sentence, then the next inaccessible cardinal can see that Vκ satisfies ψ, and this will characterize κ in a first-order manner.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Limits

Theorem If κ is inaccessible and the sententially categorical cardinals are unbounded in the inaccessible cardinals below κ, then κ is first-order theory categorical.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Limits

Theorem If κ is inaccessible and the sententially categorical cardinals are unbounded in the inaccessible cardinals below κ, then κ is first-order theory categorical. Proof.

Note that κ might not be a limit of inaccessibles.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Limits

Theorem If κ is inaccessible and the sententially categorical cardinals are unbounded in the inaccessible cardinals below κ, then κ is first-order theory categorical. Proof.

Note that κ might not be a limit of inaccessibles. Vκ can see characterizing assertions about the smaller inaccessibles.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Limits

Theorem If κ is inaccessible and the sententially categorical cardinals are unbounded in the inaccessible cardinals below κ, then κ is first-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.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Limits

Theorem If κ is inaccessible and the sententially categorical cardinals are unbounded in the inaccessible cardinals below κ, then κ is first-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 there are new sententially categorical cardinals, or else the sententially categorical cardinals will not be unbounded in the inaccessibles.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Absoluteness

Theorem Sentential categoricity in V is absolute to any sententially categorical cardinal λ, both up and down.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Absoluteness

Theorem Sentential categoricity in V is absolute to any sententially categorical cardinal λ, both up and down. Proof.

Suppose κ is categorical in Vλ via σ and λ is categorical via τ.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Absoluteness

Theorem Sentential categoricity in V is absolute to any sententially categorical cardinal λ, both up and down. Proof.

Suppose κ is categorical in Vλ via σ and λ is categorical via τ. Then κ is categorical in V by “σ and there is no inaccessible level with τ.”

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Absoluteness

Theorem Sentential categoricity in V is absolute to any sententially categorical cardinal λ, both up and down. Proof.

Suppose κ is categorical in Vλ via σ and λ is categorical via τ. Then κ is categorical in V by “σ and there is no inaccessible level with τ.” Conversely, if κ is categorical in V, then same sentence works inside any Vλ.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Absoluteness

Theorem Sentential categoricity in V is absolute to any sententially categorical cardinal λ, both up and down. Proof.

Suppose κ is categorical in Vλ via σ and λ is categorical via τ. Then κ is categorical in V by “σ and there is no inaccessible level with τ.” Conversely, if κ is categorical in V, then same sentence works inside any Vλ.

Meanwhile, sentential categoricity is not generally absolute to any inaccessible cardinal.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Eventual non-categoricity

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

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Eventual non-categoricity

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

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Eventual non-categoricity

Theorem If κ is not sententially categorical, then it is eventually not categorical in all sufficiently large Vθ. Proof. Assume κ is not sententially categorical. For each sentence σ true in Vκ, find κσ = κ such that σ also true in Vκσ.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Eventual non-categoricity

Theorem If κ is not sententially categorical, then it is eventually not categorical in all sufficiently large Vθ. Proof. Assume κ is not sententially categorical. For each sentence σ true in Vκ, find κσ = κ such that σ also true in Vκσ. If θ > κ and above all κσ, then Vθ sees κ not categorical.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Mahlo cardinals not first-order categorical

Theorem No Mahlo cardinal is first-order theory categorical.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

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, which therefore includes many inaccessible cardinals. So Vκ is not characterized by any first-order sentence or theory.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Mahlo cardinals can be second-order categorical

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

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Mahlo cardinals can be second-order categorical

Theorem The least Mahlo cardinal is second-order sententially categorical, but not first-order theory categorical. Proof. Being Mahlo is a Π1

1 property: every club C ⊆ κ has a regular

• cardinal. So the least one is second-order sententially

categorical.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Mahlo cardinals can be second-order categorical

Theorem The least Mahlo cardinal is second-order sententially categorical, but not first-order theory categorical. Proof. Being Mahlo is a Π1

1 property: every club C ⊆ κ has a regular

• cardinal. So the least one is second-order sententially

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

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Weakening Mahloness

Can weaken the hypotheses in these observations.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Weakening Mahloness

Can weaken the hypotheses in these observations. Inaccessible κ is uplifting ([HJ14]), if arbitrarily large λ with Vκ ≺ Vλ.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Weakening Mahloness

Can weaken the hypotheses in these observations. Inaccessible κ is uplifting ([HJ14]), if arbitrarily large λ with Vκ ≺ Vλ. Weaker than Mahlo. Actually only need a single nontrivial instance Vκ ≺ Vλ.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Weakening Mahloness

Can weaken the hypotheses in these observations. Inaccessible κ is uplifting ([HJ14]), if arbitrarily large λ with Vκ ≺ Vλ. Weaker than Mahlo. Actually only need a single nontrivial instance Vκ ≺ Vλ. And actually only need Vκ ≡ Vλ for non-categoricity.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

The rank elementary forest

Consider the relation κ λ if and only if Vκ ≺ Vλ, for inaccessible cardinals κ and λ.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

The rank elementary forest

Consider the relation κ λ if and only if Vκ ≺ Vλ, for inaccessible cardinals κ and λ. This is a forest order, since predecessors of any node are linearly ordered.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

The rank elementary forest

Consider the relation κ λ if and only if Vκ ≺ Vλ, for inaccessible cardinals κ and λ. This is a forest order, since predecessors of any node are linearly ordered. Observation Every first-order theory categorical cardinal is a stump in the rank elementary forest, a disconnected root node with nothing above it.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

The rank elementary forest

Consider the relation κ λ if and only if Vκ ≺ Vλ, for inaccessible cardinals κ and λ. This is a forest order, since predecessors of any node are linearly ordered. Observation Every first-order theory categorical cardinal is a stump in the rank elementary forest, a disconnected root node with nothing above it. Converse is not true, since we can have Vκ ≡ Vλ without Vκ ≺ Vλ.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Gaps in the sententially categorical cardinals

Theorem If there are uncountably many inaccessible cardinals, then there are gaps in the first-order sententially categorical cardinals. Proof. Assume uncountably many inaccessible cardinals.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Gaps in the sententially categorical cardinals

Theorem If there are uncountably many inaccessible cardinals, then there are gaps in the first-order sententially categorical cardinals. Proof. Assume uncountably many inaccessible cardinals. So there is a non sententially categorical cardinal.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Gaps in the sententially categorical cardinals

Theorem If there are uncountably many inaccessible cardinals, then there are gaps 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.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Gaps in the sententially categorical cardinals

Theorem If there are uncountably many inaccessible cardinals, then there are gaps 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.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Gaps in the sententially categorical cardinals

Theorem If there are uncountably many inaccessible cardinals, then there are gaps 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 inaccessible cardinal that is not first-order sententially categorical.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Gaps in the sententially categorical cardinals

Theorem If there are uncountably many inaccessible cardinals, then there are gaps 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 inaccessible cardinal that is not first-order sententially categorical. This is a sententially categorical characterization.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Gaps in the sententially categorical cardinals

Theorem If there are uncountably many inaccessible cardinals, then there are gaps 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 inaccessible cardinal that is not first-order sententially categorical. This is a sententially categorical characterization. So there are gaps.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

More gaps

Same analysis works with second-order sentential categoricity.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

More gaps

Same analysis works with second-order sentential categoricity. But also with theory categoricity:

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

More gaps

Same analysis works with second-order sentential categoricity. But also with theory categoricity: Theorem If enough inaccessibles, then there is first-order sententially categorical cardinal larger than some inaccessible cardinal not categorical by sentences or theories, first or second order.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

More gaps

Same analysis works with second-order sentential categoricity. But also with theory categoricity: Theorem If enough inaccessibles, then there is first-order sententially categorical cardinal larger than some inaccessible cardinal not categorical by sentences or theories, first or second order. Proof. Assume at least c+ many inaccessible cardinals.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

More gaps

Same analysis works with second-order sentential categoricity. But also with theory categoricity: Theorem If enough inaccessibles, then there is first-order sententially categorical cardinal larger than some inaccessible cardinal not categorical 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.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

More gaps

Same analysis works with second-order sentential categoricity. But also with theory categoricity: Theorem If enough inaccessibles, then there is first-order sententially categorical cardinal larger than some inaccessible cardinal not categorical 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.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

More gaps

Same analysis works with second-order sentential categoricity. But also with theory categoricity: Theorem If enough inaccessibles, then there is first-order sententially categorical cardinal larger than some inaccessible cardinal not categorical 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.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

More gaps

Same analysis works with second-order sentential categoricity. But also with theory categoricity: Theorem If enough inaccessibles, then there is first-order sententially categorical cardinal larger than some inaccessible cardinal not categorical 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.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

The number of categorical cardinals

At most countably many sententially categorical cardinals.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

The number of categorical cardinals

At most countably many sententially categorical cardinals. And if there are infinitely many inaccessibles, then there will be infinitely many sententially categorical cardinals.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

The number of categorical cardinals

At most countably many sententially categorical cardinals. And if there are infinitely many inaccessibles, then there will be infinitely many sententially categorical cardinals. The first ω many inaccessible cardinals are sententially categorical.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

The number of categorical cardinals

At most countably many sententially categorical cardinals. And if there are infinitely many inaccessibles, then there will be infinitely many sententially categorical cardinals. The first ω many inaccessible cardinals are sententially categorical. At most c many theory categorical cardinals.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

How many categorical cardinals

Question How many theory categorical cardinals must there be?

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

How many categorical cardinals

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

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

How many categorical cardinals

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

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

How many categorical cardinals

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

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Theorem It is relatively consistent with ZFC that there are a proper class of inaccessible cardinals, but only countably many theory categorical cardinals. Proof. Assume proper class of inaccessibles.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Theorem It is relatively consistent with ZFC that there are a proper class of inaccessible cardinals, but only countably many theory categorical cardinals. Proof. Assume proper class of inaccessibles. Force to V[G] collapsing cV to ω.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Theorem It is relatively consistent with ZFC that there are a proper class of inaccessible cardinals, but only countably many theory categorical cardinals. Proof. Assume proper class of inaccessibles. Force to V[G] collapsing cV to ω. Forcing is small, and hence neither creates nor destroys inaccessibles.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Theorem It is relatively consistent with ZFC that there are a proper class of inaccessible cardinals, but only countably many theory categorical cardinals. Proof. Assume proper class of inaccessibles. Force to V[G] collapsing cV to ω. Forcing is small, and hence neither creates nor destroys inaccessibles. Forcing is homogeneous and definable. So creates no categorical cardinals.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Theorem It is relatively consistent with ZFC that there are a proper class of inaccessible cardinals, but only countably many theory categorical cardinals. Proof. Assume proper class of inaccessibles. Force to V[G] collapsing cV to ω. Forcing is small, and hence neither creates nor destroys inaccessibles. Forcing is homogeneous and definable. So creates no categorical cardinals. So at most cV (now countable) many theory categorical cardinals in V[G].

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Theorem If at least c many inaccessibles, then in some forcing extension, preserving continuum and having exactly same inaccessible cardinals, the first continuum many are all first-order theory categorical.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Theorem If at least c many inaccessibles, then in some forcing extension, preserving continuum and having exactly same inaccessible cardinals, the first continuum many are all first-order theory categorical. Proof. Let κα be αth inaccessible cardinal.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Theorem If at least c many inaccessibles, then in some forcing extension, preserving continuum and having exactly same inaccessible cardinals, the first continuum many are all first-order theory categorical. Proof. Let κα be αth inaccessible cardinal. Enumerate Aα | α < c subsets Aα ⊆ ω. Force to code Aα above supremum of inaccessibles below κα.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Theorem If at least c many inaccessibles, then in some forcing extension, preserving continuum and having exactly same inaccessible cardinals, the first continuum many are all first-order theory categorical. Proof. Let κα be αth inaccessible cardinal. Enumerate Aα | α < c subsets Aα ⊆ ω. Force to code Aα above supremum of inaccessibles below κα. So Aα becomes definable in the theory of Vκα.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Theorem If at least c many inaccessibles, then in some forcing extension, preserving continuum and having exactly same inaccessible cardinals, the first continuum many are all first-order theory categorical. Proof. Let κα be αth inaccessible cardinal. Enumerate Aα | α < c subsets Aα ⊆ ω. Force to code Aα above supremum 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 and cofinalities.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Forcing categoricity

Can also arrange intermediate number:

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Forcing categoricity

Can also arrange intermediate number: Theorem It is relatively consistent that the number of first-order theory categorical cardinals is ω1, with large continuum and a proper class of inaccessible cardinals.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Forcing categoricity

Can also arrange intermediate number: Theorem It is relatively consistent that the number of first-order theory categorical cardinals is ω1, with large continuum and a proper class of inaccessible cardinals. Proof. Start with a model having CH and first ω1 many inaccessibles all theory categorical by coding reals into GCH pattern.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Forcing categoricity

Can also arrange intermediate number: Theorem It is relatively consistent that the number of first-order theory categorical cardinals is ω1, with large continuum and a proper class of inaccessible cardinals. Proof. Start with a model having CH and first ω1 many inaccessibles all theory categorical by coding reals into GCH pattern. Force with Add(ω, θ) to make continuum large.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Forcing categoricity

Can also arrange intermediate number: Theorem It is relatively consistent that the number of first-order theory categorical cardinals is ω1, with large continuum and a proper class of inaccessible cardinals. Proof. Start with a model having CH and first ω1 many inaccessibles all theory 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 ω1 many theory categorical cardinals.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Forcing categoricity

Can also arrange intermediate number: Theorem It is relatively consistent that the number of first-order theory categorical cardinals is ω1, with large continuum and a proper class of inaccessible cardinals. Proof. Start with a model having CH and first ω1 many inaccessibles all theory 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 ω1 many theory categorical cardinals. But forcing is definable and homogeneous, so no new categorical cardinals.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Variations

The argument is extremely flexible. We could have arranged to have exactly ℵ17 many first-order theory categorical cardinals, while the continuum is ℵω2+5, or whatever, in diverse other possible combinations.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Theorem Complete implication diagram for categoricity:

κ is first-order sententially categorical κ is first-order theory categorical κ is second-order sententially categorical κ is second-order theory categorical

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

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Theorem Complete implication diagram for categoricity:

κ is first-order sententially categorical κ is first-order theory categorical κ is second-order sententially categorical κ is second-order theory categorical

None of these implications are reversible and no other implications are provable. The positive implication are easy.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points κ is first-order sententially categorical κ is first-order theory categorical κ is second-order sententially categorical κ is second-order theory categorical

To prove no implications reverse, need: A second-order sententially categorical cardinal that is not first-order theory categorical.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points κ is first-order sententially categorical κ is first-order theory categorical κ is second-order sententially categorical κ is second-order theory categorical

To prove no implications reverse, need: A second-order sententially categorical cardinal that is not first-order theory categorical. A first-order theory categorical cardinal that is not second-order sententially categorical.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points κ is first-order sententially categorical κ is first-order theory categorical κ is second-order sententially categorical κ is second-order theory categorical

To prove no implications reverse, need: A second-order sententially categorical cardinal that is not first-order theory categorical. A first-order theory categorical cardinal that is not second-order sententially categorical. We have the first already: The least Mahlo cardinal (much less is ok) is second-order sententially categorical, but not first-order theory categorical.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Theorem If there are uncountably many inaccessible cardinals, then the first inaccessible cardinal above all second-order sententially categorical cardinals is first-order theory categorical, but not sententially categorical.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Theorem If there are uncountably many inaccessible cardinals, then the first inaccessible cardinal above all second-order sententially categorical cardinals is first-order theory categorical, but not sententially categorical. Proof. This is an instance of limit theorem from beginning.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Theorem If there are uncountably many inaccessible cardinals, then the first inaccessible cardinal above all second-order sententially categorical cardinals is first-order theory categorical, but not sententially categorical. Proof. This is an instance of limit theorem from beginning. It is part of the theory of that cardinal that all the smaller sententially categorical cardinals exist.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Theorem If there are uncountably many inaccessible cardinals, then the first inaccessible cardinal above all second-order sententially categorical cardinals is first-order theory categorical, but not sententially categorical. Proof. This is an instance of limit theorem from beginning. It is part of the theory of that cardinal that all the smaller sententially categorical cardinals exist. And that there are no inaccessible cardinals above all second-order sententially categorical cardinals.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Theorem If there are uncountably many inaccessible cardinals, then the first inaccessible cardinal above all second-order sententially categorical cardinals is first-order theory categorical, but not sententially categorical. Proof. This is an instance of limit theorem from beginning. It is part of the theory of that cardinal that all the smaller sententially categorical cardinals exist. And that there are no inaccessible cardinals above all second-order sententially categorical cardinals. Only Vκ thinks that exactly those sententially categorical cardinals exist with nothing above them.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Theory categorical cardinals are taller

Corollary If there are enough inaccessible cardinals, then the supremum

• f the theory categorical cardinals is strictly larger than the

supremum of the sententially categorical cardinals.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Internal vs. external categoricity

This analysis uses an internal account of categoricity, formalizing it in ZFC.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Internal vs. external categoricity

This analysis uses an internal account of categoricity, formalizing it in ZFC. But this is different from a metatheoretic account.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Internal vs. external categoricity

This analysis uses an internal account of categoricity, formalizing it in ZFC. But this is different from a metatheoretic account. To see the difference, imagine a model of ZFC with nonstandard finite n many inaccessible cardinals.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Internal vs. external categoricity

This analysis uses an internal account of categoricity, formalizing it in ZFC. But this is different from a metatheoretic account. To see the difference, imagine a model of ZFC with nonstandard finite n many inaccessible cardinals. They are all internally sententially categorical, the kth

• inaccessible. But for nonstandard k, this is not actually

expressible in the metatheory.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Internal vs. external categoricity

This analysis uses an internal account of categoricity, formalizing it in ZFC. But this is different from a metatheoretic account. To see the difference, imagine a model of ZFC with nonstandard finite n many inaccessible cardinals. They are all internally sententially categorical, the kth

• inaccessible. But for nonstandard k, this is not actually

expressible in the metatheory. Question Is categoricity a metatheoretic claim or an object-theoretic claim?

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Tension between categoricity and reflection

Mathematicians often point to categoricity as a positive feature

• f our accounts of the natural numbers, the real numbers, the

complex numbers.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Tension between categoricity and reflection

Mathematicians often point to categoricity as a positive feature

• f our accounts of the natural numbers, the real numbers, the

complex numbers. We know these structures, because we have categorical accounts of them.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Tension between categoricity and reflection

Mathematicians often point to categoricity as a positive feature

• f our accounts of the natural numbers, the real numbers, the

complex numbers. We know these structures, because we have categorical accounts of them. And yet, set theorists point to reflection principles as a fundamental expectation of the set-theoretic universe.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Tension between categoricity and reflection

Mathematicians often point to categoricity as a positive feature

• f our accounts of the natural numbers, the real numbers, the

complex numbers. We know these structures, because we have categorical accounts of them. And yet, set theorists point to reflection principles as a fundamental expectation of the set-theoretic universe. But reflection principles are at heart anti-categorical!

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Tension between categoricity and reflection

Mathematicians often point to categoricity as a positive feature

• f our accounts of the natural numbers, the real numbers, the

complex numbers. We know these structures, because we have categorical accounts of them. And yet, set theorists point to reflection principles as a fundamental 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.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Tension between categoricity and reflection

Mathematicians often point to categoricity as a positive feature

• f our accounts of the natural numbers, the real numbers, the

complex numbers. We know these structures, because we have categorical accounts of them. And yet, set theorists point to reflection principles as a fundamental 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.

Categorical cardinals Joel David Hamkins

Categoricity Categorical cardinals Gaps in categoricity Number of categorical cardinals Implications Two philosophical points

Thank you.

Slides and articles available on http://jdh.hamkins.org. Joel David Hamkins Oxford

Categorical cardinals Joel David Hamkins