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 ZF 2 satisfy fully categorical theories? In many instances we can form categorical theories by augmenting ZF 2 with a first-order sentence, forming a theory ZF 2 + σ that is true in exactly one V κ . 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 V κ is characterized by the theory ZF 2 + “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 V κ is characterized by the theory ZF 2 + “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 V κ is characterized by the theory ZF 2 + “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 V κ is categorically characterized by ZF 2 + 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 V κ is categorically characterized by ZF 2 + σ . 2 κ is first-order theory categorical , if there is a first-order theory T in the language of set theory, such that V κ is categorically characterized by ZF 2 + 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 V κ is categorically characterized by ZF 2 + σ . 2 κ is first-order theory categorical , if there is a first-order theory T in the language of set theory, such that V κ is categorically characterized by ZF 2 + 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 V κ is categorically characterized by ZF 2 + 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 V κ is categorically characterized by ZF 2 + σ . 2 κ is first-order theory categorical , if there is a first-order theory T in the language of set theory, such that V κ is categorically characterized by ZF 2 + 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 V κ is categorically characterized by ZF 2 + σ . 4 κ is second-order theory categorical , if there is a second-order theory T in the language of set theory, such that V κ is categorically characterized by ZF 2 + 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 V κ is categorically characterized by ZF 2 + σ . 2 κ is first-order theory categorical , if there is a first-order theory T in the language of set theory, such that V κ is categorically characterized by ZF 2 + 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 V κ is categorically characterized by ZF 2 + σ . 4 κ is second-order theory categorical , if there is a second-order theory T in the language of set theory, such that V κ is categorically characterized by ZF 2 + T . Generalize to Σ m n -categoricity or even Σ α 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 V κ | = ZFC 2 , 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 V κ | = 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 V κ | = ZFC 2 , 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 V κ | = σ . And similarly with the other notions. This is about categorical characterizations of V κ 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 α th 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 α th 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 α th 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 ω CK 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 ω 1 th inaccessible cardinal is sententially categorical, and the ω 2 nd, 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 ω 1 th inaccessible cardinal is sententially categorical, and the ω 2 nd, and more. The ω α th 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 ω 1 th inaccessible cardinal is sententially categorical, and the ω 2 nd, and more. The ω α th 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 V θ . If κ is categorical and θ > κ , then V θ 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 V θ . If κ is categorical and θ > κ , then V θ knows that κ is categorical. Proof. V θ can verify that V κ 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 V θ . If κ is categorical and θ > κ , then V θ knows that κ is categorical. Proof. V θ can verify that V κ has the theory that it has. And there are fewer challenges to categoricity in V θ 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 V θ . If κ is categorical and θ > κ , then V θ knows that κ is categorical. Proof. V θ can verify that V κ has the theory that it has. And there are fewer challenges to categoricity in V θ than in V . So κ is categorical inside V θ . 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 V κ satisfies ψ , and this will characterize in a first-order manner. κ 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. V κ 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. V κ can see characterizing assertions about the smaller inaccessibles. So no inaccessible δ < κ can have V δ with same theory as V κ . 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. V κ can see characterizing assertions about the smaller inaccessibles. So no inaccessible δ < κ can have V δ with same theory as V κ . And no larger θ > κ can have same theory, since in V θ 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 V λ 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 V λ 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 V λ 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 V λ . 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 V λ 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 V λ . 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 V θ . 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 V θ . 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 V θ . Proof. Assume κ is not sententially categorical. For each sentence σ true in V κ , find κ σ � = κ such that σ also true in V κ σ . 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 V θ . Proof. Assume κ is not sententially categorical. For each sentence σ true in V κ , find κ σ � = κ such that σ also true in V κ σ . If θ > κ and above all κ σ , then V θ 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 V δ ≺ V κ for a stationary set of δ , which therefore includes many inaccessible cardinals. So V κ 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 Π 1 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 Π 1 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 V κ ≺ V λ . 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 V κ ≺ V λ . Weaker than Mahlo. Actually only need a single nontrivial instance V κ ≺ V λ . 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 V κ ≺ V λ . Weaker than Mahlo. Actually only need a single nontrivial instance V κ ≺ V λ . And actually only need V κ ≡ V λ 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 V κ ≺ V λ , 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 V κ ≺ V λ , 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 V κ ≺ V λ , 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 V κ ≺ V λ , 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 V κ ≡ V λ without V κ ≺ V λ . 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 V θ 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 V θ 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 V θ 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 V θ 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
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