The ubiquity of pseudofinite metric structures (joint work with Bradd Hart) Isaac Goldbring University of California, Irvine Workshop on model theory of pseudofinite structures April 2018 Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 1 / 23

Motivation Motivation 1 2 The proof of the main theorem The general case 3 Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 2 / 23

Motivation The Urysohn space Definition The Urysohn space U is the unique (up to isometry) Polish metric space that is universal and ultrahomogeneous . The theory of the Urysohn space can be seen as the continuous analogue of either the theory of a pure infinite set or the theory of the random graph. Similarities: ω -categorical, QE, model completion of the relevant class, Fraïsse limit of the relevant class,... Differences: Not stable or simple. An infinite set and the random graph are both pseudofinite. But: Question (G. and Lopes) Is U pseudofinite? Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 3 / 23

Motivation The Urysohn space Definition The Urysohn space U is the unique (up to isometry) Polish metric space that is universal and ultrahomogeneous . The theory of the Urysohn space can be seen as the continuous analogue of either the theory of a pure infinite set or the theory of the random graph. Similarities: ω -categorical, QE, model completion of the relevant class, Fraïsse limit of the relevant class,... Differences: Not stable or simple. An infinite set and the random graph are both pseudofinite. But: Question (G. and Lopes) Is U pseudofinite? Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 3 / 23

Motivation The Urysohn space Definition The Urysohn space U is the unique (up to isometry) Polish metric space that is universal and ultrahomogeneous . The theory of the Urysohn space can be seen as the continuous analogue of either the theory of a pure infinite set or the theory of the random graph. Similarities: ω -categorical, QE, model completion of the relevant class, Fraïsse limit of the relevant class,... Differences: Not stable or simple. An infinite set and the random graph are both pseudofinite. But: Question (G. and Lopes) Is U pseudofinite? Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 3 / 23

Motivation A crash course in continuous logic Structures are complete metric spaces equipped with distinguished constants, functions, and [ 0 , 1 ] -valued predicates. Signatures, besides providing arities, also must provide moduli of uniform continuity ∆ F and ∆ P for function symbols F and predicate symbols P . Terms are built as usual. (Predictable modulus of uniform continuity ∆ t .) Atomic formulae: P ( t 1 , . . . , t n ) and d ( t 1 , t 2 ) . General formulae: close under continuous connectives u : [ 0 , 1 ] n → [ 0 , 1 ] and quantifiers sup and inf. (Predictable modulus of uniform continuity ∆ ϕ .) Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 4 / 23

Motivation Pseudofiniteness in continuous logic Theorem Suppose that L is a metric signature and M is an L-structure. Then the following are equivalent: 1 If σ is an L-sentence such that σ A = 0 for every finite (resp. compact) L-structure A, then σ M = 0 . 2 If σ is an L-sentence with σ M = 0 , then for every ǫ > 0 , there is a finite (resp. compact) L-structure A such that σ A < ǫ . 3 There is a family ( A i : i ∈ I ) of finite (resp. compact) L-structures and an ultrafilter U on I such that M ≡ � U A i . Definition If M satisfies any of the equivalent conditions in the previous theorem, we say that M is pseudofinite (resp. pseudocompact ). Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 5 / 23

Motivation Pseudofiniteness in continuous logic Theorem Suppose that L is a metric signature and M is an L-structure. Then the following are equivalent: 1 If σ is an L-sentence such that σ A = 0 for every finite (resp. compact) L-structure A, then σ M = 0 . 2 If σ is an L-sentence with σ M = 0 , then for every ǫ > 0 , there is a finite (resp. compact) L-structure A such that σ A < ǫ . 3 There is a family ( A i : i ∈ I ) of finite (resp. compact) L-structures and an ultrafilter U on I such that M ≡ � U A i . Definition If M satisfies any of the equivalent conditions in the previous theorem, we say that M is pseudofinite (resp. pseudocompact ). Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 5 / 23

Motivation Basic facts and examples Facts and examples (G. and Lopes) For classical structures viewed as metric structures, the notions classical pseudofiniteness, pseudofiniteness, and pseudocompactness all coincide. If L is a relational metric signature, then the notions of pseudofiniteness and pseudocompactness coincide. For arbitrary signatures, they “almost” coincide. Atomless probability algebras (even expanded by a generic automorphism) are pseudofinite. Infinite-dimensional Hilbert spaces (even expanded by a generic automorphism) are pseudocompact. Richly branching R -trees are pseudofinite. Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 6 / 23

Motivation Basic facts and examples Facts and examples (G. and Lopes) For classical structures viewed as metric structures, the notions classical pseudofiniteness, pseudofiniteness, and pseudocompactness all coincide. If L is a relational metric signature, then the notions of pseudofiniteness and pseudocompactness coincide. For arbitrary signatures, they “almost” coincide. Atomless probability algebras (even expanded by a generic automorphism) are pseudofinite. Infinite-dimensional Hilbert spaces (even expanded by a generic automorphism) are pseudocompact. Richly branching R -trees are pseudofinite. Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 6 / 23

Motivation Basic facts and examples Facts and examples (G. and Lopes) For classical structures viewed as metric structures, the notions classical pseudofiniteness, pseudofiniteness, and pseudocompactness all coincide. If L is a relational metric signature, then the notions of pseudofiniteness and pseudocompactness coincide. For arbitrary signatures, they “almost” coincide. Atomless probability algebras (even expanded by a generic automorphism) are pseudofinite. Infinite-dimensional Hilbert spaces (even expanded by a generic automorphism) are pseudocompact. Richly branching R -trees are pseudofinite. Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 6 / 23

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