The ubiquity of pseudofinite metric structures (joint work with - - PowerPoint PPT Presentation

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

1

Motivation

2

The proof of the main theorem

3

The general case

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

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(t1, . . . , tn) and d(t1, t2). 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 σA = 0 for every finite (resp.

compact) L-structure A, then σM = 0.

2 If σ is an L-sentence with σM = 0, then for every ǫ > 0, there is a

finite (resp. compact) L-structure A such that σA < ǫ.

3 There is a family (Ai : i ∈ I) of finite (resp. compact) L-structures

and an ultrafilter U on I such that M ≡

U Ai.

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 σA = 0 for every finite (resp.

compact) L-structure A, then σM = 0.

2 If σ is an L-sentence with σM = 0, then for every ǫ > 0, there is a

finite (resp. compact) L-structure A such that σA < ǫ.

3 There is a family (Ai : i ∈ I) of finite (resp. compact) L-structures

and an ultrafilter U on I such that M ≡

U Ai.

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

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

Main theorem

Theorem (G. and Hart) U is pseudofinite. Recall that a topological space is perfect if it has no isolated points. We call a metric structure perfect if its underlying metric space is perfect. Theorem If L is a relational continuous signature and M is a perfect L-structure, then M is pseudofinite!

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 7 / 23

Motivation

Main theorem

Theorem (G. and Hart) U is pseudofinite. Recall that a topological space is perfect if it has no isolated points. We call a metric structure perfect if its underlying metric space is perfect. Theorem If L is a relational continuous signature and M is a perfect L-structure, then M is pseudofinite!

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 7 / 23

Motivation

Main theorem

Theorem (G. and Hart) U is pseudofinite. Recall that a topological space is perfect if it has no isolated points. We call a metric structure perfect if its underlying metric space is perfect. Theorem If L is a relational continuous signature and M is a perfect L-structure, then M is pseudofinite!

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 7 / 23

The proof of the main theorem

1

Motivation

2

The proof of the main theorem

3

The general case

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 8 / 23

The proof of the main theorem

A DST prerequisite-reasonable measures

Definition A Borel probability measure on a Hausdorff space X is called reasonable if it is atomless and strictly positive, that is, every nonempty open subset of X has positive measure. Fact A Polish space admits a reasonable measure if and only if it is perfect. Fact Suppose that µ is an atomless measure on X with product measure µm on X m. Then µm(X m

= ) = 1, where

X m

= := {(x1, . . . , xm) ∈ X m : xi = xj for i = j}.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 9 / 23

The proof of the main theorem

A DST prerequisite-reasonable measures

Definition A Borel probability measure on a Hausdorff space X is called reasonable if it is atomless and strictly positive, that is, every nonempty open subset of X has positive measure. Fact A Polish space admits a reasonable measure if and only if it is perfect. Fact Suppose that µ is an atomless measure on X with product measure µm on X m. Then µm(X m

= ) = 1, where

X m

= := {(x1, . . . , xm) ∈ X m : xi = xj for i = j}.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 9 / 23

The proof of the main theorem

A DST prerequisite-reasonable measures

Definition A Borel probability measure on a Hausdorff space X is called reasonable if it is atomless and strictly positive, that is, every nonempty open subset of X has positive measure. Fact A Polish space admits a reasonable measure if and only if it is perfect. Fact Suppose that µ is an atomless measure on X with product measure µm on X m. Then µm(X m

= ) = 1, where

X m

= := {(x1, . . . , xm) ∈ X m : xi = xj for i = j}.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 9 / 23

The proof of the main theorem

Kind sentences

Throughout this section, L is a relational signature. Definition A kind L-sentence is one of the form sup

• x

infy min  

i=j

d(xi, xj), ϕ(x, y)   , where ϕ is a quantifier-free L-formula. The point: Asking that a kind sentence vanish is the same as asking that tuples of distinct points have approximate witnesses to ϕ = 0.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 10 / 23

The proof of the main theorem

Kind sentences

Throughout this section, L is a relational signature. Definition A kind L-sentence is one of the form sup

• x

infy min  

i=j

d(xi, xj), ϕ(x, y)   , where ϕ is a quantifier-free L-formula. The point: Asking that a kind sentence vanish is the same as asking that tuples of distinct points have approximate witnesses to ϕ = 0.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 10 / 23

The proof of the main theorem

The key result

Theorem Suppose that L is a relational signature, M is a separable, perfect L-structure, and σ is a kind sentence such that σM = 0. Then for any ǫ > 0 and any reasonable measure µ on M, we have lim

m→∞ µm{(a1, . . . , am) ∈ Mm = : σ¯ a < ǫ} = 1,

where ¯ a represents the substructure of M formed by a1, . . . , am.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 11 / 23

The proof of the main theorem

Proof of the key result

Suppose that σ and ϕ(x, y) are as in the definition of kind sentence with x = (x1, . . . , xn) and y = (y1, . . . , yk). Fix m ≥ n + k. Write m = n + qk + r with r < k. For i = 1, . . . , q, set Ai := { a ∈ Mm

= : ϕM(a1, . . . , an, an+(i−1)k+1, . . . , an+ik) < ǫ}.

Since Ai is a nonempty open subset of Mm (as σM = 0 and M is perfect) and µ is reasonable, we have that µm(Ai) > 0. By Sm-invariance of µm, it follows that µm(Ai) = µm(Aj) for all i, j = 1, . . . , q; call this common value p. It is important to note that p is independent of m. Let B := q

i=1(Mm = \ Ai). Since the Ai’s are independent, we have

that µm(B) = (1 − p)q = (1 − p)⌊ m−n

k

⌋.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 12 / 23

The proof of the main theorem

Proof of the key result

Suppose that σ and ϕ(x, y) are as in the definition of kind sentence with x = (x1, . . . , xn) and y = (y1, . . . , yk). Fix m ≥ n + k. Write m = n + qk + r with r < k. For i = 1, . . . , q, set Ai := { a ∈ Mm

= : ϕM(a1, . . . , an, an+(i−1)k+1, . . . , an+ik) < ǫ}.

Since Ai is a nonempty open subset of Mm (as σM = 0 and M is perfect) and µ is reasonable, we have that µm(Ai) > 0. By Sm-invariance of µm, it follows that µm(Ai) = µm(Aj) for all i, j = 1, . . . , q; call this common value p. It is important to note that p is independent of m. Let B := q

i=1(Mm = \ Ai). Since the Ai’s are independent, we have

that µm(B) = (1 − p)q = (1 − p)⌊ m−n

k

⌋.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 12 / 23

The proof of the main theorem

Proof of the key result

Suppose that σ and ϕ(x, y) are as in the definition of kind sentence with x = (x1, . . . , xn) and y = (y1, . . . , yk). Fix m ≥ n + k. Write m = n + qk + r with r < k. For i = 1, . . . , q, set Ai := { a ∈ Mm

= : ϕM(a1, . . . , an, an+(i−1)k+1, . . . , an+ik) < ǫ}.

Since Ai is a nonempty open subset of Mm (as σM = 0 and M is perfect) and µ is reasonable, we have that µm(Ai) > 0. By Sm-invariance of µm, it follows that µm(Ai) = µm(Aj) for all i, j = 1, . . . , q; call this common value p. It is important to note that p is independent of m. Let B := q

i=1(Mm = \ Ai). Since the Ai’s are independent, we have

that µm(B) = (1 − p)q = (1 − p)⌊ m−n

k

⌋.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 12 / 23

The proof of the main theorem

Proof of the key result

Suppose that σ and ϕ(x, y) are as in the definition of kind sentence with x = (x1, . . . , xn) and y = (y1, . . . , yk). Fix m ≥ n + k. Write m = n + qk + r with r < k. For i = 1, . . . , q, set Ai := { a ∈ Mm

= : ϕM(a1, . . . , an, an+(i−1)k+1, . . . , an+ik) < ǫ}.

Since Ai is a nonempty open subset of Mm (as σM = 0 and M is perfect) and µ is reasonable, we have that µm(Ai) > 0. By Sm-invariance of µm, it follows that µm(Ai) = µm(Aj) for all i, j = 1, . . . , q; call this common value p. It is important to note that p is independent of m. Let B := q

i=1(Mm = \ Ai). Since the Ai’s are independent, we have

that µm(B) = (1 − p)q = (1 − p)⌊ m−n

k

⌋.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 12 / 23

The proof of the main theorem

Proof of the key result

Suppose that σ and ϕ(x, y) are as in the definition of kind sentence with x = (x1, . . . , xn) and y = (y1, . . . , yk). Fix m ≥ n + k. Write m = n + qk + r with r < k. For i = 1, . . . , q, set Ai := { a ∈ Mm

= : ϕM(a1, . . . , an, an+(i−1)k+1, . . . , an+ik) < ǫ}.

Since Ai is a nonempty open subset of Mm (as σM = 0 and M is perfect) and µ is reasonable, we have that µm(Ai) > 0. By Sm-invariance of µm, it follows that µm(Ai) = µm(Aj) for all i, j = 1, . . . , q; call this common value p. It is important to note that p is independent of m. Let B := q

i=1(Mm = \ Ai). Since the Ai’s are independent, we have

that µm(B) = (1 − p)q = (1 − p)⌊ m−n

k

⌋.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 12 / 23

The proof of the main theorem

Proof of the key result

Suppose that σ and ϕ(x, y) are as in the definition of kind sentence with x = (x1, . . . , xn) and y = (y1, . . . , yk). Fix m ≥ n + k. Write m = n + qk + r with r < k. For i = 1, . . . , q, set Ai := { a ∈ Mm

= : ϕM(a1, . . . , an, an+(i−1)k+1, . . . , an+ik) < ǫ}.

Since Ai is a nonempty open subset of Mm (as σM = 0 and M is perfect) and µ is reasonable, we have that µm(Ai) > 0. By Sm-invariance of µm, it follows that µm(Ai) = µm(Aj) for all i, j = 1, . . . , q; call this common value p. It is important to note that p is independent of m. Let B := q

i=1(Mm = \ Ai). Since the Ai’s are independent, we have

that µm(B) = (1 − p)q = (1 − p)⌊ m−n

k

⌋.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 12 / 23

The proof of the main theorem

Proof of the key result

Suppose that σ and ϕ(x, y) are as in the definition of kind sentence with x = (x1, . . . , xn) and y = (y1, . . . , yk). Fix m ≥ n + k. Write m = n + qk + r with r < k. For i = 1, . . . , q, set Ai := { a ∈ Mm

= : ϕM(a1, . . . , an, an+(i−1)k+1, . . . , an+ik) < ǫ}.

Since Ai is a nonempty open subset of Mm (as σM = 0 and M is perfect) and µ is reasonable, we have that µm(Ai) > 0. By Sm-invariance of µm, it follows that µm(Ai) = µm(Aj) for all i, j = 1, . . . , q; call this common value p. It is important to note that p is independent of m. Let B := q

i=1(Mm = \ Ai). Since the Ai’s are independent, we have

that µm(B) = (1 − p)q = (1 − p)⌊ m−n

k

⌋.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 12 / 23

The proof of the main theorem

Proof of the key result (cont’d)

For G = {i1, . . . , in} ∈ [m]n, let BG be defined exactly as B except replacing {1, . . . , n} with {i1, . . . , in}. Again, by Sm-invariance of µm, we have that µm(BG) = (1 − p)⌊ m−n

k

⌋.

The set that we are really interested in is C := {(a1, . . . , am) ∈ Mm

= : σ¯ a ≥ ǫ}.

It follows that µm(C) ≤ µm  

G∈[m]n

BG   ≤ m n

• (1 − p)⌊ m−n

k

⌋.

The right-hand side goes to 0 as m → ∞, yielding the desired result.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 13 / 23

The proof of the main theorem

Proof of the key result (cont’d)

For G = {i1, . . . , in} ∈ [m]n, let BG be defined exactly as B except replacing {1, . . . , n} with {i1, . . . , in}. Again, by Sm-invariance of µm, we have that µm(BG) = (1 − p)⌊ m−n

k

⌋.

The set that we are really interested in is C := {(a1, . . . , am) ∈ Mm

= : σ¯ a ≥ ǫ}.

It follows that µm(C) ≤ µm  

G∈[m]n

BG   ≤ m n

• (1 − p)⌊ m−n

k

⌋.

The right-hand side goes to 0 as m → ∞, yielding the desired result.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 13 / 23

The proof of the main theorem

Proof of the key result (cont’d)

For G = {i1, . . . , in} ∈ [m]n, let BG be defined exactly as B except replacing {1, . . . , n} with {i1, . . . , in}. Again, by Sm-invariance of µm, we have that µm(BG) = (1 − p)⌊ m−n

k

⌋.

The set that we are really interested in is C := {(a1, . . . , am) ∈ Mm

= : σ¯ a ≥ ǫ}.

It follows that µm(C) ≤ µm  

G∈[m]n

BG   ≤ m n

• (1 − p)⌊ m−n

k

⌋.

The right-hand side goes to 0 as m → ∞, yielding the desired result.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 13 / 23

The proof of the main theorem

Proof of the key result (cont’d)

For G = {i1, . . . , in} ∈ [m]n, let BG be defined exactly as B except replacing {1, . . . , n} with {i1, . . . , in}. Again, by Sm-invariance of µm, we have that µm(BG) = (1 − p)⌊ m−n

k

⌋.

The set that we are really interested in is C := {(a1, . . . , am) ∈ Mm

= : σ¯ a ≥ ǫ}.

It follows that µm(C) ≤ µm  

G∈[m]n

BG   ≤ m n

• (1 − p)⌊ m−n

k

⌋.

The right-hand side goes to 0 as m → ∞, yielding the desired result.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 13 / 23

The proof of the main theorem

Proof of the key result (cont’d)

For G = {i1, . . . , in} ∈ [m]n, let BG be defined exactly as B except replacing {1, . . . , n} with {i1, . . . , in}. Again, by Sm-invariance of µm, we have that µm(BG) = (1 − p)⌊ m−n

k

⌋.

The set that we are really interested in is C := {(a1, . . . , am) ∈ Mm

= : σ¯ a ≥ ǫ}.

It follows that µm(C) ≤ µm  

G∈[m]n

BG   ≤ m n

• (1 − p)⌊ m−n

k

⌋.

The right-hand side goes to 0 as m → ∞, yielding the desired result.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 13 / 23

The proof of the main theorem

Proof of the main theorem

Theorem Suppose that L is a relational signature and M is a perfect L-structure. Then M is pseudofinite. Proof. Without loss of generality, we may assume that M is ∀∃-axiomatizable. (Morelyize!) Without loss of generality, we may further assume that the axiomatization is kind. Without loss of generality, we may further assume that M is separable. By the key proposition and compactness, we have M is pseudofinite.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 14 / 23

The proof of the main theorem

Proof of the main theorem

Theorem Suppose that L is a relational signature and M is a perfect L-structure. Then M is pseudofinite. Proof. Without loss of generality, we may assume that M is ∀∃-axiomatizable. (Morelyize!) Without loss of generality, we may further assume that the axiomatization is kind. Without loss of generality, we may further assume that M is separable. By the key proposition and compactness, we have M is pseudofinite.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 14 / 23

The proof of the main theorem

An approximate 0-1 law

Let L- Strm be the set of L-structures with universe {1, . . . , m} and L- StrM

m be the subset of L- Strm consisting of those structures that

embed into M. Given a reasonable measure µ on M, we let Φ∗

mµm denote the

pushforward of the measure µm via Φm : Mm

= → L- StrM m.

Corollary (Approximate 0-1 law) Suppose that L is a relational signature and that M is a perfect L-structure. Fix an L-sentence σ and let r := σM. Then for any reasonable measure µ on M and any ǫ > 0, we have lim

m→∞ µm{(a1, . . . , am) ∈ Mm = : |σ − r|¯ a < ǫ} = 1,

whence lim

m→∞ Φ∗ mµm{A ∈ L- StrM m : |σ − r|A < ǫ} = 1.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 15 / 23

The proof of the main theorem

An approximate 0-1 law

Let L- Strm be the set of L-structures with universe {1, . . . , m} and L- StrM

m be the subset of L- Strm consisting of those structures that

embed into M. Given a reasonable measure µ on M, we let Φ∗

mµm denote the

pushforward of the measure µm via Φm : Mm

= → L- StrM m.

Corollary (Approximate 0-1 law) Suppose that L is a relational signature and that M is a perfect L-structure. Fix an L-sentence σ and let r := σM. Then for any reasonable measure µ on M and any ǫ > 0, we have lim

m→∞ µm{(a1, . . . , am) ∈ Mm = : |σ − r|¯ a < ǫ} = 1,

whence lim

m→∞ Φ∗ mµm{A ∈ L- StrM m : |σ − r|A < ǫ} = 1.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 15 / 23

The proof of the main theorem

The case of finite metric spaces

Let Metm denote the set of metric spaces with underlying set {1, . . . , m}. Let Ψm : Um

= → Metm be the natural surjection.

For a reasonable measure µ on U, let Ψ∗

mµm denote the

pushforward measure. Corollary (Approximate 0-1 law for finite metric spaces) Let σ be any sentence in the empty continuous signature and let r := σU. Then for any reasonable measure µ on U and any ǫ > 0, we have lim

m→∞ Ψ∗ mµm{A ∈ Metm : |σ − r|A < ǫ} = 1.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 16 / 23

The proof of the main theorem

The case of finite metric spaces

Let Metm denote the set of metric spaces with underlying set {1, . . . , m}. Let Ψm : Um

= → Metm be the natural surjection.

For a reasonable measure µ on U, let Ψ∗

mµm denote the

pushforward measure. Corollary (Approximate 0-1 law for finite metric spaces) Let σ be any sentence in the empty continuous signature and let r := σU. Then for any reasonable measure µ on U and any ǫ > 0, we have lim

m→∞ Ψ∗ mµm{A ∈ Metm : |σ − r|A < ǫ} = 1.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 16 / 23

The general case

1

Motivation

2

The proof of the main theorem

3

The general case

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 17 / 23

The general case

Function symbols?

Let L be a countable metric signature (possibly with function symbols). Let (rk) be an enumeration of Q ∩ [0, 1] and let (fk) be an enumeration of the L-terms. For m ∈ N, we define a language L(m) which only differs from L by changing the modulus of uniform continuity for f1, . . . , fm by declaring ∆fj,L(m)(rk + 1

m) := ∆fj,L(rk) for j, k = 1, . . . , m.

Lemma Suppose that p : N → N is such that limm→∞ p(m) = ∞. Further suppose that, for each m, Am is an L(p(m))-structure. Then for any nonprincipal ultrafilter U on N, there is a well-defined ultraproduct

• U Am which is, moreover, an actual L-structure.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 18 / 23

The general case

Function symbols?

Let L be a countable metric signature (possibly with function symbols). Let (rk) be an enumeration of Q ∩ [0, 1] and let (fk) be an enumeration of the L-terms. For m ∈ N, we define a language L(m) which only differs from L by changing the modulus of uniform continuity for f1, . . . , fm by declaring ∆fj,L(m)(rk + 1

m) := ∆fj,L(rk) for j, k = 1, . . . , m.

Lemma Suppose that p : N → N is such that limm→∞ p(m) = ∞. Further suppose that, for each m, Am is an L(p(m))-structure. Then for any nonprincipal ultrafilter U on N, there is a well-defined ultraproduct

• U Am which is, moreover, an actual L-structure.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 18 / 23

The general case

A theorem for function symbols

Here is a version of our preceding theorem for arbitrary signatures: Theorem Suppose that M is a perfect L-structure. Then there is p : N → N with limm→∞ p(m) = ∞ and finite L(p(m))-structures Am such that, for any nonprincipal ultrafilter U on N, we have M ≡

U Am.

Idea of proof Relationalize. Quote the relational version of the theorem. Figure out a way to sensibly make the finite relational structures L(p(m))-structures for suitable p(m)’s. Awkward!

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 19 / 23

The general case

A theorem for function symbols

Here is a version of our preceding theorem for arbitrary signatures: Theorem Suppose that M is a perfect L-structure. Then there is p : N → N with limm→∞ p(m) = ∞ and finite L(p(m))-structures Am such that, for any nonprincipal ultrafilter U on N, we have M ≡

U Am.

Idea of proof Relationalize. Quote the relational version of the theorem. Figure out a way to sensibly make the finite relational structures L(p(m))-structures for suitable p(m)’s. Awkward!

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 19 / 23

The general case

A theorem for function symbols

Here is a version of our preceding theorem for arbitrary signatures: Theorem Suppose that M is a perfect L-structure. Then there is p : N → N with limm→∞ p(m) = ∞ and finite L(p(m))-structures Am such that, for any nonprincipal ultrafilter U on N, we have M ≡

U Am.

Idea of proof Relationalize. Quote the relational version of the theorem. Figure out a way to sensibly make the finite relational structures L(p(m))-structures for suitable p(m)’s. Awkward!

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 19 / 23

The general case

General structures

Definition A vocabulary V is a set of constant, function, and predicate symbols, the latter two coming with arities. Given a vocabulary V, a general V-structure is a universe together with interpretations of the symbols in V, noting that the interpretation of a predicate symbol is simply a function from the appropriate cartesian power of the universe into [0, 1]. Definition Given a vocabulary V, a metric signature L over V is a specification

• f a distinguished binary predicate d ∈ V together with moduli of

uniform continuity for each of the function and predicate symbols in V. Given a metric signature L, one then defines the notion of an L-structure as in (ordinary) continuous logic.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 20 / 23

The general case

General structures

Definition A vocabulary V is a set of constant, function, and predicate symbols, the latter two coming with arities. Given a vocabulary V, a general V-structure is a universe together with interpretations of the symbols in V, noting that the interpretation of a predicate symbol is simply a function from the appropriate cartesian power of the universe into [0, 1]. Definition Given a vocabulary V, a metric signature L over V is a specification

• f a distinguished binary predicate d ∈ V together with moduli of

uniform continuity for each of the function and predicate symbols in V. Given a metric signature L, one then defines the notion of an L-structure as in (ordinary) continuous logic.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 20 / 23

The general case

General theorems

Corollary Suppose that L is a metric signature over the vocabulary V and M is a perfect L-structure. Then M is pseudofinite as a general V-structure. In fact: Corollary Suppose that V is a countable vocabulary and that M is a perfect general V-structure. Then M is pseudofinite. Uses the notion of a metric expansion of a general structure.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 21 / 23

The general case

General theorems

Corollary Suppose that L is a metric signature over the vocabulary V and M is a perfect L-structure. Then M is pseudofinite as a general V-structure. In fact: Corollary Suppose that V is a countable vocabulary and that M is a perfect general V-structure. Then M is pseudofinite. Uses the notion of a metric expansion of a general structure.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 21 / 23

The general case

Open questions

Question Are all perfect metric structures pseudofinite? Question Let G be the Gurarij Banach space (the Banach space analogue of U). Is G pseudofinite? Pseudocompact? Pseudo-finite dimensional? A positive solution to the latter question could give some sort of approximate 0-1 law for finite-dimensional Banach spaces.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 22 / 23

The general case

Open questions

Question Are all perfect metric structures pseudofinite? Question Let G be the Gurarij Banach space (the Banach space analogue of U). Is G pseudofinite? Pseudocompact? Pseudo-finite dimensional? A positive solution to the latter question could give some sort of approximate 0-1 law for finite-dimensional Banach spaces.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 22 / 23

The general case

References

ISAAC GOLDBRING AND BRADD HART, The Urysohn space is pseudofinite, preprint. arXiv 1708.05321. ISAAC GOLDBRING AND VINICIUS CIFU LOPES, Pseudofinite and pseudocompact metric structures, Notre Dame Journal of Formal Logic 56 (2015), 493-510.

Isaac Goldbring (UCI) Pseudofinite metric strucures Pseudofinite-fest April 2018 23 / 23