Many-Sorted First-Order Model Theory Lecture 8 25 th June, 2020 1 / - - PowerPoint PPT Presentation

Many-Sorted First-Order Model Theory

Lecture 8 25th June, 2020

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Applications of ultraproducts

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Fancy models via ultraproducts

Example 1 (Growing chains)

Let Cn be the n-element chain. Let U be a non-principal filter on ω. Then,

• n<ω Cn/U is an infinite chain consisting of a copy of ω at the bottom, a

copy of dual ω at the top, and uncountably many copies of Z in between.

Example 2 (Non-standard natural numbers)

Let U be a non-principal ultrafilter on ω. Consider Nω/U. It is elementarily equivalent to N. Take the element (1, 2, 3, 4, . . . )/U. It is strictly greater than any standard natural number.

Example 3 (Infinitesimals)

Let U be a non-principal ultrafilter on ω. Consider Rω/U. It is elementarily equivalent to R. Take the element (1, 1

2, 1 3, 1 4, . . . )/U. It is

strictly greater than 0, yet strictly smaller than any standard real number.

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Fancy models via ultraproducts

Example 1 (Growing chains)

Let Cn be the n-element chain. Let U be a non-principal filter on ω. Then,

• n<ω Cn/U is an infinite chain consisting of a copy of ω at the bottom, a

copy of dual ω at the top, and uncountably many copies of Z in between.

Example 2 (Non-standard natural numbers)

Let U be a non-principal ultrafilter on ω. Consider Nω/U. It is elementarily equivalent to N. Take the element (1, 2, 3, 4, . . . )/U. It is strictly greater than any standard natural number.

Example 3 (Infinitesimals)

Let U be a non-principal ultrafilter on ω. Consider Rω/U. It is elementarily equivalent to R. Take the element (1, 1

2, 1 3, 1 4, . . . )/U. It is

strictly greater than 0, yet strictly smaller than any standard real number.

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Fancy models via ultraproducts

Example 1 (Growing chains)

Let Cn be the n-element chain. Let U be a non-principal filter on ω. Then,

• n<ω Cn/U is an infinite chain consisting of a copy of ω at the bottom, a

copy of dual ω at the top, and uncountably many copies of Z in between.

Example 2 (Non-standard natural numbers)

Let U be a non-principal ultrafilter on ω. Consider Nω/U. It is elementarily equivalent to N. Take the element (1, 2, 3, 4, . . . )/U. It is strictly greater than any standard natural number.

Example 3 (Infinitesimals)

Let U be a non-principal ultrafilter on ω. Consider Rω/U. It is elementarily equivalent to R. Take the element (1, 1

2, 1 3, 1 4, . . . )/U. It is

strictly greater than 0, yet strictly smaller than any standard real number.

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Elementary classes and ultraproducts

Definition 4

A class C of models is called elementary, if C is the class of models of some set T of sentences, that is, C = Mod(T).

Lemma 5

If C is an elementary class, then C is closed under ultraproducts. ◮ The next theorem was first proved by Keisler, under the assumption

• f GCH. Later Shelah proved it with no special assumptions.

◮ The proof requires sophisticated infinite combinatorics, and is beyond the scope if the subject.

Theorem 6 (Keisler, Shelah)

Two similar structures A and B are elementarily equivalent if and only if there is a set I and an ultrafilter U on I such that AI/U ∼ = BI/U.

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Elementary classes and ultraproducts

Definition 4

A class C of models is called elementary, if C is the class of models of some set T of sentences, that is, C = Mod(T).

Lemma 5

If C is an elementary class, then C is closed under ultraproducts. ◮ The next theorem was first proved by Keisler, under the assumption

• f GCH. Later Shelah proved it with no special assumptions.

◮ The proof requires sophisticated infinite combinatorics, and is beyond the scope if the subject.

Theorem 6 (Keisler, Shelah)

Two similar structures A and B are elementarily equivalent if and only if there is a set I and an ultrafilter U on I such that AI/U ∼ = BI/U.

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Elementary classes and ultraproducts

Definition 4

A class C of models is called elementary, if C is the class of models of some set T of sentences, that is, C = Mod(T).

Lemma 5

If C is an elementary class, then C is closed under ultraproducts. ◮ The next theorem was first proved by Keisler, under the assumption

• f GCH. Later Shelah proved it with no special assumptions.

◮ The proof requires sophisticated infinite combinatorics, and is beyond the scope if the subject.

Theorem 6 (Keisler, Shelah)

Two similar structures A and B are elementarily equivalent if and only if there is a set I and an ultrafilter U on I such that AI/U ∼ = BI/U.

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Elementary classes and ultraproducts

Definition 4

A class C of models is called elementary, if C is the class of models of some set T of sentences, that is, C = Mod(T).

Lemma 5

If C is an elementary class, then C is closed under ultraproducts. ◮ The next theorem was first proved by Keisler, under the assumption

• f GCH. Later Shelah proved it with no special assumptions.

◮ The proof requires sophisticated infinite combinatorics, and is beyond the scope if the subject.

Theorem 6 (Keisler, Shelah)

Two similar structures A and B are elementarily equivalent if and only if there is a set I and an ultrafilter U on I such that AI/U ∼ = BI/U.

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Elementary classes characterised

Theorem 7

A ≡ B if and only if there is an elementary embedding of A into an ultrapower of B. More generally, the following is true for any class K of structures of the same signature Σ:

• 1. K is the class of all Σ-models of some set of first order sentences.
• 2. K is closed under elementary embeddings and ultraproducts.
• 3. K = EPU(K′) for some class of Σ-structures K′ (E denotes

elementary embeddings and PU ultraproducts).

Proof.

◮ (1) ⇒ (2) is immediate, as elementary embeddings and ultraproducts preserve all sentences (n.b. closure under elementary embeddings implies closure under isomorphism). ◮ (2) ⇒ (3) by taking K′ to be K. ◮ Now assume (3) and consider Th(K′). If A | = Th(K′), then A ≡ B for some model B ∈ K′. By Theorem 7, we have AI /U ∼ = BI /U for some I and U. ◮ As A AI /U, we have A BI /U; moreover BI /U ∈ PU(K′). ◮ So, A ∈ EPU(K′), proving (1).

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Elementary classes characterised

Theorem 7

A ≡ B if and only if there is an elementary embedding of A into an ultrapower of B. More generally, the following is true for any class K of structures of the same signature Σ:

• 1. K is the class of all Σ-models of some set of first order sentences.
• 2. K is closed under elementary embeddings and ultraproducts.
• 3. K = EPU(K′) for some class of Σ-structures K′ (E denotes

elementary embeddings and PU ultraproducts).

Proof.

◮ (1) ⇒ (2) is immediate, as elementary embeddings and ultraproducts preserve all sentences (n.b. closure under elementary embeddings implies closure under isomorphism). ◮ (2) ⇒ (3) by taking K′ to be K. ◮ Now assume (3) and consider Th(K′). If A | = Th(K′), then A ≡ B for some model B ∈ K′. By Theorem 7, we have AI /U ∼ = BI /U for some I and U. ◮ As A AI /U, we have A BI /U; moreover BI /U ∈ PU(K′). ◮ So, A ∈ EPU(K′), proving (1).

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Digression: finite models

Lemma 8

Let C be a class of finite models of some signature Σ. If C contains arbitrarily large finite models, then C is not elementary.

Proof.

◮ Suppose C = Mod(T) for some theory T. ◮ Let (Ai : i ∈ ω) be a sequence of models from C with strictly increasing sizes of their

• universes. Take A =

i∈ω Ai/U, for a non-principal U.

◮ Then, A | = T, but A is infinite, so C ⊂ Mod(T), contradicting the assumption.

Finite Model Theory

Lemma 8 can be viewed as dashing all hope for a reasonable model theory of finite structures: completeness fails. Compactness also fails: say, the set {n < c : n ∈ N} (in the signature of natural numbers with an additional costant c) is finitely satisfiable in the class of finite models, but has no finite model. Yet, Finite Model Theory is alive and well. This is mostly because Ehrenfeucht-Fra¨ ıss´ e games work. We will look at them later.

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Digression: finite models

Lemma 8

Let C be a class of finite models of some signature Σ. If C contains arbitrarily large finite models, then C is not elementary.

Proof.

◮ Suppose C = Mod(T) for some theory T. ◮ Let (Ai : i ∈ ω) be a sequence of models from C with strictly increasing sizes of their

• universes. Take A =

i∈ω Ai/U, for a non-principal U.

◮ Then, A | = T, but A is infinite, so C ⊂ Mod(T), contradicting the assumption.

Finite Model Theory

Lemma 8 can be viewed as dashing all hope for a reasonable model theory of finite structures: completeness fails. Compactness also fails: say, the set {n < c : n ∈ N} (in the signature of natural numbers with an additional costant c) is finitely satisfiable in the class of finite models, but has no finite model. Yet, Finite Model Theory is alive and well. This is mostly because Ehrenfeucht-Fra¨ ıss´ e games work. We will look at them later.

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Digression: finite models

Lemma 8

Let C be a class of finite models of some signature Σ. If C contains arbitrarily large finite models, then C is not elementary.

Proof.

◮ Suppose C = Mod(T) for some theory T. ◮ Let (Ai : i ∈ ω) be a sequence of models from C with strictly increasing sizes of their

• universes. Take A =

i∈ω Ai/U, for a non-principal U.

◮ Then, A | = T, but A is infinite, so C ⊂ Mod(T), contradicting the assumption.

Finite Model Theory

Lemma 8 can be viewed as dashing all hope for a reasonable model theory of finite structures: completeness fails. Compactness also fails: say, the set {n < c : n ∈ N} (in the signature of natural numbers with an additional costant c) is finitely satisfiable in the class of finite models, but has no finite model. Yet, Finite Model Theory is alive and well. This is mostly because Ehrenfeucht-Fra¨ ıss´ e games work. We will look at them later.

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Ultraproducts and axiomatisability

Lemma 9

Let (ϕn : n < ω) be a sequence of sentences of some signature Σ, and let T be a Σ-theory. Let C be the subclass of Mod(T) such that C ∈ C iff C | = ϕn for some n < ω. Assume that C contains a set {Ci : i < ω} of models such that for every n < ω the set {i ∈ ω : Ci | = ϕn} is finite. Then, C is not an elementary class.

Proof.

◮ Let C =

n<ω Cn/U, where U is an ultrafilter containing the Fr´

echet filter. So, C | = T. ◮ Moreover, since [[ϕn]] is finite, we have C | = ¬ϕn for every n. ◮ Hence C / ∈ C, so C is not closed under ultraproducts.

Corollary 10

The following classes are not elementary: (1) all torsion groups; (2) all connected graphs; (3) all fields of non-zero characteristic.

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Ultraproducts and axiomatisability

Lemma 9

Let (ϕn : n < ω) be a sequence of sentences of some signature Σ, and let T be a Σ-theory. Let C be the subclass of Mod(T) such that C ∈ C iff C | = ϕn for some n < ω. Assume that C contains a set {Ci : i < ω} of models such that for every n < ω the set {i ∈ ω : Ci | = ϕn} is finite. Then, C is not an elementary class.

Proof.

◮ Let C =

n<ω Cn/U, where U is an ultrafilter containing the Fr´

echet filter. So, C | = T. ◮ Moreover, since [[ϕn]] is finite, we have C | = ¬ϕn for every n. ◮ Hence C / ∈ C, so C is not closed under ultraproducts.

Corollary 10

The following classes are not elementary: (1) all torsion groups; (2) all connected graphs; (3) all fields of non-zero characteristic.

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Ultraproducts and axiomatisability

Lemma 9

Let (ϕn : n < ω) be a sequence of sentences of some signature Σ, and let T be a Σ-theory. Let C be the subclass of Mod(T) such that C ∈ C iff C | = ϕn for some n < ω. Assume that C contains a set {Ci : i < ω} of models such that for every n < ω the set {i ∈ ω : Ci | = ϕn} is finite. Then, C is not an elementary class.

Proof.

◮ Let C =

n<ω Cn/U, where U is an ultrafilter containing the Fr´

echet filter. So, C | = T. ◮ Moreover, since [[ϕn]] is finite, we have C | = ¬ϕn for every n. ◮ Hence C / ∈ C, so C is not closed under ultraproducts.

Corollary 10

The following classes are not elementary: (1) all torsion groups; (2) all connected graphs; (3) all fields of non-zero characteristic.

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Failures of finite axiomatisability

Exercise 1

Prove Corollary 10 applying Lemma 9.

A technique for showing failures of finite axiomatisability

Let C = Mod(T) for some finite set T of sentences. ◮ By Lemma 5, we have that C is closed under ultraproducts. ◮ But note that also the complement of C is an elementary class, namely, A / ∈ C iff A | = ¬ T. ◮ So, C is also closed under ultraproducts. ◮ Hence, if C is not closed under ultraproducts, then C is not finitely axiomatised. Monk used this technique to prove that the class RRA of Representable Relation Algebras is not finitely axiomatised.

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Failures of finite axiomatisability

Exercise 1

Prove Corollary 10 applying Lemma 9.

A technique for showing failures of finite axiomatisability

Let C = Mod(T) for some finite set T of sentences. ◮ By Lemma 5, we have that C is closed under ultraproducts. ◮ But note that also the complement of C is an elementary class, namely, A / ∈ C iff A | = ¬ T. ◮ So, C is also closed under ultraproducts. ◮ Hence, if C is not closed under ultraproducts, then C is not finitely axiomatised. Monk used this technique to prove that the class RRA of Representable Relation Algebras is not finitely axiomatised.

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Failures of finite axiomatisability

Exercise 1

Prove Corollary 10 applying Lemma 9.

A technique for showing failures of finite axiomatisability

Let C = Mod(T) for some finite set T of sentences. ◮ By Lemma 5, we have that C is closed under ultraproducts. ◮ But note that also the complement of C is an elementary class, namely, A / ∈ C iff A | = ¬ T. ◮ So, C is also closed under ultraproducts. ◮ Hence, if C is not closed under ultraproducts, then C is not finitely axiomatised. Monk used this technique to prove that the class RRA of Representable Relation Algebras is not finitely axiomatised.

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Example: Algebras of Binary Relations

We view them as two-sorted structures: ◮ Algebra sort. We will use a, b, c, . . . for its elements. They form a Boolean algebra with additional structure consisting of a constant 1′, a unary operation ˘, and a binary operation ;, which intuitively represent the identity relation, relational converse, and relational composition, respectively. ◮ Base set sort. We will use x, y, z, . . . for its elements. They are vertices of a complete graph whose edges are labelled with the elements of the algebra sort. Intuitively, the algebra elements are names of some binary relations on the base set. ◮ Labelling. A ternary relation λ(a, x, y). Intuitively, λ(a, x, y) says a(x, y), or written in infix notation xay. ◮ Formally an Algebra of Binary Relations is a structure (A, X, +, ·, ;, −, ˘, 0, 1, 1′, λ) where λ ∈ A × X × X (all operations are of algebra sort), such that:

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Algebras of Binary Relations

• A1. (A, +, ·, −, 0, 1) is a Boolean Algebra (+ is join, · is meet, − is complement).
• A2. (A, ;, ˘, 1′) is an involutive monoid.
• A3. (a + b) ; c = a ; c + b ; c

and a ; (b + c) = a ; b + a ; c.

• A4. a ; b · c = 0

⇔ a˘; c · b = 0 ⇔ b ; c˘· a˘ = 0

• R1. λ(a + b, x, y) ↔ λ(a, x, y) ∨ λ(b, x, y)
• R2. λ(¬a, x, y) ↔ λ(1, x, y) ∧ ¬λ(a, x, y)
• R3. λ(1′, x, y) ↔ x = y
• R4. λ(a˘, x, y) ↔ λ(a, y, x)
• R5. λ(a ; b, x, y) ↔ ∃z · λ(a, x, z) ∧ λ(b, z, y)
• R6. a = 0 → ∃x, y · λ(a, x, y).

Comments on the axioms

◮ All apparently free variables are universally quantified. ◮ All purely algebraic axioms are (equivalent to) equations.

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Algebras of Binary Relations

• A1. (A, +, ·, −, 0, 1) is a Boolean Algebra (+ is join, · is meet, − is complement).
• A2. (A, ;, ˘, 1′) is an involutive monoid.
• A3. (a + b) ; c = a ; c + b ; c

and a ; (b + c) = a ; b + a ; c.

• A4. a ; b · c = 0

⇔ a˘; c · b = 0 ⇔ b ; c˘· a˘ = 0

• R1. λ(a + b, x, y) ↔ λ(a, x, y) ∨ λ(b, x, y)
• R2. λ(¬a, x, y) ↔ λ(1, x, y) ∧ ¬λ(a, x, y)
• R3. λ(1′, x, y) ↔ x = y
• R4. λ(a˘, x, y) ↔ λ(a, y, x)
• R5. λ(a ; b, x, y) ↔ ∃z · λ(a, x, z) ∧ λ(b, z, y)
• R6. a = 0 → ∃x, y · λ(a, x, y).

Comments on the axioms

◮ All apparently free variables are universally quantified. ◮ All purely algebraic axioms are (equivalent to) equations.

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Algebras of Binary Relations

Definition 11

An algebra (A, +, ·, ;, ¬, ˘, 0, 1, 1′) is a Relation Algebra (RA) if it satisfies all equations in A1–A4. An algebra (A, +, ·, ;, ¬, ˘, 0, 1, 1′) is a Representable Relation Algebra (RRA) if it is isomorphic to an algebra reduct of an Algebra of Binary Relations. We will write (A, X, λ) for a two-sorted algebra of binary relations whose algebra reduct is A.

Historical comments

◮ RRA is closed under HSP, hence definable by equations (Tarski). ◮ Not all RAs are RRAs (Lyndon). ◮ RRA is not finitely axiomatised (Monk).

Exercise 2

Let (A1, X1, λ1) and (A2, X2, λ2) be algebras of binary relations. Prove that the structure (A1 × A2, X1 ˙ ∪ X2, µ), with µ defined by putting µ((a1, a2), x, y) iff λi(ai, x, y) and x, y ∈ Xi (i = 1, 2), is an algebra of binary relations.

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Algebras of Binary Relations

Definition 11

An algebra (A, +, ·, ;, ¬, ˘, 0, 1, 1′) is a Relation Algebra (RA) if it satisfies all equations in A1–A4. An algebra (A, +, ·, ;, ¬, ˘, 0, 1, 1′) is a Representable Relation Algebra (RRA) if it is isomorphic to an algebra reduct of an Algebra of Binary Relations. We will write (A, X, λ) for a two-sorted algebra of binary relations whose algebra reduct is A.

Historical comments

◮ RRA is closed under HSP, hence definable by equations (Tarski). ◮ Not all RAs are RRAs (Lyndon). ◮ RRA is not finitely axiomatised (Monk).

Exercise 2

Let (A1, X1, λ1) and (A2, X2, λ2) be algebras of binary relations. Prove that the structure (A1 × A2, X1 ˙ ∪ X2, µ), with µ defined by putting µ((a1, a2), x, y) iff λi(ai, x, y) and x, y ∈ Xi (i = 1, 2), is an algebra of binary relations.

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Algebras of Binary Relations

Definition 11

An algebra (A, +, ·, ;, ¬, ˘, 0, 1, 1′) is a Relation Algebra (RA) if it satisfies all equations in A1–A4. An algebra (A, +, ·, ;, ¬, ˘, 0, 1, 1′) is a Representable Relation Algebra (RRA) if it is isomorphic to an algebra reduct of an Algebra of Binary Relations. We will write (A, X, λ) for a two-sorted algebra of binary relations whose algebra reduct is A.

Historical comments

◮ RRA is closed under HSP, hence definable by equations (Tarski). ◮ Not all RAs are RRAs (Lyndon). ◮ RRA is not finitely axiomatised (Monk).

Exercise 2

Let (A1, X1, λ1) and (A2, X2, λ2) be algebras of binary relations. Prove that the structure (A1 × A2, X1 ˙ ∪ X2, µ), with µ defined by putting µ((a1, a2), x, y) iff λi(ai, x, y) and x, y ∈ Xi (i = 1, 2), is an algebra of binary relations.

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Monk’s proof

Definition 12

Let A be a complete atomic Boolean algebra, with the set of atoms {1′} ∪ {ai : i < κ} (where κ a cardinal). Define ai˘ = ai for every i, and ai ; aj =

• ¬(ai + aj)

if i = j 1′ + ai if i = j All other operations are implicitly defined by this. Then A is a relation algebra, called a Lyndon algebra. We will write Lyκ for so defined RA.

Lemma 13

(Lyκ, X, λ) is an algebra of binary relations if and only if X is the set of points of an affine plane. In particular, we have: ◮ Every infinite Lyndon algebra is representable. ◮ A finite Lyndon algebra Lyn is representable iff an affine plane of

• rder n exists.

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Monk’s proof

Definition 12

Let A be a complete atomic Boolean algebra, with the set of atoms {1′} ∪ {ai : i < κ} (where κ a cardinal). Define ai˘ = ai for every i, and ai ; aj =

• ¬(ai + aj)

if i = j 1′ + ai if i = j All other operations are implicitly defined by this. Then A is a relation algebra, called a Lyndon algebra. We will write Lyκ for so defined RA.

Lemma 13

(Lyκ, X, λ) is an algebra of binary relations if and only if X is the set of points of an affine plane. In particular, we have: ◮ Every infinite Lyndon algebra is representable. ◮ A finite Lyndon algebra Lyn is representable iff an affine plane of

• rder n exists.

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Monk’s proof

Lemma 14 (Bruck, Ryser, Chowla)

If a finite affine plane of order n exists and n = 1, 2 (mod 4), then n is a sum of two squares. Thus, affine planes of order n do not exist for infinitely many n.

Lemma 15

Let I be the set of all these n for which there is no affine plane of order n. Consider the ultraproduct

i∈I Lyi/U for some non-principal ultraproduct

• n I. Then, Lyi is not representable for every i ∈ I, but

i∈I Lyi/U is

representable.

Theorem 16

The class RRA is not finitely axiomatised. In particular, RRA has no finite equational axiomatisation.

Proof.

The complement of RRA is not closed under ultraproducts.

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Monk’s proof

Lemma 14 (Bruck, Ryser, Chowla)

If a finite affine plane of order n exists and n = 1, 2 (mod 4), then n is a sum of two squares. Thus, affine planes of order n do not exist for infinitely many n.

Lemma 15

Let I be the set of all these n for which there is no affine plane of order n. Consider the ultraproduct

i∈I Lyi/U for some non-principal ultraproduct

• n I. Then, Lyi is not representable for every i ∈ I, but

i∈I Lyi/U is

representable.

Theorem 16

The class RRA is not finitely axiomatised. In particular, RRA has no finite equational axiomatisation.

Proof.

The complement of RRA is not closed under ultraproducts.

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Monk’s proof

Lemma 14 (Bruck, Ryser, Chowla)

If a finite affine plane of order n exists and n = 1, 2 (mod 4), then n is a sum of two squares. Thus, affine planes of order n do not exist for infinitely many n.

Lemma 15

Let I be the set of all these n for which there is no affine plane of order n. Consider the ultraproduct

i∈I Lyi/U for some non-principal ultraproduct

• n I. Then, Lyi is not representable for every i ∈ I, but

i∈I Lyi/U is

representable.

Theorem 16

The class RRA is not finitely axiomatised. In particular, RRA has no finite equational axiomatisation.

Proof.

The complement of RRA is not closed under ultraproducts.

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