The Logic of Comparative Cardinality

Yifeng Ding (voidprove.com) Joint work with Matthew Harrison-Trainer and Wesley Holliday Aug. 7, 2018 @ BLAST 2018

UC Berkeley Group of Logic and the Methodology of Science Introduction • The equational theory of Boolean algebras is also the equational theory of ﬁelds of sets, if we only care about Boolean operations. • But more information can be extracted from a ﬁeld of sets. • We compare their sizes.

A ﬁeld of sets

Deﬁnition A ﬁeld of sets (X , F) is a pair where

1. X is a set and F ⊆ ℘(X ); 2. F is closed under intersection and complementation.

1 • But more information can be extracted from a ﬁeld of sets. • We compare their sizes.

A ﬁeld of sets

Deﬁnition A ﬁeld of sets (X , F) is a pair where

1. X is a set and F ⊆ ℘(X ); 2. F is closed under intersection and complementation.

• The equational theory of Boolean algebras is also the equational theory of ﬁelds of sets, if we only care about Boolean operations.

1 • We compare their sizes.

A ﬁeld of sets

Deﬁnition A ﬁeld of sets (X , F) is a pair where

1. X is a set and F ⊆ ℘(X ); 2. F is closed under intersection and complementation.

• The equational theory of Boolean algebras is also the equational theory of ﬁelds of sets, if we only care about Boolean operations. • But more information can be extracted from a ﬁeld of sets.

1 A ﬁeld of sets

Deﬁnition A ﬁeld of sets (X , F) is a pair where

1. X is a set and F ⊆ ℘(X ); 2. F is closed under intersection and complementation.

1 • A ﬁeld of sets model is hX , F, V i where V :Φ → F. • Terms are evaluated by Vb on F in the obvious way. •| s| ≥ |t|: set s is at least as large as set t: there is an injection from Vb(t) to Vb(s).

Comparing the sizes

Deﬁnition Given a countably inﬁnite set Φ of set labels, the language L is generated by the following grammar:

t ::= a | tc | (t ∩ t) ϕ ::= |t| ≥ |t| | ¬ϕ | (ϕ ∧ ϕ),

2 • Terms are evaluated by Vb on F in the obvious way. •| s| ≥ |t|: set s is at least as large as set t: there is an injection from Vb(t) to Vb(s).

Comparing the sizes

Deﬁnition Given a countably inﬁnite set Φ of set labels, the language L is generated by the following grammar:

t ::= a | tc | (t ∩ t) ϕ ::= |t| ≥ |t| | ¬ϕ | (ϕ ∧ ϕ),

• A ﬁeld of sets model is hX , F, V i where V :Φ → F.

2 •| s| ≥ |t|: set s is at least as large as set t: there is an injection from Vb(t) to Vb(s).

Comparing the sizes

Deﬁnition Given a countably inﬁnite set Φ of set labels, the language L is generated by the following grammar:

t ::= a | tc | (t ∩ t) ϕ ::= |t| ≥ |t| | ¬ϕ | (ϕ ∧ ϕ),

• A ﬁeld of sets model is hX , F, V i where V :Φ → F. • Terms are evaluated by Vb on F in the obvious way.

2 Comparing the sizes

Deﬁnition Given a countably inﬁnite set Φ of set labels, the language L is generated by the following grammar:

t ::= a | tc | (t ∩ t) ϕ ::= |t| ≥ |t| | ¬ϕ | (ϕ ∧ ϕ),

• A ﬁeld of sets model is hX , F, V i where V :Φ → F. • Terms are evaluated by Vb on F in the obvious way. •| s| ≥ |t|: set s is at least as large as set t: there is an injection from Vb(t) to Vb(s).

2 s t

• • For ﬁnite sets s, t, |s| ≥ |t| ↔ |s ∩ tc | ≥ |t ∩ sc |. • For inﬁnite sets s, t, u •| s| ≥ |t| → |s ∩ tc | ≥ |t ∩ sc | is not valid; • (|s| ≥ |t| ∧ |s| ≥ |u|) → |s| ≥ |t ∪ u| is valid.

Finite sets and inﬁnite sets

• Finite sets and inﬁnite sets obey very diﬀerent laws.

3 Finite sets and inﬁnite sets

• Finite sets and inﬁnite sets obey very diﬀerent laws.

s t

3 • The sentences in L valid on inﬁnite sets have been axiomatized, with size interpreted as likelihood or possibilities. • We want to combine them: with no extra constraint on (X , F), what is the logic?

More background

• The sentences in L valid on ﬁnite sets have been axiomatized, with size interpreted as probability, credence, etc..

4 • We want to combine them: with no extra constraint on (X , F), what is the logic?

More background

• The sentences in L valid on ﬁnite sets have been axiomatized, with size interpreted as probability, credence, etc.. • The sentences in L valid on inﬁnite sets have been axiomatized, with size interpreted as likelihood or possibilities.

4 More background

4 Outline

Introduction

Laws common to ﬁnite and inﬁnite sets

A representation theorem

Logic with predicates for ﬁnite and inﬁnite sets

Eliminating extra predicates

Further questions

5 Laws common to ﬁnite and inﬁnite sets • if t = 0 is provable in the equational theory of Boolean algebras, then |∅| ≥ |t| is a theorem.

•| s| ≥ |t| ∨ |t| ≥ |s|;(|s| ≥ |t| ∧ |t| ≥ |u|) → |s| ≥ |u|; c •| ∅| ≥ |s ∩ t | → |t| ≥ |s|;

c •¬| ∅| ≥ |∅ |; • (|∅| ≥ |s| ∧ |∅| ≥ |t|) → |∅| ≥ |s ∪ t|;

BasicCompLogic

Deﬁnition (BasicCompLogic) Boolean reasoning on the sentence level. Boolean Reasoning on the set level.

≥ is a total preorder extending ⊇.

≥ works well with ∅.

6 • if t = 0 is provable in the equational theory of Boolean algebras, then |∅| ≥ |t| is a theorem.

•| s| ≥ |t| ∨ |t| ≥ |s|;(|s| ≥ |t| ∧ |t| ≥ |u|) → |s| ≥ |u|; c •| ∅| ≥ |s ∩ t | → |t| ≥ |s|;

c •¬| ∅| ≥ |∅ |; • (|∅| ≥ |s| ∧ |∅| ≥ |t|) → |∅| ≥ |s ∪ t|;

BasicCompLogic

Deﬁnition (BasicCompLogic) Boolean reasoning on the sentence level. Boolean Reasoning on the set level.

≥ is a total preorder extending ⊇.

≥ works well with ∅.

6 •| s| ≥ |t| ∨ |t| ≥ |s|;(|s| ≥ |t| ∧ |t| ≥ |u|) → |s| ≥ |u|; c •| ∅| ≥ |s ∩ t | → |t| ≥ |s|;

c •¬| ∅| ≥ |∅ |; • (|∅| ≥ |s| ∧ |∅| ≥ |t|) → |∅| ≥ |s ∪ t|;

BasicCompLogic

≥ works well with ∅.

6 c •¬| ∅| ≥ |∅ |; • (|∅| ≥ |s| ∧ |∅| ≥ |t|) → |∅| ≥ |s ∪ t|;

BasicCompLogic

Deﬁnition (BasicCompLogic) Boolean reasoning on the sentence level. Boolean Reasoning on the set level. • if t = 0 is provable in the equational theory of Boolean algebras, then |∅| ≥ |t| is a theorem. ≥ is a total preorder extending ⊇. •| s| ≥ |t| ∨ |t| ≥ |s|;(|s| ≥ |t| ∧ |t| ≥ |u|) → |s| ≥ |u|; c •| ∅| ≥ |s ∩ t | → |t| ≥ |s|; ≥ works well with ∅.

6 BasicCompLogic

6 Any formula ϕ consistent with BasicCompLogic is satisﬁable in a ﬁnite comparison algebra, with interpreting | · | ≥ | · |.

ϕ ⇒ Σ ⇒ hB, , V i ⇒ h V (T (var(ϕ))) , , V i | {z } relevant terms, a ﬁnite set

6⇒ hX , F, V i

From logic to algebra

Deﬁnition A comparison algebra is a pair hB, i where B is a Boolean algebra and is a total preorder on B such that

• for all a, b ∈ B, a ≥B b implies a b,

•⊥ B 6 b for all b ∈ B \ {⊥B }.

7 6⇒ hX , F, V i

From logic to algebra

Deﬁnition A comparison algebra is a pair hB, i where B is a Boolean algebra and is a total preorder on B such that

• for all a, b ∈ B, a ≥B b implies a b,

•⊥ B 6 b for all b ∈ B \ {⊥B }.

Any formula ϕ consistent with BasicCompLogic is satisﬁable in a ﬁnite comparison algebra, with interpreting | · | ≥ | · |.

ϕ ⇒ Σ ⇒ hB, , V i ⇒ h V (T (var(ϕ))) , , V i | {z } relevant terms, a ﬁnite set

7 From logic to algebra

Deﬁnition A comparison algebra is a pair hB, i where B is a Boolean algebra and is a total preorder on B such that

• for all a, b ∈ B, a ≥B b implies a b,

•⊥ B 6 b for all b ∈ B \ {⊥B }.

Any formula ϕ consistent with BasicCompLogic is satisﬁable in a ﬁnite comparison algebra, with interpreting | · | ≥ | · |.

ϕ ⇒ Σ ⇒ hB, , V i ⇒ h V (T (var(ϕ))) , , V i | {z } relevant terms, a ﬁnite set

6⇒ hX , F, V i

7 • (010) and (100) should be ﬁnite. • Then all must be ﬁnite. • But |(011)| = |(101)| while |(010)| < |(100)|.

Not enough constraints

(111) : 4

(011) : 4 (101) : 4 (110) : 4

(001) : 1 (010) : 2 (100) : 3

(000) : 0

8 Not enough constraints

(111) : 4

• (010) and (100) should be (011) : 4 (101) : 4 (110) : 4 ﬁnite. • Then all must be ﬁnite. • But |(011)| = |(101)| while (001) : 1 (010) : 2 (100) : 3 |(010)| < |(100)|.

(000) : 0

8 • But the ordering in this B might not be based on any cardinality comparison. • We need to know when the ordering arise from cardinality comparison, and add the constraints to the logic.

Message

• Any formula ϕ consistent with BasicCompLogic is satisﬁable in a ﬁnite comparison algebra B, with interpreting | · | ≥ | · |.

9 • We need to know when the ordering arise from cardinality comparison, and add the constraints to the logic.

Message

9 Message

• Any formula ϕ consistent with BasicCompLogic is satisﬁable in a ﬁnite comparison algebra B, with interpreting | · | ≥ | · |. • But the ordering in this B might not be based on any cardinality comparison. • We need to know when the ordering arise from cardinality comparison, and add the constraints to the logic.

9 Plan

Introduction

Laws common to ﬁnite and inﬁnite sets

A representation theorem

Logic with predicates for ﬁnite and inﬁnite sets

Eliminating extra predicates

Further questions

10 Deﬁnition A comparison algebra hB, i is represented by a measure algebra hB, µi if for all a, b ∈ B, we have a b iﬀ µ(a) ≥ µ(b).

Deﬁnitions

Deﬁnition

A measure algebra is a pair hB, µi, where B is a Boolean algebra and µ is a function assigning a cardinal to each element of B such that

• if a ∧ b = ⊥, then µ(a ∨ b) = µ(a) + µ(b), and • µ(b) = 0 iﬀ b = ⊥.

11 Deﬁnitions

Deﬁnition

A measure algebra is a pair hB, µi, where B is a Boolean algebra and µ is a function assigning a cardinal to each element of B such that

• if a ∧ b = ⊥, then µ(a ∨ b) = µ(a) + µ(b), and • µ(b) = 0 iﬀ b = ⊥.

Deﬁnition A comparison algebra hB, i is represented by a measure algebra hB, µi if for all a, b ∈ B, we have a b iﬀ µ(a) ≥ µ(b).

11 We call this condition “ﬁnite cancellation”

A representation theorem for ﬁnite sets

Theorem (Kraft, Pratt, Seidenberg) For any ﬁnite comparison algebra hB, i, it is represented by a measure algebra hB, µi where Range(µ) = ω if and only if:

• for any two sequences of elements a1, a2,..., an and b1, b2,..., bn from B, if every atom of B is below (in the order of the Boolean algebra) exactly as many a’s as b’s, and

if ai bi for all i ∈ {1,..., n − 1}, then bn an.

12 A representation theorem for ﬁnite sets

Theorem (Kraft, Pratt, Seidenberg) For any ﬁnite comparison algebra hB, i, it is represented by a measure algebra hB, µi where Range(µ) = ω if and only if:

• for any two sequences of elements a1, a2,..., an and b1, b2,..., bn from B, if every atom of B is below (in the order of the Boolean algebra) exactly as many a’s as b’s, and

if ai bi for all i ∈ {1,..., n − 1}, then bn an.

We call this condition “ﬁnite cancellation”

12 • Then |a0| + |a1| + |a2| = |b0| + |b1| + |b2|.

• Then if |a0| ≥ |b0| and |a1| ≥ |b1|, we can’t have |a2| > |b2|, which means |b2| ≥ |a2|.

Finite cancellation illustrated

a0 + a1 + a2 = b0 + b1 + b2 1 1 0 2 0 1 1 0 1 0 1 1 0 0                 +   +   =   =   +   +   1 1 1 3 1 1 1 0 1 1 2 1 1 0

13 • Then if |a0| ≥ |b0| and |a1| ≥ |b1|, we can’t have |a2| > |b2|, which means |b2| ≥ |a2|.

Finite cancellation illustrated

• Then |a0| + |a1| + |a2| = |b0| + |b1| + |b2|.

13 Finite cancellation illustrated

• Then |a0| + |a1| + |a2| = |b0| + |b1| + |b2|.

• Then if |a0| ≥ |b0| and |a1| ≥ |b1|, we can’t have |a2| > |b2|, which means |b2| ≥ |a2|.

13 Suppose there is an F ⊆ B such that the following conditions are satisﬁed:

1. F is an ideal; 2. elements in F satisfy the ﬁnite cancellation condition; 3. for any a, b, c ∈ B such that a 6∈ F , if a b and a c then

a b ∨B c; 4. for any a, b ∈ B, if a ∈ F and b 6∈ F , then b a and not a b.

Then B is represented by a finite measure algebra m(B) = hB, µi such that a ∈ F iff µ(a) is finite.

A representation theorem

Let B = hB, i be a ﬁnite comparison algebra.

14 1. F is an ideal; 2. elements in F satisfy the ﬁnite cancellation condition; 3. for any a, b, c ∈ B such that a 6∈ F , if a b and a c then

a b ∨B c; 4. for any a, b ∈ B, if a ∈ F and b 6∈ F , then b a and not a b.

Then B is represented by a finite measure algebra m(B) = hB, µi such that a ∈ F iff µ(a) is finite.

A representation theorem

Let B = hB, i be a ﬁnite comparison algebra. Suppose there is an F ⊆ B such that the following conditions are satisﬁed:

14 2. elements in F satisfy the ﬁnite cancellation condition; 3. for any a, b, c ∈ B such that a 6∈ F , if a b and a c then

a b ∨B c; 4. for any a, b ∈ B, if a ∈ F and b 6∈ F , then b a and not a b.

Then B is represented by a finite measure algebra m(B) = hB, µi such that a ∈ F iff µ(a) is finite.

A representation theorem

Let B = hB, i be a ﬁnite comparison algebra. Suppose there is an F ⊆ B such that the following conditions are satisﬁed:

1. F is an ideal;

14 3. for any a, b, c ∈ B such that a 6∈ F , if a b and a c then

a b ∨B c; 4. for any a, b ∈ B, if a ∈ F and b 6∈ F , then b a and not a b.

Then B is represented by a finite measure algebra m(B) = hB, µi such that a ∈ F iff µ(a) is finite.

A representation theorem

Let B = hB, i be a ﬁnite comparison algebra. Suppose there is an F ⊆ B such that the following conditions are satisﬁed:

1. F is an ideal; 2. elements in F satisfy the ﬁnite cancellation condition;

14 4. for any a, b ∈ B, if a ∈ F and b 6∈ F , then b a and not a b.

Then B is represented by a finite measure algebra m(B) = hB, µi such that a ∈ F iff µ(a) is finite.

A representation theorem

Let B = hB, i be a ﬁnite comparison algebra. Suppose there is an F ⊆ B such that the following conditions are satisﬁed:

1. F is an ideal; 2. elements in F satisfy the ﬁnite cancellation condition; 3. for any a, b, c ∈ B such that a 6∈ F , if a b and a c then

a b ∨B c;

14 Then B is represented by a finite measure algebra m(B) = hB, µi such that a ∈ F iff µ(a) is finite.

A representation theorem

Let B = hB, i be a ﬁnite comparison algebra. Suppose there is an F ⊆ B such that the following conditions are satisﬁed:

1. F is an ideal; 2. elements in F satisfy the ﬁnite cancellation condition; 3. for any a, b, c ∈ B such that a 6∈ F , if a b and a c then

a b ∨B c; 4. for any a, b ∈ B, if a ∈ F and b 6∈ F , then b a and not a b.

14 A representation theorem

Let B = hB, i be a ﬁnite comparison algebra. Suppose there is an F ⊆ B such that the following conditions are satisﬁed:

1. F is an ideal; 2. elements in F satisfy the ﬁnite cancellation condition; 3. for any a, b, c ∈ B such that a 6∈ F , if a b and a c then

a b ∨B c; 4. for any a, b ∈ B, if a ∈ F and b 6∈ F , then b a and not a b.

Then B is represented by a finite measure algebra m(B) = hB, µi such that a ∈ F iff µ(a) is finite.

14 Plan

Introduction

Laws common to ﬁnite and inﬁnite sets

A representation theorem

Logic with predicates for ﬁnite and inﬁnite sets

Eliminating extra predicates

Further questions

15 The logic with extra predicates (With FC)

Add the following to BasicCompLogic, CardCompLogicFin,Inf is sound and complete w.r.t. measure algebras and ﬁelds of sets.

1. Fin(s) ⊕ Inf(s); 2.( Fin(s) ∧ Fin(t)) → Fin(s ∪ t); (Fin(t) ∧ s ⊆ t) → Fin(s); 3.( Fin(s) ∧ Inf(t)) → |t| > |s|; 4. Inf(s) → ((|s| ≥ |t| ∧ |s| ≥ |u|) → |s| ≥ |t ∪ u|); Vn 5. i=1(Fin(si ) ∧ Fin(ti )) → FC(s1, ··· , sn, t1, ··· , tn),

16 ϕ(0), ϕ(1), ϕ(2), ϕ(3), ϕ(4), ϕ(5), ϕ(6), ...

The logic with extra predicates (With Polarizability rule)

Add the following to BasicCompLogic, CardCompLogicFin,Inf is sound and complete w.r.t. measure algebras and ﬁelds of sets.

1. Fin(s) ⊕ Inf(s); 2.( Fin(s) ∧ Fin(t)) → Fin(s ∪ t); (Fin(t) ∧ s ⊆ t) → Fin(s); 3.( Fin(s) ∧ Inf(t)) → |t| > |s|; 4. Inf(s) → ((|s| ≥ |t| ∧ |s| ≥ |u|) → |s| ≥ |t ∪ u|); 5.( Fin(s) ∧ Fin(t)) → (|s| ≥ |t| ↔ |s ∩ tc | ≥ |t ∩ sc |); 6. where a|t abbreviates |t ∩ a| = |t ∩ ac | for a ∈ Φ, if a|t → φ is derivable, then φ is derivable, assuming that a does not occur in t or in φ.

17 The logic with extra predicates (With Polarizability rule)

Add the following to BasicCompLogic, CardCompLogicFin,Inf is sound and complete w.r.t. measure algebras and ﬁelds of sets.

17 ϕ(1), ϕ(3), ϕ(5),

The logic with extra predicates (With Polarizability rule)

Add the following to BasicCompLogic, CardCompLogicFin,Inf is sound and complete w.r.t. measure algebras and ﬁelds of sets.

17 • Axioms (rules) for finite sets. • Axioms for infinite sets. • Some simple interaction between finite and infinite sets.

But that’s assuming that we can distinguish ﬁnite and inﬁnite sets, which uses two extra predicates.

Summary

In sum, the a complete logic can be made from the following:

• Basic comparison rules.

18 • Axioms for infinite sets. • Some simple interaction between finite and infinite sets.

But that’s assuming that we can distinguish ﬁnite and inﬁnite sets, which uses two extra predicates.

Summary

In sum, the a complete logic can be made from the following:

• Basic comparison rules. • Axioms (rules) for ﬁnite sets.

18 • Some simple interaction between ﬁnite and inﬁnite sets.

But that’s assuming that we can distinguish ﬁnite and inﬁnite sets, which uses two extra predicates.

Summary

In sum, the a complete logic can be made from the following:

• Basic comparison rules. • Axioms (rules) for ﬁnite sets. • Axioms for inﬁnite sets.

18 But that’s assuming that we can distinguish ﬁnite and inﬁnite sets, which uses two extra predicates.

Summary

In sum, the a complete logic can be made from the following:

• Basic comparison rules. • Axioms (rules) for finite sets. • Axioms for infinite sets. • Some simple interaction between finite and infinite sets.

18 Summary

In sum, the a complete logic can be made from the following:

• Basic comparison rules. • Axioms (rules) for finite sets. • Axioms for infinite sets. • Some simple interaction between finite and infinite sets.

But that’s assuming that we can distinguish ﬁnite and inﬁnite sets, which uses two extra predicates.

18 Plan

Introduction

Laws common to ﬁnite and inﬁnite sets

A representation theorem

Logic with predicates for ﬁnite and inﬁnite sets

Eliminating extra predicates

Further questions

19 No. There are models that satisfy exactly the same formulas in L, but one has only inﬁnite sets and the other has a ﬁnite set.

Defining finiteness and infiniteness

Can we deﬁne Fin and Inf in the language of pure cardinality comparison?

20 Defining finiteness and infiniteness

Can we deﬁne Fin and Inf in the language of pure cardinality comparison?

No. There are models that satisfy exactly the same formulas in L, but one has only inﬁnite sets and the other has a ﬁnite set.

20 Undeﬁnability

|(111)| = ℵ3

|(011)| = ℵ2 |(101)| = ℵ3 |(110)| = ℵ3

|(001)| = ℵ1 |(010)| = ℵ2 |(100)| = ℵ3

|(000)| = 0

21 Undeﬁnability

|(111)| = ℵ3

|(011)| = ℵ2 |(101)| = ℵ3 |(110)| = ℵ3

|(001)| = 1 |(010)| = ℵ2 |(100)| = ℵ3

|(000)| = 0

22 For any ﬂexible measure algebra hB, µi and any cardinal κ, there exists a ﬂexible measure algebra hB, µ0i such that

• µ(the smallest atom) = κ; • For any a, b ∈ B, µ(a) ≥ µ(b) iﬀ µ0(a) ≥ µ0(b); 0 •h B, µ, V i ≡L hB, µ , V i for any valuation V .

Flexible algebras

We call a ﬁnite measure algebra hB, µi ﬂexible when:

• There is a strictly smallest atom. • Any element (except the bottom element) is equally large to the largest atom below it. • Equivalently, there is at most one ﬁnite but non-empty element (which must be the strictly smallest atom).

23 Flexible algebras

We call a ﬁnite measure algebra hB, µi ﬂexible when:

For any ﬂexible measure algebra hB, µi and any cardinal κ, there exists a ﬂexible measure algebra hB, µ0i such that

• µ(the smallest atom) = κ; • For any a, b ∈ B, µ(a) ≥ µ(b) iﬀ µ0(a) ≥ µ0(b); 0 •h B, µ, V i ≡L hB, µ , V i for any valuation V .

23 Perhaps flexible models are the only models where finiteness can’t be defined?

Deﬁning Fin and Inf

We must do our best.

24 Deﬁning Fin and Inf

We must do our best. Perhaps flexible models are the only models where finiteness can’t be defined?

24 Deﬁning Fin and Inf

u is the

When ∆ ⊆ Φ is ﬁnite, deﬁne Fin∆(u) forunion any of setri ’s term u ∈ T (∆) as:  |R| u = S r  i=1 i _   V|R| |si ∪ ti | > |si | ≥ |ti | R⊆T0(∆)  i=1 |R|  |si ∪ ti | ≥ |ri | S,T ∈T0(∆)

Here ri ranges over elements in R, and si , ti range over the si , ti and also elementsThere exists in sequences S and T , respectively. ri , si , ti ’s ri ’s are ﬁnite

25 Deﬁning Fin and Inf

u is the

Here ri ranges over elements in R, and si , ti range over the si , ti and also elementsThere exists in sequences S and T , respectively. ri , si , ti ’s ri ’s are ﬁnite

25 Deﬁning Fin and Inf

u is the

Here ri ranges over elements in R, and si , ti range over the si , ti and also elementsThere exists in sequences S and T , respectively. ri , si , ti ’s ri ’s are ﬁnite

25 Deﬁning Fin and Inf

u is the

Here ri ranges over elements in R, and si , ti range over the si , ti and also elementsThere exists in sequences S and T , respectively. ri , si , ti ’s ri ’s are ﬁnite

25 Deﬁning Fin and Inf

When ∆ ⊆ Φ is ﬁnite, deﬁne Inf∆(u) := for any set term u ∈ T (∆) as: _ (t 6⊆ s ∧ |u| ≥ |s| ≥ |s ∪ t|)

s,t∈T0(∆)

26 Deﬁnition works

For any measure algebra model hB, µ, V i such that every element is named by a term in T (∆), namely V (T (∆)) = B:

• If Fin∆(u) is true, then µ(Vb(u)) is ﬁnite.

• If Inf∆(u) is true, then µ(Vb(u)) is inﬁnite.

• Fin∆(u) and Inf∆(u) can’t be both true. • If they are both false, then hB, µi is ﬂexible, and Vb(u) is the smallest atom. • (s ⊆ t ∧ Fin(t)) → Fin(s) and (Fin(s) ∧ Fin(t)) → Fin(s ∪ t) are derivable in BasicCompLogic.

27 The axioms

Definition Where ∆ ⊆ Φ is finite, define Axiom(∆) as the set containing all

of the following formulas for all u, s, t ∈ T0(∆):

1. ¬(Fin∆(u) ∧ Inf∆(u));

2.( ¬Fin∆(u) ∧ ¬Inf∆(u)) → V (|u| ≥ |t| → (t = ∨ t = u)); t∈T0(∆) ∅ c c 3. (Fin∆(s) ∧ Fin∆(t)) → (|s| ≥ |t| ↔ |s ∩ t | ≥ |t ∩ s |);

4. Inf∆(u) → ((|u| ≥ |s| ∧ |u| ≥ |t|) → |u| ≥ |s ∪ t|);

5. (Inf∆(s) ∧ Fin∆(t)) → |s| > |t|.

28 The logic

Deﬁnition

Let CardCompLogic be the logic for L with the following axioms and rules:

1. all axioms and rules in BasicCompLogic; 2. for any ﬁnite ∆ ⊆ Φ, all formulas in Axioms(∆); 3. the polarizability rule (A7).

29 Proof sketch

Pick a ϕ consistent with CardCompLogic, take ∆ = var(ϕ):

1. Extend it to Σ maximally consistent in CardCompLogic. 2. Σ is also maximally consistent with BasicCompLogic. Get canonical comparison model C. 3. Restrict C to terms in T (∆), get B. 4. B |= Axiom(∆) and also Fin(~s) → FC(~s). Use the terms with Fin as ﬁnite elements. Apply representation theorem and get

measure algebra model M ≡L B. 5. M |= ϕ. So ϕ is satisﬁable.

30 Conclusion

The logic of cardinal comparison on arbitrary ﬁelds of sets can be axiomatized by putting together

• a basic system for orderings extending the inclusion ordering; • a working definition for finiteness and infiniteness based on witnesses; • characteristic axioms and rules for finite and infinite sets.

The axiomatization is weak; the logic is non-compact. We use ﬁnite Boolean algebras in an essential way.

31 Plan

Introduction

Laws common to ﬁnite and inﬁnite sets

A representation theorem

Logic with predicates for ﬁnite and inﬁnite sets

Eliminating extra predicates

Further questions

32 • A ﬁeld of sets hX , Fi (F possibly inﬁnite) naturally give rise to a measure algebra hB, µi. The B part can be arbitrary due to Stone duality. But what about µ? • Same question for hB, i. What conditions must satisfy?

Our logic is not strongly complete, as it is ﬁnitary but not compact. What is the strongly complete logic?

Further questions

Representation theorems in the inﬁnite:

33 • Same question for hB, i. What conditions must satisfy?

Our logic is not strongly complete, as it is ﬁnitary but not compact. What is the strongly complete logic?

Further questions

Representation theorems in the inﬁnite:

• A ﬁeld of sets hX , Fi (F possibly inﬁnite) naturally give rise to a measure algebra hB, µi. The B part can be arbitrary due to Stone duality. But what about µ?

33 Our logic is not strongly complete, as it is ﬁnitary but not compact. What is the strongly complete logic?

Further questions

Representation theorems in the inﬁnite:

• A ﬁeld of sets hX , Fi (F possibly inﬁnite) naturally give rise to a measure algebra hB, µi. The B part can be arbitrary due to Stone duality. But what about µ? • Same question for hB, i. What conditions must satisfy?

33 Further questions

Representation theorems in the inﬁnite:

Our logic is not strongly complete, as it is ﬁnitary but not compact. What is the strongly complete logic?

33 Thank You.

34 Non-compactness

The problem of ﬁniteness:

{|sn| < |sn+1| | n ∈ ω} ∪ {|sn| ≤ |t| | n ∈ ω} ∪ {Fin(t)}.

The problem of well-foundedness:

{|sn+1| < |sn| | n ∈ ω}.

The problem of discreteness:

Disjoint{ti , si | i ∈ ω}∪

{|ti | = |tj |, |si | = |sj | | i, j ∈ ω}∪ m m √ {| ∪ t | < | ∪ s | < | ∪ t | | ( 1 , 2 ) →lim 2}. i

35 Representation in the inﬁnite

Can we have a reasonably nice list of conditions for an arbitrary comparison algebra (ﬁnite or inﬁnite) to be represented by a cardinal-valued measure function?

36 Deﬁnition of FC

Deﬁnition

n For each sequence of n terms ~s = hs0, ··· , sn−1i and f ∈ 2, deﬁne

\ \ c ~s[f ] = {si | f (i) = 1} ∩ {si | f (i) = 0}, [ −1 Nm(~s) = {~s[f ] | f : n → 2 and |f (1)| = m}.

Given two sequences ~s and ~t of n terms, deﬁne ^ ~s E~t = (Ni (~s) = Ni (~t)), 0≤i≤n ^ FC(~s,~t) = ~s E~t → (( |si | ≥ |ti |) → |tn−1| ≥ |sn−1|). i

37 Polarization

With polarization, we can almost do set addition:

38 Polarization

With polarization, we can almost do set addition:

39 Polarization

With polarization, we can almost do set addition: