The Logic of Comparative Cardinality Yifeng Ding ( voidprove.com ) - - PowerPoint PPT Presentation

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

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

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

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.
• 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.

1

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

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 X, F, V where V : Φ → F.

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 X, F, V where V : Φ → F.
• Terms are evaluated by

V 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 X, F, V where V : Φ → F.
• Terms are evaluated by

V on F in the obvious way.

• |s| ≥ |t|: set s is at least as large as set t: there is an

injection from V (t) to V (s).

2

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

• 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.

3

More background

• The sentences in L valid on finite sets have been axiomatized,

with size interpreted as probability, credence, etc..

4

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

• 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.

• We want to combine them: with no extra constraint on

(X, F), what is the logic?

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

BasicCompLogic

Definition (BasicCompLogic) Boolean reasoning on the sentence level. Boolean Reasoning on the set level. ≥ is a total preorder extending ⊇. ≥ works well with ∅.

6

BasicCompLogic

Definition (BasicCompLogic) Boolean reasoning on the sentence level. Boolean Reasoning on the set level. ≥ is a total preorder extending ⊇. ≥ works well with ∅.

6

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 ⊇. ≥ works well with ∅.

6

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|;
• |∅| ≥ |s ∩ tc| → |t| ≥ |s|;

≥ works well with ∅.

6

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|;
• |∅| ≥ |s ∩ tc| → |t| ≥ |s|;

≥ works well with ∅.

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

6

From logic to algebra

Definition A comparison algebra is a pair B, 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 b for all b ∈ B \ {⊥B}.

7

From logic to algebra

Definition A comparison algebra is a pair B, 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 b for all b ∈ B \ {⊥B}.

Any formula ϕ consistent with BasicCompLogic is satisfiable in a finite comparison algebra, with interpreting | · | ≥ | · |. ϕ ⇒ Σ ⇒ B, , V ⇒ V (T(var(ϕ)))

• relevant terms, a finite set

, , V

7

From logic to algebra

Definition A comparison algebra is a pair B, 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 b for all b ∈ B \ {⊥B}.

Any formula ϕ consistent with BasicCompLogic is satisfiable in a finite comparison algebra, with interpreting | · | ≥ | · |. ϕ ⇒ Σ ⇒ B, , V ⇒ V (T(var(ϕ)))

• relevant terms, a finite set

, , V ⇒ X, F, V

7

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 (011) : 4 (101) : 4 (110) : 4 (001) : 1 (010) : 2 (100) : 3 (000) : 0

• (010) and (100) should be

finite.

• Then all must be finite.
• But |(011)| = |(101)| while

|(010)| < |(100)|.

8

Message

• Any formula ϕ consistent with BasicCompLogic is satisfiable

in a finite comparison algebra B, with interpreting | · | ≥ | · |.

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.

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

Definitions

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

11

A representation theorem for finite sets

Theorem (Kraft, Pratt, Seidenberg) For any finite comparison algebra B, , it is represented by a measure algebra B, µ 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

• rder 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 B, , it is represented by a measure algebra B, µ 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

• rder 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

Finite cancellation illustrated

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

13

Finite cancellation illustrated

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

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

13

Finite cancellation illustrated

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

• 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

A representation theorem

Let B = B, be a finite comparison algebra.

14

A representation theorem

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

14

A representation theorem

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

• 1. F is an ideal;

14

A representation theorem

Let B = B, 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

A representation theorem

Let B = B, 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 ∈ F, if a b and a c then

a b ∨B c;

14

A representation theorem

Let B = B, 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 ∈ F, if a b and a c then

a b ∨B c;

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

a b.

14

A representation theorem

Let B = B, 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 ∈ F, if a b and a c then

a b ∨B c;

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

a b. Then B is represented by a finite measure algebra m(B) = B, µ 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|);
• 5. n

i=1(Fin(si) ∧ Fin(ti)) → FC(s1, · · · , sn, t1, · · · , tn), 16

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.

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

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.

• 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. ϕ(0), ϕ(2), ϕ(4), ϕ(6), ...

17

Summary

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

• Basic comparison rules.

18

Summary

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

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

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.

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

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

Flexible algebras

We call a finite measure algebra B, µ 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 B, µ 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). For any flexible measure algebra B, µ and any cardinal κ, there exists a flexible measure algebra B, µ′ such that

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

23

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

When ∆ ⊆ Φ is finite, define Fin∆(u) for any set term u ∈ T(∆) as:

• R⊆T0(∆)

S,T∈T0(∆)|R|

         u = |R|

i=1 ri

|R|

i=1

   |si ∪ ti| > |si| ≥ |ti| |si ∪ ti| ≥ |ri| Here ri ranges over elements in R, and si, ti range over the elements in sequences S and T, respectively. There exists ri, si, ti’s u is the union of ri’s si, ti and also ri’s are finite

25

Defining Fin and Inf

When ∆ ⊆ Φ is finite, define Fin∆(u) for any set term u ∈ T(∆) as:

• R⊆T0(∆)

S,T∈T0(∆)|R|

         u = |R|

i=1 ri

|R|

i=1

   |si ∪ ti| > |si| ≥ |ti| |si ∪ ti| ≥ |ri| Here ri ranges over elements in R, and si, ti range over the elements in sequences S and T, respectively. There exists ri, si, ti’s u is the union of ri’s si, ti and also ri’s are finite

25

Defining Fin and Inf

When ∆ ⊆ Φ is finite, define Fin∆(u) for any set term u ∈ T(∆) as:

• R⊆T0(∆)

S,T∈T0(∆)|R|

         u = |R|

i=1 ri

|R|

i=1

   |si ∪ ti| > |si| ≥ |ti| |si ∪ ti| ≥ |ri| Here ri ranges over elements in R, and si, ti range over the elements in sequences S and T, respectively. There exists ri, si, ti’s u is the union of ri’s si, ti and also ri’s are finite

25

Defining Fin and Inf

When ∆ ⊆ Φ is finite, define Fin∆(u) for any set term u ∈ T(∆) as:

• R⊆T0(∆)

S,T∈T0(∆)|R|

         u = |R|

i=1 ri

|R|

i=1

   |si ∪ ti| > |si| ≥ |ti| |si ∪ ti| ≥ |ri| Here ri ranges over elements in R, and si, ti range over the elements in sequences S and T, respectively. There exists ri, si, ti’s u is the union of ri’s si, ti and also ri’s are finite

25

Defining Fin and Inf

When ∆ ⊆ Φ is finite, define Inf∆(u) := for any set term u ∈ T(∆) as:

• s,t∈T0(∆)

(t ⊆ s ∧ |u| ≥ |s| ≥ |s ∪ t|)

26

Definition works

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

• If Fin∆(u) is true, then µ(

V (u)) is finite.

• If Inf∆(u) is true, then µ(

V (u)) is infinite.

• Fin∆(u) and Inf∆(u) can’t be both true.
• If they are both false, then B, µ is flexible, and

V (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

• f the following formulas for all u, s, t ∈ T0(∆):
• 1. ¬(Fin∆(u) ∧ Inf∆(u));
• 2. (¬Fin∆(u) ∧ ¬Inf∆(u)) →
• t∈T0(∆)(|u| ≥ |t| → (t = ∅ ∨ t = u));
• 3. (Fin∆(s) ∧ Fin∆(t)) → (|s| ≥ |t| ↔ |s ∩ tc| ≥ |t ∩ sc|);
• 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.

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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

Further questions

Representation theorems in the infinite:

33

Further questions

Representation theorems in the infinite:

• A field of sets X, F (F possibly infinite) naturally give rise

to a measure algebra B, µ. The B part can be arbitrary due to Stone duality. But what about µ?

33

Further questions

Representation theorems in the infinite:

• A field of sets X, F (F possibly infinite) naturally give rise

to a measure algebra B, µ. The B part can be arbitrary due to Stone duality. But what about µ?

• Same question for B, . What conditions must satisfy?

33

Further questions

Representation theorems in the infinite:

• A field of sets X, F (F possibly infinite) naturally give rise

to a measure algebra B, µ. The B part can be arbitrary due to Stone duality. But what about µ?

• Same question for B, . What conditions must satisfy?

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}∪ {| ∪i<m1 ti| < | ∪i<n si| < | ∪i<m2 ti| | (m1 n , m2 n ) lim → √ 2}.

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 For each sequence of n terms s = s0, · · · , sn−1 and f ∈ n2, define

• s[f ] =
• {si | f (i) = 1} ∩
• {sc

i | f (i) = 0},

Nm( s) =

• {

s[f ] | f : n → 2 and |f −1(1)| = m}. Given two sequences s and t of n terms, define

• s E

t =

• 0≤i≤n

(Ni( s) = Ni(

• t)),

FC( s, t) = s E t → ((

• i<n−1

|si| ≥ |ti|) → |tn−1| ≥ |sn−1|).

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:

40