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11ヶ月前 (2015/11/18)にアップロードin学び

This is a slide of My talk at Kyoto Nonclassical Logic Workshop (19, November, 2015). This is ba...

This is a slide of My talk at Kyoto Nonclassical Logic Workshop (19, November, 2015). This is based on my paper "A constructive naive set theory and infinity" which was accepted to Notre Dame Journal of Formal Logic.

- A constructive

naive set

theory and

Infinity

Shunsuke

Yatabe

Outline

A constructive naive set theory and Infinity

Preliminaries

Motivation

The logic & set theory

Shunsuke Yatabe

Features

The cut-elimination

The syntactical nature

Arithmetic

Center for Applied Philosophy and Ethics,

Overcome

Graduate School of Letters,

non-extensionality

Kyoto University

(A)Noncrispness

shunsuke.yatabe@gmail.com

Theorem (A)

Non-crispness of ˜

ω

Non-crispness of ω

Kyoto Nonclassical Logic Workshop

(A’) The

corollaries of

19, November, 2015

theorem (A)

(B)ω-rule ⊥

Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

1 / 35

Conclusion - 1">A constructive

Note

naive set

theory and

Infinity

Shunsuke

Yatabe

Outline

Preliminaries

Motivation

The logic & set theory

Features

The cut-elimination

This talk is based on my paper “A constructive naive set theory

The syntactical nature

Arithmetic

and infinity” which is accepted to Notre Dame Journal of

Overcome

non-extensionality

Formal Logic in September.

(A)Noncrispness

Theorem (A)

Non-crispness of ˜

ω

Non-crispness of ω

(A’) The

corollaries of

theorem (A)

(B)ω-rule ⊥

Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

2 - 1">A constructive

Outline

naive set

theory and

Infinity

Shunsuke

Known: Naive set theories do not imply a contradiction in

Yatabe

contraction-free logics,

Outline

Preliminaries

Set theory: CONS, a constructive naive set theory in

Motivation

FL

The logic & set theory

ew∀ (int. logic − the contraction rule),

Features

Theme: we examine the nature of ω,

The cut-elimination

The syntactical nature

• CONS is strongly circular,

Arithmetic

Overcome

• we do not know much about circularly

non-extensionality

(A)Noncrispness

defined infinite sets,

Theorem (A)

Non-crispness of ˜

ω

Results: negative answers to the standardness of ω:

Non-crispness of ω

(A) CONS does not prove the crispness of ω,

(A’) The

corollaries of

theorem (A)

(∀x)[x ∈ ω ∨ x

ω]

(B)ω-rule ⊥

Theorem (B)

Hidden Motivation

(B) A strong version of ω-rule (roughly

(1) Simulating !

∪

(2a) Contractivity

ω =

(2b) Necessitation

n∈N{ ¯

n}) implies a contradiction.

(3) Russell-like

paradox

3 - 1">A constructive

Motivation(1): Two source of infinity

naive set

theory and

Infinity

Shunsuke

Yatabe

Outline

The foundational theme: the study of infinity

Preliminaries

Motivation

The logic & set theory

Features

• Actual infinity

The cut-elimination

The syntactical nature

•

Arithmetic

Example: ZFC (by Well-founded sets),

Overcome

•

non-extensionality

proof theoretically very strong theories are necessary,

(A)Noncrispness

Theorem (A)

Non-crispness of ˜

ω

• Potential infinity (the limit of a process)

Non-crispness of ω

• Example: co-inductive objects, · · ·

(A’) The

corollaries of

• proof theoretically weak theories are enough [Rat04],

theorem (A)

(B)ω-rule ⊥

Theorem (B)

What can we do in such weak theories?

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

4 - 1">A constructive

Theme: “potential” infinity

naive set

theory and

Infinity

Shunsuke

• CONS proves the fixed point theorem:

Yatabe

(∃X)(∀x)x ∈ X ≡ ϕ(x, X)

Outline

Preliminaries

• Example:

Motivation

θ =

The logic & set theory

ext {θ}

Features

potentially infinite objects generate infinity, infinite

The cut-elimination

The syntactical nature

descending sequence.

Arithmetic

Overcome

non-extensionality

Θ

Θ

(A)Noncrispness

Theorem (A)

Non-crispness of ˜

ω

Non-crispness of ω

(A’) The

corollaries of

Θ

theorem (A)

(B)ω-rule ⊥

Theorem (B)

Hidden Motivation

(1) Simulating !

Θ

circular process

unfolding

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

5 - 1">A constructive

Motivation(2): study of circulary defined infinite sets

naive set

theory and

Infinity

Shunsuke

Yatabe

The fixed point theorem allows us to define ω:

Outline

Preliminaries

(∀x)x ∈ ω ≡ [x = ¯0 ∨ (∃y)[y ∈ ω ⊗ x = suc(y)]]

Motivation

The logic & set theory

Features

The cut-elimination

The syntactical nature

Question: the nature of circularly defined sets is not

Arithmetic

Overcome

well-known,

non-extensionality

(A)Noncrispness

• Is ω crisp, i.e. (∀x)[(x ∈ ω) ∨ (x ω)]?

Theorem (A)

Non-crispness of ˜

ω

• Is ω standard, i.e. can we exclude the possibility of

Non-crispness of ω

non-standard natural numbers?

(A’) The

corollaries of

theorem (A)

• In case the standardness is not provable, what happens if

(B)ω-rule ⊥

we add an infinitary rule which implies the standardness of

Theorem (B)

ω

Hidden Motivation

?

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

6 - 1">A constructive

Substructural logics

naive set

theory and

Infinity

Shunsuke

Yatabe

Outline

Preliminaries

Motivation

The logic & set theory

Features

The cut-elimination

The syntactical nature

Arithmetic

Overcome

non-extensionality

(A)Noncrispness

Theorem (A)

Non-crispness of ˜

ω

Non-crispness of ω

(A’) The

corollaries of

theorem (A)

(B)ω-rule ⊥

Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

7 - 1">A constructive

FL

naive set

ew∀

theory and

Infinity

Shunsuke

FLew∀= intuitionistic logic minus the contraction rule

Yatabe

Outline

Γ1 α Γ2, α, Θ β

Preliminaries

α α

⊥

Γ1, Γ2, Θ β

cut

Motivation

The logic & set theory

Features

Γ α β, Π γ

Γ, α β

The cut-elimination

The syntactical nature

α → β, Γ, Π γ

Γ α → β

Arithmetic

Overcome

non-extensionality

Multiplicative connectives:

(A)Noncrispness

Theorem (A)

Γ, α, β, Σ δ

Γ α Σ β

Non-crispness of ˜

ω

Non-crispness of ω

Γ, α ⊗ β, Σ δ

Γ, Σ α ⊗ β

(A’) The

corollaries of

theorem (A)

Additive connectives:

(B)ω-rule ⊥

Theorem (B)

Γ, α

Γ, α Γ β

Γ, α, Σ δ Γ, β, Σ δ

Γ α

Hidden Motivation

i, Σ

δ

i

(1) Simulating !

Γ, α

Γ, α ∧ β

Γ, α ∨ β, Σ δ

Γ α

(2a) Contractivity

1 ∧ α2, Σ

δ

1 ∨ α2

(2b) Necessitation

(3) Russell-like

paradox

8 - 1">A constructive

FL

naive set

ew∀ (conti.)

theory and

Infinity

Shunsuke

Yatabe

Quantifiers (y is not a free variable in Γ

∀xα and Γ, ∃xα β

Outline

respectively; s is a term):

Preliminaries

Motivation

The logic & set theory

Γ, α[x := s] β

Γ α[x := y]

Features

Γ, ∀xα β

Γ ∀xα

The cut-elimination

The syntactical nature

Arithmetic

Overcome

non-extensionality

Γ, α[x := y] β

Γ α[x := s]

(A)Noncrispness

Γ, ∃

Γ ∃

Theorem (A)

xα

β

xα

Non-crispness of ˜

ω

Non-crispness of ω

Structural rules:

(A’) The

corollaries of

theorem (A)

Γ, β, α, Σ δ

Γ δ

(B)ω-rule ⊥

Γ, α, β, Σ δ e

Γ, α, Σ δ w

Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

9 - 1">A constructive

A constructive naive set theory

naive set

theory and

Infinity

Shunsuke

Yatabe

• Let CONS be a set theory within FLew∀, which has a

Outline

binary predicate ∈ and terms of the form {x : ϕ(x)}, and

Preliminaries

Motivation

the following two ∈-rules:

The logic & set theory

Features

α[x := s], Γ β

Γ α[x := s]

The cut-elimination

The syntactical nature

s ∈ {x : α}, Γ

β

Γ s ∈ {x : α}

Arithmetic

Overcome

non-extensionality

(A)Noncrispness

Theorem (A)

• We define the following relations and terms as usual:

Non-crispness of ˜

ω

Non-crispness of ω

Leibniz equality x = y iff (∀z)[x ∈ z ↔ y ∈ z],

(A’) The

corollaries of

Extensional equality x =

theorem (A)

ext y iff

(∀z)[z ∈ x ↔ z ∈ y],

(B)ω-rule ⊥

Theorem (B)

The empty set ∅ = {x : x

x}.

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

10 - 1">A constructive

The cut elimination

naive set

theory and

Infinity

Shunsuke

Yatabe

The cut elimination theorem:

CONS enjoys the cut elimination.

Outline

Preliminaries

Motivation

The situation is essentially the same to [C03]:

The logic & set theory

Features

• since CONS is “highly selfreferential .., it is not possible

The cut-elimination

to eliminate cuts by progressively decreasing the

The syntactical nature

Arithmetic

complexity of the cut formulas”,

Overcome

non-extensionality

(A)Noncrispness

Theorem (A)

• but “lack of contraction still allows to apply a standard

Non-crispness of ˜

ω

elimination procedure, the induction on the length of the

Non-crispness of ω

(A’) The

number of logical inferences” (∈-level [C03], grade

corollaries of

[Pet00]),

theorem (A)

•

(B)ω-rule ⊥

if S’s upper sequents’s ∈-levels are i, j, then there is a

Theorem (B)

cut-free deduction of S whose ∈-level is ≤ i + j,

Hidden Motivation

(1) Simulating !

• actually, the cut elimination is easier than standard ones.

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

11 - 1">A constructive

The syntactical nature

naive set

theory and

Infinity

Shunsuke

Yatabe

Outline

Naive set theories have a syntactic nature [Pet00][C03] [Tr04]:

Preliminaries

Motivation

The logic & set theory

• the axiom of extensionality implies a contradiction in

Features

The cut-elimination

CONS,

The syntactical nature

•

Arithmetic

the axiom of extensionality says (∀x, y)[x =ext y → x = y],

Overcome

•

non-extensionality

it implies full contraction rule, so it implies a contradiction,

(A)Noncrispness

Theorem (A)

Non-crispness of ˜

ω

• Furthermore, CONS is very syntactical (and weak) [Tr04]:

Non-crispness of ω

(A’) The

t = u iff t is syntactically equivalent to u,

corollaries of

theorem (A)

The proof is an easy application of the cut-elimination.

(B)ω-rule ⊥

Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

12 - 1">A constructive

Arithmetic in CONS

naive set

theory and

Infinity

Shunsuke

Yatabe

Testbed: termination judgement

Outline

on propositions about ω

Preliminaries

Motivation

The fixed point theorem allows us to define ω:

The logic & set theory

Features

The cut-elimination

(∀x)x ∈ ω ≡ [x = ¯0 ∨ (∃y)[y ∈ ω ∧ x = suc(y)]]

The syntactical nature

Arithmetic

Overcome

non-extensionality

Judgement of the membership of ω

(A)Noncrispness

if s is in ω, then either

Theorem (A)

Non-crispness of ˜

ω

•

Non-crispness of ω

the bottom case: s = ¯0,

(A’) The

• the successor step: s = suc(t) for some t ∈ ω.

corollaries of

theorem (A)

(Note: ¯0 = ∅ and suc(n) = {n})

(B)ω-rule ⊥

Theorem (B)

Hidden Motivation

Our intension: the judgement process should terminate

(1) Simulating !

(2a) Contractivity

eventually.

(2b) Necessitation

(3) Russell-like

paradox

13 - 1">A constructive

Arithmetic in CONS (conti.)

naive set

theory and

Infinity

Shunsuke

Yatabe

However, the fixed point theorem is not strong enough to show:

Outline

Preliminaries

• ω is a crisp set, i.e. CONS proves tertium non

Motivation

The logic & set theory

datur holds for ω

Features

The cut-elimination

The syntactical nature

(∀x)[x ∈ ω ∨ x

ω]

Arithmetic

Overcome

non-extensionality

• Plus is a crisp relation,

(A)Noncrispness

Theorem (A)

• whether we can define a function

Non-crispness of ˜

ω

Non-crispness of ω

plus : ω × ω → ω,

(A’) The

corollaries of

• the totality of the function plus (if we can define it),

theorem (A)

(B)ω-rule ⊥

• (ω, ≤) becomes a linear ordering.

Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

14 - 1">A constructive

The problem

naive set

theory and

Infinity

Shunsuke

Yatabe

The non-extensionality prevents developing arithmetic:

• both =, =

Outline

ext are too strict to develop arithmetic.

Preliminaries

Motivation

• = is too strict because it is syntactical though natural

The logic & set theory

numbers are defined by using =,

Features

The cut-elimination

The syntactical nature

• =

Arithmetic

ext is still too strict to develop arithmetic: Let us assume

Overcome

ζ =

non-extensionality

ext 0. However two series,

(A)Noncrispness

Theorem (A)

(I)

0, suc(0), suc(suc(0)), suc(suc(suc(0))), · · ·

Non-crispness of ˜

ω

Non-crispness of ω

(II)

ζ, suc(ζ), suc(suc(ζ)), suc(suc(suc(ζ))), · · ·

(A’) The

corollaries of

might be completely different with respect to =

theorem (A)

ext .

(B)ω-rule ⊥

Theorem (B)

Hidden Motivation

• =, =ext are good for well-founded sets, but not good for

(1) Simulating !

(2a) Contractivity

potentially infinite objects.

(2b) Necessitation

(3) Russell-like

paradox

15 - 1">A constructive

Bisimulation ∼

naive set

theory and

Infinity

Shunsuke

The alternative identity relation for arithmetic (and co-inductive

Yatabe

objects):

Outline

• ∼ is a bisimulation relation as follows:

Preliminaries

Motivation

• (∀x, y)[x =ext y → x ∼ y],

The logic & set theory

• suc(a) ∼ suc(b) ≡ a ∼ b.

Features

The cut-elimination

• a ∼ b represents that a’s behavior with respect to suc is

The syntactical nature

Arithmetic

the same to that of b: the number of iteration of suc in a

Overcome

non-extensionality

is equal to that of b.

(A)Noncrispness

Theorem (A)

iterating

Non-crispness of ˜

ω

0

S0

a

Non-crispness of ω

suc

suc

(A’) The

corollaries of

theorem (A)

∼

∼

∼

(B)ω-rule ⊥

iterating

Theorem (B)

Hidden Motivation

ζ

Sζ

b

(1) Simulating !

suc

suc

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

16 - 1">A constructive

The definition of ˜

ω

naive set

theory and

Infinity

Shunsuke

Yatabe

Outline

Preliminaries

Motivation

We define an equivalence class [m] by ∼ of m for any m ∈ ω

The logic & set theory

and set of equivalent classes ˜

ω, ∼ as follows:

Features

The cut-elimination

• For any a, [a] = {x : x ∼ a},

The syntactical nature

Arithmetic

•

Overcome

˜

ω is a set of ∼-equivalence classes whose representative

non-extensionality

element is a natural number:

(A)Noncrispness

Theorem (A)

Non-crispness of ˜

ω

˜

ω = {[n] : n ∈ ω}

Non-crispness of ω

(A’) The

corollaries of

theorem (A)

(B)ω-rule ⊥

Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

17 - 1">A constructive

Theorem (A)

naive set

theory and

Infinity

Shunsuke

Yatabe

Theorem (A): CONS does not prove the crispness of

ω, i.e.

Outline

(∀x)[x ∈ ω ∨ x

ω]

Preliminaries

Motivation

The logic & set theory

Features

• We will construct a term t such that t ∈ ω and t

ω,

The cut-elimination

The syntactical nature

• The rough idea:

Arithmetic

Overcome

fix is a fixed point of the successor function suc with

non-extensionality

(A)Noncrispness

respect to ∼: suc(fix) ∼ fix: this is a conterexample.

Theorem (A)

therefore there is no finite proof of fix ∈ ω.

Non-crispness of ˜

ω

Non-crispness of ω

(A’) The

corollaries of

theorem (A)

(B)ω-rule ⊥

Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

• This is a negative answer to the standardness of ω!

(3) Russell-like

paradox

18 - 1">A constructive

Proof of the non-crispness of ˜

ω

naive set

theory and

Infinity

Shunsuke

Yatabe

Outline

Preliminaries

lemma

Motivation

The logic & set theory

(1) CONS does not prove fix ∈ ˜

ω,

Features

The cut-elimination

(2) CONS does not prove fix

˜

ω

The syntactical nature

Arithmetic

Overcome

non-extensionality

(A)Noncrispness

Theorem (A)

• This shows CONS does not prove (fix ∈ ˜ω) ∨ (fix

˜

ω) by

Non-crispness of ˜

ω

Non-crispness of ω

disjunction property,

(A’) The

corollaries of

• This will prove that CONS does not prove the crispness of

theorem (A)

ω (later!)

(B)ω-rule ⊥

Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

19 - 1">A constructive

Proof of the non-crispness of ˜

ω

naive set

theory and

Infinity

(1) Assume otherwise:

“fix ∈ ˜

ω”.

Shunsuke

Yatabe

• The proof should be of the form (existential property &

normal proof!)

Outline

.

Preliminaries

..

Motivation

.

.

The logic & set theory

t

.

1 = ¯

0 ∨ (∃y ∈ ω)t1 = suc(y)

..

Features

t

t

The cut-elimination

1 ∈ ω

0 = suc( t1)

The syntactical nature

t1 ∈ ω ⊗ t0 = suc(t1)

Arithmetic

.

Overcome

(∃x ∈ ω)t

..

non-extensionality

0 = suc(x)

.

(A)Noncrispness

t0 = ¯0 ∨ (∃x ∈ ω)t0 = suc(x)

t0 ∼ fix

Theorem (A)

t

Non-crispness of ˜

ω

0 ∈ ω ⊗ t0 ∼ fix

Non-crispness of ω

(∃x ∈ ω)fix ∼ x

(A’) The

corollaries of

fix ∈ ˜

ω

theorem (A)

(B)ω-rule ⊥

Theorem (B)

• in this way, the proof is an infinite regress, and the

Hidden Motivation

(1) Simulating !

proof never achieves the bottom case, tn ∼ 0 for some

(2a) Contractivity

(2b) Necessitation

n, in finite steps.

(3) Russell-like

paradox

Therefore there is no finite proof of

“fix ∈ ˜

ω”.

20 - 1">A constructive

Proof of the non-crispness of ˜

ω (conti.)

naive set

theory and

Infinity

Shunsuke

Yatabe

(2) Assume otherwise:

“fix

˜

ω”.

Outline

Similarly, the proof should be of the form

Preliminaries

.

Motivation

.

.

The logic & set theory

.

.

.

.

x

Features

.

1 ∈ ˜

ω, x0 ∼ suc(x1), fix ∼ suc(x0) ⊥

The cut-elimination

x0 ∼ ¯0, fix ∼ suc(x0)

⊥ (∃x1 ∈ ω)x0 ∼ suc(x1)], fix ∼ suc(x0) ⊥

The syntactical nature

Arithmetic

[x0 ∼ ¯0 ∨ (∃x1 ∈ ω)x0 ∼ suc(x1)], fix ∼ suc(x0)

⊥

.

.

Overcome

x

.

non-extensionality

0 ∈ ω, fix ∼ suc(x0)

⊥

.

(A)Noncrispness

(∃x0 ∈ ω)fix ∼ suc(x0)

⊥

fix ∼ ¯0

⊥

Theorem (A)

fix ∼ ¯0 ∨ (∃x0 ∈ ω)fix ∼ suc(x0)

⊥

Non-crispness of ˜

ω

Non-crispness of ω

fix ∈ ˜

ω ⊥

(A’) The

fix

˜

ω

corollaries of

theorem (A)

(B)ω-rule ⊥

Theorem (B)

Infinite regress!

Hidden Motivation

(1) Simulating !

Therefore there is no finite proof of

fix ∈ ˜

ω → ⊥.

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

21 - 1">A constructive

Proof of the non-crispness of ω

naive set

theory and

Infinity

Shunsuke

Yatabe

Outline

• rk is a relation over set such that rk ⊆ ˜ω × ω:

Preliminaries

Motivation

x, y ∈ rk ≡ [(x ∼ 0 ∧ y = 0) ∨

The logic & set theory

Features

(∃z0, z1)[ z0, z1 ∈ rk

The cut-elimination

The syntactical nature

∧x =

Arithmetic

ext { z0} ∧ y = suc( z1)]]

Overcome

non-extensionality

(A)Noncrispness

Roughly speaking, rk unfolds the nested box, and

Theorem (A)

counts how many suc are nested.

Non-crispness of ˜

ω

Non-crispness of ω

(A’) The

corollaries of

• If CONS proves fix ∈ dom(rk) , i.e. (∃x)[ fix, x ∈ rk] (or

theorem (A)

its negation), this means CONS proves fix ∈ ˜

ω (fix

˜

ω):

(B)ω-rule ⊥

Theorem (B)

contradiction!

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

22 - 1">A constructive

Non-tatality of Plus

naive set

theory and

Infinity

Shunsuke

Yatabe

Outline

Preliminaries

Motivation

Question: if fix ∈ ω, what is fix + fix =?

The logic & set theory

• we would like to say fix + fix = fix, i.e.

Features

The cut-elimination

The syntactical nature

Arithmetic

plus(fix, fix, fix)

Overcome

non-extensionality

(A)Noncrispness

but there is no finite proof of it!

Theorem (A)

Non-crispness of ˜

ω

• In this sense we cannot prove that the value of +

Non-crispness of ω

(A’) The

calculation is always determined.

corollaries of

theorem (A)

(B)ω-rule ⊥

Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

23 - 1">A constructive

Proof theoretic ordinal of CONS?

naive set

theory and

Infinity

Shunsuke

Yatabe

Outline

•

Preliminaries

Remember: α is a proof theoretic ordinal of a system Γ if

Motivation

α = sup{β : Γ β is a W.O},

The logic & set theory

Features

• since we do not know whether fix ∈ ω or not, we cannot

The cut-elimination

The syntactical nature

prove that < is a W.O. over ω where

Arithmetic

Overcome

non-extensionality

n < m ⇐⇒ (∃x)plus(n, x, m)

(A)Noncrispness

Theorem (A)

Non-crispness of ˜

ω

• This suggests that ω is a proof theoretic ordinal of CONS,

Non-crispness of ω

(A’) The

• Question: since its proof theoretic ordinal is very low, can I

corollaries of

theorem (A)

say ∈-terms are logical constants?

(B)ω-rule ⊥

Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

24 - 1">A constructive

Theorem (B)

naive set

theory and

Infinity

Shunsuke

Yatabe

Outline

• Theorem (B):

Preliminaries

A strong version of ω-rule, which is an infinitary

Motivation

The logic & set theory

rule saying ω consists of numerals only (roughly

Features

∪

ω =

The cut-elimination

n∈N{ ¯

n}), implies a contradiction in CONS.

The syntactical nature

Arithmetic

Overcome

non-extensionality

• A partial and negative answer to the claim of the

(A)Noncrispness

standardness of ω in CONS:

Theorem (A)

Non-crispness of ˜

ω

• the ω-rule implies the ω-consistency, i.e. if ϕ( ¯n) holds for

Non-crispness of ω

any numeral ¯n then (∀x)[x ∈ ω → ϕ(x)] holds for any ϕ(x),

(A’) The

corollaries of

• a theory (which is consistent with the ω-rule) has a

theorem (A)

standard model, i.e. any natural number in that model is a

(B)ω-rule ⊥

numeral.

Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

25 - 1">A constructive

Hidden Motivation: Simulating a logical operator

naive set

theory and

Infinity

Shunsuke

• Remember: a naive set theory, e.g. CONS,

Yatabe

• doesn’t imply a contradiction without the contraction,

Outline

• imply a strong form of circularity:

Preliminaries

Motivation

(∃X)(∀x)x ∈ X ≡ ϕ(x, X)

The logic & set theory

Features

• Simulating logical operators in a naive set theory

The cut-elimination

The syntactical nature

• well-known: we can simulate connectives [Pet00] [Tr04]

Arithmetic

• well-known: circularity forgives copying [H04]:

Overcome

non-extensionality

(A)Noncrispness

suc( ¯

n) ∈ θϕ ≡ ϕ ⊗ ( ¯n ∈ θϕ)

Theorem (A)

Non-crispness of ˜

ω

(∀x)[x ∈ ω → x ∈ θϕ]

: infinitary conjunction?

Non-crispness of ω

(A’) The

• Question: can we simulate Girard’s ! in CONS?

corollaries of

theorem (A)

!ϕ, !ϕ

ψ

ψ

(B)ω-rule ⊥

!ϕ

ψ

!ψ

Theorem (B)

Hidden Motivation

(1) Simulating !

Impossible!: adding ! to CONS implies a contradiction!

(2a) Contractivity

(2b) Necessitation

What principle allow to define this?

(3) Russell-like

paradox

26 - 1">A constructive

Theorem (B)

naive set

theory and

Infinity

Shunsuke

Yatabe

Outline

• Theorem (B):

Preliminaries

A strong version of ω-rule, which is an infinitary

Motivation

The logic & set theory

rule saying ω consists of numerals only (roughly

∪

Features

ω =

The cut-elimination

n∈N{ ¯

n}), implies a contradiction in CONS.

The syntactical nature

Arithmetic

Overcome

non-extensionality

• Proof:

(A)Noncrispness

(1) defining !-like operator !? by using coding and a total

Theorem (A)

Non-crispness of ˜

ω

truth predicate by circularity,

Non-crispness of ω

(2) the strong version of ω-rule implies that !? is

(A’) The

corollaries of

contractive and satisfies necessitation rule,

theorem (A)

(3) so this implies the contraction rule: Russell-like paradox

(B)ω-rule ⊥

implies a contradiction.

Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

27 - 1">A constructive

Coding

naive set

theory and

Infinity

Shunsuke

Yatabe

For any formula ϕ, the code of ϕ, ϕ , is inductively defined:

Outline

• ⊥ = {z : ⊥} and

= {z : },

Preliminaries

Motivation

• x ∈ y = {z : x ∈ y},

The logic & set theory

Features

• x ∈ t = {z : x ∈ t}, s ∈ y = {z : s ∈ y} and

The cut-elimination

The syntactical nature

s ∈ t = {z : s ∈ t} for some term s, t such that

∪

Arithmetic

Overcome

z

FV(s)

FV(t),

non-extensionality

∪

(A)Noncrispness

• ϕ ◦ ψ = {z : z ∈ ϕ ◦ z ∈ ψ } for z FV(ϕ)

FV(ψ)

Theorem (A)

for any connective ◦,

Non-crispness of ˜

ω

Non-crispness of ω

∪

• Qxϕ[x] = {z : Qx(z ∈ ϕ[x] )} for z FV(ϕ)

FV(ψ)for

(A’) The

corollaries of

any quantifier Q.

theorem (A)

(B)ω-rule ⊥

We can prove

Theorem (B)

ϕ ≡ (∅ ∈ ϕ )

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

28 - 1">A constructive

Simulating !

naive set

theory and

Infinity

Shunsuke

Yatabe

• Analogy of Girard’s ! (!ϕ means ϕ is contractive):

Outline

• Let π be a relation defined by recursion:

Preliminaries

• 0, ϕ ∈ π ≡ ,

Motivation

The logic & set theory

• suc(x), ϕ ∈ π ≡ (∅ ∈ ϕ ) ⊗ ( x, ϕ ∈ π)

Features

• Let Π be a relation defined by recursion:

The cut-elimination

The syntactical nature

Arithmetic

X, ϕ

∈ Π ≡ (∀x)[x ∈ X → x, ϕ ∈ π]

Overcome

non-extensionality

(A)Noncrispness

It’s easy:

¯

m, ϕ

∈ π ≡ { ¯m}, ϕ ∈ Π

Theorem (A)

•

Non-crispness of ˜

ω

!?ϕ ≡ ω, ϕ

∈ Π

Non-crispness of ω

• Intuitive meaning:

(A’) The

corollaries of

theorem (A)

!?ϕ ≡ ϕ ⊗ ϕ ⊗ ϕ ⊗ · · ·

(B)ω-rule ⊥

Theorem (B)

ω many ϕ’s

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

29 - 1">A constructive

ω-rule and the revenge

naive set

theory and

Infinity

Shunsuke

Yatabe

• If we ca prove !? satisfies the following, it simulate Girards

Outline

! perfectly, i.e. it implies a contradiction!

Preliminaries

• Contractive:

Motivation

!?ϕ, !?ϕ

ψ

The logic & set theory

Features

!?ϕ

ψ

The cut-elimination

The syntactical nature

• Necessitation:

Arithmetic

ϕ

Overcome

non-extensionality

!?ϕ

(A)Noncrispness

Theorem (A)

Non-crispness of ˜

ω

• We will show that a strong version of the ω-rule implies

Non-crispness of ω

them!:

(A’) The

corollaries of

theorem (A)

if Γ

Ψ[{ ¯n}] for any n, then Γ Ψ[ω]

(B)ω-rule ⊥

Theorem (B)

for any Ψ[x] of the form (∀x)[x ∈ X → ψ[x]] where X

Hidden Motivation

(1) Simulating !

does not occur in ψ[x],

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

30 - 1">A constructive

(2a) Contractivity

naive set

theory and

Infinity

Shunsuke

Yatabe

lemma: the ω-rule proves the contractivity:

Outline

!?ϕ, !?ϕ

ψ

Preliminaries

Motivation

!?ϕ

ψ

The logic & set theory

Features

The cut-elimination

proof This proof uses the ω-rule essentially.

The syntactical nature

Arithmetic

Fix any natural number m and n.

Overcome

non-extensionality

(A)Noncrispness

m+n many

m+n many

Theorem (A)

Non-crispness of ˜

ω

ϕ ⊗ · · · ⊗ ϕ

ϕ ⊗ · · · ⊗ ϕ

Non-crispness of ω

(A’) The

{m + n}, ϕ ∈ Π

( { ¯

m}, ϕ

∈ Π) ⊗ ( { ¯n}, ϕ ∈ Π)

corollaries of

theorem (A)

!?ϕ

( { ¯

m}, ϕ

∈ Π) ⊗ ( { ¯n}, ϕ ∈ Π)

(B)ω-rule ⊥

Theorem (B)

Hidden Motivation

Therefore, the ω-rule implies

(1) Simulating !

(2a) Contractivity

!?ϕ

( ω, ϕ

∈ Π) ⊗ ( ω, ϕ ∈ Π), i.e. !?ϕ !?ϕ ⊗ !?ϕ.

(2b) Necessitation

(3) Russell-like

paradox

31 - 1">A constructive

Contractivity: why do we need the strengthen ω-rule?

naive set

theory and

Infinity

Shunsuke

Yatabe

ϕ ⊗ · · · ⊗ ϕ

ϕ ⊗ · · · ⊗ ϕ

Outline

{m + n}, ϕ ∈ Π

( { ¯

m}, ϕ

∈ Π) ⊗ ( { ¯n}, ϕ ∈ Π)

Preliminaries

!?ϕ

( { ¯

m}, ϕ

∈ Π) ⊗ ( { ¯n}, ϕ ∈ Π)

Motivation

The logic & set theory

Features

Therefore, the ω-rule implies

The cut-elimination

The syntactical nature

Arithmetic

!?ϕ

( ω, ϕ

∈ Π) ⊗ ( ω, ϕ ∈ Π)

Overcome

non-extensionality

(A)Noncrispness

Theorem (A)

Non-crispness of ˜

ω

Note: the standard form of the ω-rule just implies

Non-crispness of ω

(A’) The

(∀x, y) [ x, ϕ

∈ π ⊗ y, ϕ ∈ π]

corollaries of

theorem (A)

this is not enough since it is not equivalent to

(B)ω-rule ⊥

Theorem (B)

Hidden Motivation

(∀x)[ x, ϕ

∈ π] ⊗ (∀y)[ y, ϕ ∈ π]

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

Smuggling of distribution law!

(3) Russell-like

paradox

32 - 1">A constructive

(2b) Necessitation

naive set

theory and

Infinity

Shunsuke

Yatabe

ϕ

Outline

Preliminaries

lemma: ω-rule proves

!?ϕ

Motivation

The logic & set theory

Features

proof This proof also uses the ω-rule essentially.

The cut-elimination

Assume

ϕ, and fix any natural number n.

The syntactical nature

Arithmetic

Overcome

ϕ

ϕ

non-extensionality

(A)Noncrispness

ϕ ⊗ ϕ

Theorem (A)

· · ·

Non-crispness of ˜

ω

ϕ ⊗ · · · ⊗ ϕ

Non-crispness of ω

(A’) The

n many

corollaries of

theorem (A)

{ ¯n}, ϕ ∈ Π

(B)ω-rule ⊥

Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

33 - 1">A constructive

A paradox

naive set

theory and

Infinity

Shunsuke

Let us define the following Russell-like set:

Yatabe

Outline

R = {x : !?(x

x)}

Preliminaries

Motivation

The logic & set theory

Features

lemma: The ω-rule proves ⊥

The cut-elimination

The syntactical nature

Arithmetic

proof

Overcome

non-extensionality

(A)Noncrispness

!? R

R

R ∈ R ⊥

⊥

Theorem (A)

Non-crispness of ˜

ω

R

R, !?R

R

⊥

Non-crispness of ω

..

(A’) The

!? R

R, !?R

R

⊥

.

corollaries of

.

theorem (A)

R ∈ R

!? R

R

!? R

R

⊥

R

R

(B)ω-rule ⊥

R ∈ R

⊥

!? R

R

Theorem (B)

Hidden Motivation

R

R

R ∈ R

(1) Simulating !

(2a) Contractivity

⊥

(2b) Necessitation

(3) Russell-like

paradox

34 - 1">A constructive

Conclusion

naive set

theory and

Infinity

Shunsuke

Yatabe

Set theory:

• CONS, a constructive naive set theory in

Outline

FLew∀ (intuitionistic logic minus the

Preliminaries

contraction rule),

Motivation

The logic & set theory

Motivation:

• CONS is strongly circular: it allows to

Features

“infinitary copying” of formulae,

The cut-elimination

The syntactical nature

• CONS cannot proves the crispness of ω:

Arithmetic

Overcome

how about the consistency with the ω-rule?

non-extensionality

(A)Noncrispness

Theorem:

• A strong version of ω-rule (roughly

Theorem (A)

∪

Non-crispness of ˜

ω

ω = n∈N{ ¯n}) implies a contradiction in

Non-crispness of ω

CONS.

(A’) The

corollaries of

theorem (A)

Proof

• We can define !? which simulate Girard’s !,

(B)ω-rule ⊥

• The strong version of ω-rule implies !?

Theorem (B)

satisfies the contractivity and the

Hidden Motivation

(1) Simulating !

necessitation.

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

35 - 1">A constructive

Andrea Cantini. “The undecidability of Gris˘ın’s set theory” Studia

naive set

theory and

logica 74 (2003) pp.345-368

Infinity

Shunsuke

Griˇsin, V. N. 1982. Predicate and set-theoretic caliculi based on

Yatabe

logic without contractions. Math. USSR Izvestija 18: 41-59.

Outline

Petr Hajek. “On arithmetic in the Cantor-Łukasiewicz fuzzy set

Preliminaries

Motivation

theory” AML (2004).

The logic & set theory

Features

Uwe Petersen. Logic Without Contraction as Based on Inclusion

The cut-elimination

and Unrestricted Abstraction. Studia Logica 64(3): 365-403

The syntactical nature

Arithmetic

(2000)

Overcome

non-extensionality

(A)Noncrispness

M. Rathjen. Predicativity, circularity, and anti-foundation. In G.

Theorem (A)

Link, editor, One hundred years of Russell’s paradox, volume 6

Non-crispness of ˜

ω

Non-crispness of ω

of Logic and its Applications, pages 191-219. de Gruyter, Berlin,

(A’) The

2004

corollaries of

theorem (A)

Kazushige Terui. Light affine set theory: A naive set theory of

(B)ω-rule ⊥

polynomial time. Studia Logica, Vol. 77, No. 1, pp. 9-40, 2004.

Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-like

paradox

35