A constructive naive set theory and infinity

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A constructive 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 Conclusion A constructive naive set theory and Infinity Shunsuke Yatabe Center for Applied Philosophy and Ethics, Graduate School of Letters, Kyoto University [email protected] Kyoto Nonclassical Logic Workshop 19, November, 2015 1 / 35

Transcript of A constructive naive set theory and infinity

Page 1: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

A constructive naive set theory and Infinity

Shunsuke Yatabe

Center for Applied Philosophy and Ethics,Graduate School of Letters,

Kyoto [email protected]

Kyoto Nonclassical Logic Workshop19, November, 2015

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Page 2: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Note

This talk is based on my paper “A constructive naive set theoryand infinity” which is accepted to Notre Dame Journal ofFormal Logic in September.

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Page 3: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Outline

Known: Naive set theories do not imply a contradiction incontraction-free logics,

Set theory: CONS, a constructive naive set theory inFLew∀ (int. logic − the contraction rule),

Theme: we examine the nature of ω,• CONS is strongly circular,• we do not know much about circularly

defined infinite sets,

Results: negative answers to the standardness of ω:(A) CONS does not prove the crispness of ω,

0 (∀x)[x ∈ ω ∨ x < ω]

(B) A strong version of ω-rule (roughlyω =

∪n∈N{n}) implies a contradiction.

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Page 4: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Motivation(1): Two source of infinity

The foundational theme: the study of infinity

• Actual infinity• Example: ZFC (by Well-founded sets),• proof theoretically very strong theories are necessary,

• Potential infinity (the limit of a process)• Example: co-inductive objects, · · ·• proof theoretically weak theories are enough [Rat04],

What can we do in such weak theories?

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Page 5: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Theme: “potential” infinity

• CONS proves the fixed point theorem:

(∃X)(∀x)x ∈ X ≡ ϕ(x, X)

• Example:θ =ext {θ}

potentially infinite objects generate infinity, infinitedescending sequence.

Θ Θ

Θ

Θcircular process unfolding

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Page 6: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Motivation(2): study of circulary defined infinite sets

The fixed point theorem allows us to define ω:

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

Question: the nature of circularly defined sets is notwell-known,

• Is ω crisp, i.e. (∀x)[(x ∈ ω) ∨ (x < ω)]?• Is ω standard, i.e. can we exclude the possibility of

non-standard natural numbers?

• In case the standardness is not provable, what happens ifwe add an infinitary rule which implies the standardness ofω?

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Page 7: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Substructural logics

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Page 8: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

FLew∀

FLew∀= intuitionistic logic minus the contraction rule

α ` α ⊥ ` ` >Γ1 ` α Γ2, α,Θ ` βΓ1, Γ2,Θ ` β

cut

Γ ` α β,Π ` γα → β, Γ,Π ` γ

Γ, α ` βΓ ` α → β

Multiplicative connectives:

Γ, α, β,Σ ` δΓ, α ⊗ β,Σ ` δ

Γ ` α Σ ` βΓ,Σ ` α ⊗ β

Additive connectives:

Γ, αi,Σ ` δΓ, α1 ∧ α2,Σ ` δ

Γ, ` α Γ ` βΓ, ` α ∧ β

Γ, α,Σ ` δ Γ, β,Σ ` δΓ, α ∨ β,Σ ` δ

Γ ` αiΓ ` α1 ∨ α2

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Page 9: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

FLew∀ (conti.)

Quantifiers (y is not a free variable in Γ ` ∀xα and Γ, ∃xα ` βrespectively; s is a term):

Γ, α[x := s] ` βΓ, ∀xα ` β

Γ ` α[x := y]Γ ` ∀xα

Γ, α[x := y] ` βΓ, ∃xα ` β

Γ ` α[x := s]Γ ` ∃xα

Structural rules:

Γ, β, α,Σ ` δΓ, α, β,Σ ` δ e Γ ` δ

Γ, α,Σ ` δ w

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Page 10: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

A constructive naive set theory

• Let CONS be a set theory within FLew∀, which has abinary predicate ∈ and terms of the form {x : ϕ(x)}, andthe following two ∈-rules:

α[x := s], Γ ` βs ∈ {x : α}, Γ ` β

Γ ` α[x := s]Γ ` s ∈ {x : α}

• We define the following relations and terms as usual:

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

(∀z)[z ∈ x ↔ z ∈ y],The empty set ∅ = {x : x , x}.

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Page 11: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

The cut elimination

The cut elimination theorem:CONS enjoys the cut elimination.

The situation is essentially the same to [C03]:

• since CONS is “highly selfreferential .., it is not possibleto eliminate cuts by progressively decreasing thecomplexity of the cut formulas”,

• but “lack of contraction still allows to apply a standardelimination procedure, the induction on the length of thenumber of logical inferences” (∈-level [C03], grade[Pet00]),• if S’s upper sequents’s ∈-levels are i, j, then there is a

cut-free deduction of S whose ∈-level is ≤ i + j,• actually, the cut elimination is easier than standard ones.

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Page 12: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

The syntactical nature

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

• the axiom of extensionality implies a contradiction inCONS,• the axiom of extensionality says (∀x, y)[x =ext y → x = y],• it implies full contraction rule, so it implies a contradiction,

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

` t = u iff t is syntactically equivalent to u,

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

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Page 13: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Arithmetic in CONS

Testbed: termination judgementon propositions about ω

The fixed point theorem allows us to define ω:

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

Judgement of the membership of ωif s is in ω, then either

• the bottom case: s = 0,

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

(Note: 0 = ∅ and suc(n) = {n})

Our intension: the judgement process should terminateeventually.

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Page 14: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Arithmetic in CONS (conti.)

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

• ω is a crisp set, i.e. CONS proves tertium nondatur holds for ω

` (∀x)[x ∈ ω ∨ x < ω]

• Plus is a crisp relation,

• whether we can define a functionplus : ω × ω → ω,

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

• (ω,≤) becomes a linear ordering.

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Page 15: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

The problem

The non-extensionality prevents developing arithmetic:• both =,=ext are too strict to develop arithmetic.

• = is too strict because it is syntactical though naturalnumbers are defined by using =,

• =ext is still too strict to develop arithmetic: Let us assumeζ =ext 0. However two series,

(I) 0, suc(0), suc(suc(0)), suc(suc(suc(0))), · · ·(II) ζ, suc(ζ), suc(suc(ζ)), suc(suc(suc(ζ))), · · ·

might be completely different with respect to =ext .

• =,=ext are good for well-founded sets, but not good forpotentially infinite objects.

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Page 16: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Bisimulation ∼

The alternative identity relation for arithmetic (and co-inductiveobjects):• ∼ is a bisimulation relation as follows:

• (∀x, y)[x =ext y → x ∼ y],• suc(a) ∼ suc(b) ≡ a ∼ b.

• a ∼ b represents that a’s behavior with respect to suc isthe same to that of b: the number of iteration of suc in ais equal to that of b.

S00

Sζζ

~ ~suc

suc

a

b

~suc

suc

iterating

iterating

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Page 17: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

The definition of ω

We define an equivalence class [m] by ∼ of m for any m ∈ ωand set of equivalent classes 〈ω,∼〉 as follows:

• For any a, [a] = {x : x ∼ a},• ω is a set of ∼-equivalence classes whose representative

element is a natural number:

ω = {[n] : n ∈ ω}

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Page 18: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Theorem (A)

Theorem (A): CONS does not prove the crispness ofω, i.e.

0 (∀x)[x ∈ ω ∨ x < ω]

• We will construct a term t such that 0 t ∈ ω and 0 t < ω,• The rough idea:

fix is a fixed point of the successor function suc withrespect to ∼: suc(fix) ∼ fix: this is a conterexample.therefore there is no finite proof of fix ∈ ω.

• This is a negative answer to the standardness of ω!18 / 35

Page 19: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Proof of the non-crispness of ω

lemma

(1) CONS does not prove fix ∈ ω,

(2) CONS does not prove fix < ω

• This shows CONS does not prove (fix ∈ ω) ∨ (fix < ω) bydisjunction property,

• This will prove that CONS does not prove the crispness ofω (later!)

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Page 20: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Proof of the non-crispness of ω

(1) Assume otherwise: ` “fix ∈ ω”.• The proof should be of the form (existential property &

normal proof!)....

` t1 = 0 ∨ (∃y ∈ ω)t1 = suc(y)` t1 ∈ ω

....` t0 = suc(t1)

` t1 ∈ ω ⊗ t0 = suc(t1)` (∃x ∈ ω)t0 = suc(x)

` t0 = 0 ∨ (∃x ∈ ω)t0 = suc(x)

....` t0 ∼ fix

` t0 ∈ ω ⊗ t0 ∼ fix` (∃x ∈ ω)fix ∼ x` fix ∈ ω

• in this way, the proof is an infinite regress, and theproof never achieves the bottom case, tn ∼ 0 for somen, in finite steps.

Therefore there is no finite proof of ` “fix ∈ ω”.20 / 35

Page 21: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Proof of the non-crispness of ω (conti.)

(2) Assume otherwise: ` “fix < ω”.Similarly, the proof should be of the form

....x0 ∼ 0, fix ∼ suc(x0) ` ⊥

....x1 ∈ ω, x0 ∼ suc(x1), fix ∼ suc(x0) ` ⊥

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

x0 ∈ ω, fix ∼ suc(x0) ` ⊥(∃x0 ∈ ω)fix ∼ suc(x0) ` ⊥

....fix ∼ 0 ` ⊥

fix ∼ 0 ∨ (∃x0 ∈ ω)fix ∼ suc(x0) ` ⊥fix ∈ ω ` ⊥` fix < ω

Infinite regress!Therefore there is no finite proof of ` fix ∈ ω → ⊥.

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Page 22: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Proof of the non-crispness of ω

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

〈x, y〉 ∈ rk ≡ [(x ∼ 0 ∧ y = 0) ∨(∃z0, z1)[〈z0, z1〉 ∈ rk∧x =ext {z0} ∧ y = suc(z1)]]

Roughly speaking, rk unfolds the nested box, andcounts how many suc are nested.

• If CONS proves fix ∈ dom(rk) , i.e. (∃x)[〈fix, x〉 ∈ rk] (orits negation), this means CONS proves fix ∈ ω (fix < ω):contradiction!

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Page 23: A constructive naive set theory and infinity

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theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Non-tatality of Plus

Question: if fix ∈ ω, what is fix + fix =?• we would like to say fix + fix = fix, i.e.

plus(fix, fix, fix)

but there is no finite proof of it!• In this sense we cannot prove that the value of +

calculation is always determined.

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Page 24: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Proof theoretic ordinal of CONS?

• Remember: α is a proof theoretic ordinal of a system Γ ifα = sup{β : Γ ` β is a W.O},

• since we do not know whether fix ∈ ω or not, we cannotprove that < is a W.O. over ω where

n < m ⇐⇒ (∃x)plus(n, x, m)

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

• Question: since its proof theoretic ordinal is very low, can Isay ∈-terms are logical constants?

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Page 25: A constructive naive set theory and infinity

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theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Theorem (B)

• Theorem (B):A strong version of ω-rule, which is an infinitaryrule saying ω consists of numerals only (roughlyω =

∪n∈N{n}), implies a contradiction in CONS.

• A partial and negative answer to the claim of thestandardness of ω in CONS:• the ω-rule implies the ω-consistency, i.e. if ϕ(n) holds for

any numeral n then (∀x)[x ∈ ω → ϕ(x)] holds for any ϕ(x),• a theory (which is consistent with the ω-rule) has a

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

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Page 26: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Hidden Motivation: Simulating a logical operator

• Remember: a naive set theory, e.g. CONS,• doesn’t imply a contradiction without the contraction,• imply a strong form of circularity:

(∃X)(∀x)x ∈ X ≡ ϕ(x, X)

• Simulating logical operators in a naive set theory• well-known: we can simulate connectives [Pet00] [Tr04]• well-known: circularity forgives copying [H04]:

suc(n) ∈ θϕ ≡ ϕ ⊗ (n ∈ θϕ)(∀x)[x ∈ ω → x ∈ θϕ] : infinitary conjunction?

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

!ϕ, !ϕ ` ψ!ϕ ` ψ

` ψ`!ψ

Impossible!: adding ! to CONS implies a contradiction!

What principle allow to define this?26 / 35

Page 27: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Theorem (B)

• Theorem (B):A strong version of ω-rule, which is an infinitaryrule saying ω consists of numerals only (roughlyω =

∪n∈N{n}), implies a contradiction in CONS.

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

truth predicate by circularity,(2) the strong version of ω-rule implies that !? is

contractive and satisfies necessitation rule,(3) so this implies the contraction rule: Russell-like paradox

implies a contradiction.

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Page 28: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Coding

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

• d⊥e = {z : ⊥} and d>e = {z : >},• dx ∈ ye = {z : x ∈ y},• dx ∈ te = {z : x ∈ t}, ds ∈ ye = {z : s ∈ y} andds ∈ te = {z : s ∈ t} for some term s, t such thatz < FV(s)

∪FV(t),

• dϕ ◦ ψe = {z : z ∈ dϕe ◦ z ∈ dψe} for z < FV(ϕ)∪

FV(ψ)for any connective ◦,

• dQxϕ[x]e = {z : Qx(z ∈ dϕ[x]e)} for z < FV(ϕ)∪

FV(ψ)forany quantifier Q.

We can prove` ϕ ≡ (∅ ∈ dϕe)

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Page 29: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Simulating !

• Analogy of Girard’s ! (!ϕ means ϕ is contractive):• Let π be a relation defined by recursion:

• 〈0, dϕe〉 ∈ π ≡ >,• 〈suc(x), dϕe〉 ∈ π ≡ (∅ ∈ dϕe) ⊗ (〈x, dϕe〉 ∈ π)

• Let Π be a relation defined by recursion:

〈X, dϕe〉 ∈ Π ≡ (∀x)[x ∈ X → 〈x, dϕe〉 ∈ π]

It’s easy: 〈m, dϕe〉 ∈ π ≡ 〈{m}, dϕe〉 ∈ Π• !?ϕ ≡ 〈ω, dϕe〉 ∈ Π

• Intuitive meaning:

!?ϕ ≡ ϕ ⊗ ϕ ⊗ ϕ ⊗ · · ·︸ ︷︷ ︸ω many ϕ’s

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Page 30: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

ω-rule and the revenge

• If we ca prove !? satisfies the following, it simulate Girards! perfectly, i.e. it implies a contradiction!• Contractive:

!?ϕ, !?ϕ ` ψ!?ϕ ` ψ

• Necessitation: ` ϕ` !?ϕ

• We will show that a strong version of the ω-rule impliesthem!:

if Γ ` Ψ[{n}] for any n, then Γ ` Ψ[ω]

for any Ψ[x] of the form (∀x)[x ∈ X → ψ[x]] where Xdoes not occur in ψ[x],

30 / 35

Page 31: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

(2a) Contractivity

lemma: the ω-rule proves the contractivity:

!?ϕ, !?ϕ ` ψ!?ϕ ` ψ

proof This proof uses the ω-rule essentially.Fix any natural number m and n.

m+n many︷ ︸︸ ︷ϕ ⊗ · · · ⊗ ϕ `

m+n many︷ ︸︸ ︷ϕ ⊗ · · · ⊗ ϕ

〈{m + n}, dϕe〉 ∈ Π ` (〈{m}, dϕe〉 ∈ Π) ⊗ (〈{n}, dϕe〉 ∈ Π)!?ϕ ` (〈{m}, dϕe〉 ∈ Π) ⊗ (〈{n}, dϕe〉 ∈ Π)

Therefore, the ω-rule implies!?ϕ ` (〈ω, dϕe〉 ∈ Π) ⊗ (〈ω, dϕe〉 ∈ Π), i.e. !?ϕ ` !?ϕ ⊗ !?ϕ. �

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Page 32: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

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

ϕ ⊗ · · · ⊗ ϕ ` ϕ ⊗ · · · ⊗ ϕ〈{m + n}, dϕe〉 ∈ Π ` (〈{m}, dϕe〉 ∈ Π) ⊗ (〈{n}, dϕe〉 ∈ Π)

!?ϕ ` (〈{m}, dϕe〉 ∈ Π) ⊗ (〈{n}, dϕe〉 ∈ Π)

Therefore, the ω-rule implies

!?ϕ ` (〈ω, dϕe〉 ∈ Π) ⊗ (〈ω, dϕe〉 ∈ Π)

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

` (∀x, y) [〈x, dϕe〉 ∈ π ⊗ 〈y, dϕe〉 ∈ π]

this is not enough since it is not equivalent to

` (∀x)[〈x, dϕe〉 ∈ π] ⊗ (∀y)[〈y, dϕe〉 ∈ π]

Smuggling of distribution law!

32 / 35

Page 33: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

(2b) Necessitation

lemma: ω-rule proves

` ϕ` !?ϕ

proof This proof also uses the ω-rule essentially.Assume ` ϕ, and fix any natural number n.

` ϕ ` ϕ` ϕ ⊗ ϕ· · ·

` ϕ ⊗ · · · ⊗ ϕ︸ ︷︷ ︸n many

` 〈{n}, dϕe〉 ∈ Π �

33 / 35

Page 34: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

A paradox

Let us define the following Russell-like set:

R = {x : !?(x < x)}

lemma: The ω-rule proves ⊥

proof

R ∈ R ` !?R < R

!?R < R ` R ∈ R ⊥ ` ⊥R < R, !?R < R ` ⊥

!?R < R, !?R < R ` ⊥!?R < R ` ⊥

R ∈ R ` ⊥` R < R

....` R < R` !?R < R` R ∈ R

⊥ �

34 / 35

Page 35: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Conclusion

Set theory: • CONS, a constructive naive set theory inFLew∀ (intuitionistic logic minus thecontraction rule),

Motivation: • CONS is strongly circular: it allows to“infinitary copying” of formulae,

• CONS cannot proves the crispness of ω:how about the consistency with the ω-rule?

Theorem: • A strong version of ω-rule (roughlyω =

∪n∈N{n}) implies a contradiction in

CONS.

Proof • We can define !? which simulate Girard’s !,• The strong version of ω-rule implies !?

satisfies the contractivity and thenecessitation.

35 / 35

Page 36: A constructive naive set theory and infinity

A constructivenaive set

theory andInfinity

ShunsukeYatabe

Outline

PreliminariesMotivation

The logic & set theory

FeaturesThe cut-elimination

The syntactical nature

Arithmetic

Overcomenon-extensionality

(A)NoncrispnessTheorem (A)

Non-crispness of ω

Non-crispness of ω

(A’) Thecorollaries oftheorem (A)

(B)ω-rule` ⊥Theorem (B)

Hidden Motivation

(1) Simulating !

(2a) Contractivity

(2b) Necessitation

(3) Russell-likeparadox

Conclusion

Andrea Cantini. “The undecidability of Grisın’s set theory” Studialogica 74 (2003) pp.345-368

Grisin, V. N. 1982. Predicate and set-theoretic caliculi based onlogic without contractions. Math. USSR Izvestija 18: 41-59.

Petr Hajek. “On arithmetic in the Cantor-Łukasiewicz fuzzy settheory” AML (2004).

Uwe Petersen. Logic Without Contraction as Based on Inclusionand Unrestricted Abstraction. Studia Logica 64(3): 365-403(2000)

M. Rathjen. Predicativity, circularity, and anti-foundation. In G.Link, editor, One hundred years of Russell’s paradox, volume 6of Logic and its Applications, pages 191-219. de Gruyter, Berlin,2004

Kazushige Terui. Light affine set theory: A naive set theory ofpolynomial time. Studia Logica, Vol. 77, No. 1, pp. 9-40, 2004.

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