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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
1 / 35
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.
2 / 35
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.
3 / 35
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?
4 / 35
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
5 / 35
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?
6 / 35
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
7 / 35
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
8 / 35
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
9 / 35
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}.
10 / 35
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.
11 / 35
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.
12 / 35
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.
13 / 35
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.
14 / 35
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.
15 / 35
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
16 / 35
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}
17 / 35
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
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!)
19 / 35
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
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 → ⊥.
21 / 35
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!
22 / 35
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
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.
23 / 35
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?
24 / 35
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.
• 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.
25 / 35
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
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.
27 / 35
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)
28 / 35
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
29 / 35
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
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. �
31 / 35
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
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
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
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
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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