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On the “finitary” infinite Ramsey’s theorem
Florian Pelupessy
Tohoku University
Overview:
1 Motivation 2 “finitary” Ramsey 3 Inserting the parameter in “finitary” Ramsey 4 Some logical strengths for different values
On the finitary infinite Ramseys theorem Florian Pelupessy Tohoku - - PowerPoint PPT Presentation
On the finitary infinite Ramseys theorem Florian Pelupessy Tohoku - - PowerPoint PPT Presentation
On the finitary infinite Ramseys theorem Florian Pelupessy Tohoku University CTFM, Tokyo, 8 September 2015 Overview: 1 Motivation 2 finitary Ramsey 3 Inserting the parameter in finitary Ramsey 4 Some logical strengths for
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Motivation
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Motivation
From the study of first order concrete mathematical incomplete- ness there is the phenomenon of the phase transition: Given some T ϕ, examine the parametrised version ϕf . Classify parameter values f according to the provability of ϕf . Results in this programme follow certain heuristics.
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Motivation
From the study of first order concrete mathematical incomplete- ness there is the phenomenon of the phase transition: Given some T ϕ, examine the parametrised version ϕf . Classify parameter values f according to the provability of ϕf . Results in this programme follow certain heuristics. Question: Do we have something similar in Reverse Mathemat- ics?
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Remarks
Instead of unprovability we will examine equivalences.
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Remarks
Instead of unprovability we will examine equivalences. Theorems examined in reverse mathematics have no obvious parametrisation.
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The “finitary” infinite Ramsey’s theorem
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“finitary” Ramsey
The “finitary” pigeonhole principle was introduced by Tao, ex- amined by Gaspar and Kohlenbach. We examine the generalisation to Ramsey’s theorem.
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“finitary” Ramsey
Definition (AS)
F : {(codes of) finite sets} → N is asymptotically stable if for every sequence X1, X2, . . . of finite sets there is i such that F(Xi) = F(Xj) for all j > i.
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“finitary” Ramsey
Definition (AS)
F : {(codes of) finite sets} → N is asymptotically stable if for every sequence X1, X2, . . . of finite sets there is i such that F(Xi) = F(Xj) for all j > i.
Definition
1 [X]d = set of d-element subsets of X 2 [a, b]d = [{a, . . . , b}]d 3 For C : [X]d → [0, c] a set H ⊆ X is C-homogeneous if C
restricted to [H]d is constant.
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“finitary” Ramsey
Definition (RTk
d)
For every C : Nd → [0, k] there exists infinite C-homogeneous H ⊆ N.
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“finitary” Ramsey
Definition (RTk
d)
For every C : Nd → [0, k] there exists infinite C-homogeneous H ⊆ N.
Definition (FRTk
d)
For every F ∈ AS there exists R such that for all C : [0, R]d → [0, k] there exists C-homogeneous H of size > F(H).
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“finitary” Ramsey
Definition (RTk
d)
For every C : Nd → [0, k] there exists infinite C-homogeneous H ⊆ N.
Definition (FRTk
d)
For every F ∈ AS there exists R such that for all C : [0, R]d → [0, k] there exists C-homogeneous H of size > F(H).
Theorem
WKL0 ⊢ FRTk
d ↔ RTk d.
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Inserting the parameter
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Inserting the parameter:
We can use subsets of AS as parameter values:
Definition (FRTk
d(G))
For every F ∈ G there exists R such that for all C : [0, R]d → [0, k] there exists C-homogeneous H of size > F(H).
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Some values:
Definition (CF)
Constant functions
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Some values:
Definition (CF)
Constant functions
Definition (UI)
{F ∈ AS : ∃m∀X.F(X) ≤ max{min X, m}}
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Some values:
Definition (CF)
Constant functions
Definition (UI)
{F ∈ AS : ∃m∀X.F(X) ≤ max{min X, m}}
Definition (MD)
{F ∈ AS : min X = min Y → F(X) = F(Y )}
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Some logical strengths for different values
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Strengths for values
FRT(CF) is the finite Ramsey’s theorem, which is known to be provable in RCA0.
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Strengths for values
FRT(CF) is the finite Ramsey’s theorem, which is known to be provable in RCA0. FRT(UI) is equivalent to the Paris–Harrington principle, which is known to be equivalent to 1-Con(IΣd) when the dimension is fixed to d + 1.
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Strengths for values
Definition
1 WO(α) is the statement “α is well-founded”. 2 ω0 = 1 and ωn+1 = ωωn.
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Strengths for values
Definition
1 WO(α) is the statement “α is well-founded”. 2 ω0 = 1 and ωn+1 = ωωn.
Theorem
RCA0 ⊢ FRTd(MD) ↔ WO(ωd)
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Summary
RCA0 proves the following: FRT ↔ RT FRTk
d
← RTk
d for (d > 2)
FRTk
d
→ RTk
d
FRT(MD) ↔ AR ↔ WO(ε0) FRTd+1(MD) ↔ ARd ↔ WO(ωd+1) FRT(UI) ↔ 1-consistency of PA FRTd+1(UI) ↔ 1-consistency of IΣd FRT(CF) Furthermore, WKL0 ⊢ RTk
d → FRTk d.
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- Thank you for listening.