Recursively defined trees and their maximal order types Jeroen Van - - PowerPoint PPT Presentation

Recursively defined trees and their maximal order types Jeroen Van der Meeren 2 CTFM 2013 2 Work related with a program between Michael Rathjen and Andreas Weiermann

Introduction Recursively defined trees Conclusions Structure presentation 1 Introduction 2 Recursively defined trees 3 Conclusions Jeroen Van der Meeren Recursively defined trees and their maximal order types

Introduction Recursively defined trees Conclusions The theorem of Kruskal Theorem (Kruskal) T is a wpo. Jeroen Van der Meeren Recursively defined trees and their maximal order types

Introduction Recursively defined trees Conclusions The theorem of Kruskal Theorem (Kruskal) T is a wpo. What is T ? T is the set of finite planar rooted trees: • is an element of T , Jeroen Van der Meeren Recursively defined trees and their maximal order types

Introduction Recursively defined trees Conclusions The theorem of Kruskal Theorem (Kruskal) T is a wpo. What is T ? T is the set of finite planar rooted trees: • is an element of T , If T 1 , . . . , T n ∈ T , then ❆ ✁ ❆ ✁ T 1 T n ❆ ✁ ❆ ✁ . . . ❆ ✁ ❆ ✁ t t ◗ ✑✑✑ ❭ ✜ ❭ ✜ ◗ ◗ ❭ ✜ t is also an element of T . Jeroen Van der Meeren Recursively defined trees and their maximal order types

Introduction Recursively defined trees Conclusions Tree-embeddability t � t ▲ � ▲ t ❅ ▲ t t t ❅ ☞☞ ▲ ☞☞ t t t t t ▲ ❅ � ❆ ✁ ≤ T ☞ ▲ ☞ ❅ � ❆ ✁ t t t t t t t ◗ ✑✑✑ ❭ ✜ ❆ ✁ ❭ ✜ ◗ ◗ ❆ ✁ ❭ ✜ t t ◗ ✑✑✑ t ◗ ◗ t Jeroen Van der Meeren Recursively defined trees and their maximal order types

Introduction Recursively defined trees Conclusions Tree-embeddability t � t ▲ � ▲ t ✐ ❅ ▲ t t t ❅ ☞☞ ▲ ☞☞ t t ✐ t t ✐ t ✐ ▲ ❅ � ❆ ✁ ≤ T ☞ ▲ ☞ ❅ � ❆ ✁ t t t t ✐ t ✐ t ✐ t ◗ ✑✑✑ ❭ ✜ ❆ ✁ ❭ ✜ ◗ ◗ ❆ ✁ ❭ ✜ ✐ t t ◗ ✑✑✑ t ◗ ◗ t Jeroen Van der Meeren Recursively defined trees and their maximal order types

Introduction Recursively defined trees Conclusions Tree-embeddability t � t ▲ � ▲ ✐ t ❅ ▲ t t t ❅ ☞☞ ▲ ☞☞ t t t ✐ ✐ t t ✐ ▲ ❅ � ❆ ✁ ≤ T ☞ ▲ ☞ ❅ � ❆ ✁ t t t t ✐ t ✐ t t ✐ ◗ ✑✑✑ ❭ ✜ ❆ ✁ ❭ ✜ ◗ ◗ ❆ ✁ ❭ ✜ t ✐ t ◗ ✑✑✑ t ◗ ◗ t t ▲ ▲ t t ❆ ✁ ▲ t t t ❆ ✁ ☞☞ ▲ ☞☞ t t t t ▲ ❆ ❆ ❆ ✁ ≤ T ☞ ▲ ☞ ❆ ❆ ❆ ✁ t t t t t t t ◗ ✑✑✑ ❭ ✜ ❆ ✁ ◗ ❭ ✜ ◗ ❆ ✁ ❭ ✜ t t Jeroen Van der Meeren Recursively defined trees and their maximal order types

Introduction Recursively defined trees Conclusions Tree-embeddability t � t ▲ � ▲ ✐ t ❅ ▲ t t t ❅ ☞☞ ▲ ☞☞ t t ✐ t t ✐ t ✐ ▲ ❅ � ❆ ✁ ≤ T ☞ ▲ ☞ ❅ � ❆ ✁ t t t ✐ t t t ✐ ✐ t ◗ ✑✑✑ ❭ ✜ ❆ ✁ ❭ ✜ ◗ ◗ ❆ ✁ ❭ ✜ ✐ t t ◗ ✑✑✑ t ◗ ◗ t t ▲ ▲ t ✐ t ❆ ✁ ▲ t t t ❆ ✁ ☞☞ ▲ ☞☞ ✐ t t t ✐ t ✐ ▲ ❆ ❆ ❆ ✁ ≤ T ☞ ▲ ☞ ❆ ❆ ❆ ✁ t t t t ✐ t ✐ t ✐ t ◗ ✑✑✑ ❭ ✜ ❆ ✁ ◗ ❭ ✜ ◗ ❆ ✁ ❭ ✜ ✐ t t Jeroen Van der Meeren Recursively defined trees and their maximal order types

Introduction Recursively defined trees Conclusions Tree-embeddability: definition ❆ ✁ ❆ ✁ T 1 T n ❆ ✁ ❆ ✁ ❆ ✁ T i . . . ❆ ✁ ❆ ✁ ❆ ✁ ≤ T t t ◗ ✑✑✑ ❭ ✜ ❆ ✁ ❭ ✜ ◗ t ◗ ❭ ✜ t Jeroen Van der Meeren Recursively defined trees and their maximal order types

Introduction Recursively defined trees Conclusions Tree-embeddability: definition ❆ ✁ ❆ ✁ T 1 T n ❆ ✁ ❆ ✁ ❆ ✁ T i . . . ❆ ✁ ❆ ✁ ❆ ✁ ≤ T t t ◗ ✑✑✑ ❭ ✜ ❆ ✁ ❭ ✜ ◗ t ◗ ❭ ✜ t If k 1 < k 2 < · · · < k n and T i ≤ T T ′ k i for every i , then ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ T ′ T ′ T ′ T 1 T n ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ k 1 k i k n . . . . . . . . . . . . . . . ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ≤ T ❳ ✘✘✘✘✘✘✘✘ ❛ ✦✦✦✦✦ t t ❳ t t t ◗ ✑✑✑ ❝ ★ ❳ ❛ ❭ ✜ ❭ ✜ ❳ ❛ ❝ ★★ ❳ ◗ ❳ ❛ ❳ ◗ ❭ ✜ ❳ ❝ ❛ t t Jeroen Van der Meeren Recursively defined trees and their maximal order types

Introduction Recursively defined trees Conclusions The theorem of Kruskal Theorem (Kruskal) T is a wpo. What is a wpo? A well-partial-ordering (wpo) is a partial ordering that is well-founded, has no infinite antichain. Jeroen Van der Meeren Recursively defined trees and their maximal order types

Introduction Recursively defined trees Conclusions The theorem of Kruskal Theorem (Kruskal) T is a wpo. What is a wpo? A well-partial-ordering (wpo) is a partial ordering that is well-founded, has no infinite antichain. Definition A well-partial-ordering ( X , ≤ X ) is a partial ordering such that for every infinite sequence x 1 , x 2 , . . . of elements in X , indices i < j exists such that x i ≤ X x j . Jeroen Van der Meeren Recursively defined trees and their maximal order types

Introduction Recursively defined trees Conclusions The theorem of Kruskal Theorem (Kruskal) T is a wpo. = Theorem (Kruskal) For every infinite sequence T 1 , T 2 , . . . of elements in T , there exists indices i < j such that T i ≤ T T j . Jeroen Van der Meeren Recursively defined trees and their maximal order types

Introduction Recursively defined trees Conclusions Theorem T is wpo. ⇓ New tree-class T ( W ) Theorem T ( W ) is wpo. Interested in: Is this theorem true? What is the maximal order type of T ( W )? Which theories T can (and which cannot) prove ‘ T ( W ) is wpo’? Jeroen Van der Meeren Recursively defined trees and their maximal order types

Introduction Recursively defined trees Conclusions Why interested in this? Trying to obtain the strength of trees with gap-condition. A natural generalization of the notion ‘tree’ and of Kruskal’s theorem. Relations with ordinal notation systems. Jeroen Van der Meeren Recursively defined trees and their maximal order types

Introduction Recursively defined trees Conclusions Maximal order type Definition The maximal order type of a well-partial-ordering ( X , ≤ X ) is defined as sup { α |≤ X ⊆≤ + with ≤ + a linear ordering on X o ( X , ≤ X ) = and α = otype ( X , ≤ + ) } . Every extension of a well-partial-ordering to a linear ordering is a well-ordering. Jeroen Van der Meeren Recursively defined trees and their maximal order types

Introduction Definition Recursively defined trees The maximal order type of T ( W ) Conclusions Proof-theoretical strength Let us introduce T ( W )! Jeroen Van der Meeren Recursively defined trees and their maximal order types

Introduction Definition Recursively defined trees The maximal order type of T ( W ) Conclusions Proof-theoretical strength Definition X ∗ : the Higman ordering X ∗ is the set of finite sequence over X with the Higman ordering: ( x 1 , . . . , x n ) ≤ ∗ ( y 1 , . . . , y m ) ⇔ ∃ 1 ≤ i 1 < · · · < i n ≤ m such that x j ≤ X y i j for every j = 1 , . . . , n . Theorem If X is a well-partial-ordering, then X ∗ is also a well-partial-ordering. Jeroen Van der Meeren Recursively defined trees and their maximal order types

Introduction Definition Recursively defined trees The maximal order type of T ( W ) Conclusions Proof-theoretical strength Definition X ∗ : the Higman ordering Theorem (De Jongh & Parikh; D. Schmidt) If X is a well-partial-ordering, then  ω ω o ( X )+1 if o ( X ) is equal to e + n     with e an epsilon number and n < ω , o ( X ∗ ) = ω ω o ( X ) − 1 if o ( X ) is finite,    ω ω o ( X )  otherwise. Jeroen Van der Meeren Recursively defined trees and their maximal order types

Introduction Definition Recursively defined trees The maximal order type of T ( W ) Conclusions Proof-theoretical strength W ( X ) = X + W ( X ) = X + = X ∗ \{ () } . Jeroen Van der Meeren Recursively defined trees and their maximal order types

Introduction Definition Recursively defined trees The maximal order type of T ( W ) Conclusions Proof-theoretical strength Example: T ( X + ) T is the set of finite planar rooted trees: • is an element of T , If T 1 , . . . , T n ∈ T , then ❆ ✁ ❆ ✁ T 1 T n ❆ ✁ ❆ ✁ . . . ❆ ✁ ❆ ✁ t t ◗ ✑✑✑ ❭ ✜ ❭ ✜ ◗ ◗ ❭ ✜ t is also an element of T . Jeroen Van der Meeren Recursively defined trees and their maximal order types