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Finite trees as ordinals Herman Ruge Jervell University of Oslo - - PowerPoint PPT Presentation
Finite trees as ordinals Herman Ruge Jervell University of Oslo - - PowerPoint PPT Presentation
Finite trees as ordinals Herman Ruge Jervell University of Oslo Honouring Wilfried M unchen April 5, 2008 Typical trees The natural numbers: 0 = Typical trees The natural numbers: 0 = 1 = Typical trees The natural numbers:
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Typical trees
The natural numbers: 0 = · 1 = · ·
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Typical trees
The natural numbers: 0 = · 1 = · · 2 = · · ·
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Typical trees
The natural numbers: 0 = · 1 = · · 2 = · · · 3 = · · · ·
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Typical trees
The natural numbers: 0 = · 1 = · · 2 = · · · 3 = · · · · And some ordinals:
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Typical trees
The natural numbers: 0 = · 1 = · · 2 = · · · 3 = · · · · And some ordinals: ω = · · ·
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Typical trees
The natural numbers: 0 = · 1 = · · 2 = · · · 3 = · · · · And some ordinals: ω = · · · ωω = · · · ·
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Typical trees
The natural numbers: 0 = · 1 = · · 2 = · · · 3 = · · · · And some ordinals: ω = · · · ωω = · · · · ǫ0 = · · · ·
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Typical trees
The natural numbers: 0 = · 1 = · · 2 = · · · 3 = · · · · And some ordinals: ω = · · · ωω = · · · · ǫ0 = · · · · Γ0 = · · · · ·
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The ordering
A < B ⇔ A ≤ B ∨ (A < B ∧ A < B)
◮ A — sequence of immediate subtrees
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The ordering
A < B ⇔ A ≤ B ∨ (A < B ∧ A < B)
◮ A — sequence of immediate subtrees ◮ A ≤ B
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The ordering
A < B ⇔ A ≤ B ∨ (A < B ∧ A < B)
◮ A — sequence of immediate subtrees ◮ A ≤ B
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The ordering
A < B ⇔ A ≤ B ∨ (A < B ∧ A < B)
◮ A — sequence of immediate subtrees ◮ A ≤ B A ≤ some immediate subtree of B
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The ordering
A < B ⇔ A ≤ B ∨ (A < B ∧ A < B)
◮ A — sequence of immediate subtrees ◮ A ≤ B A ≤ some immediate subtree of B ◮ A < B
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The ordering
A < B ⇔ A ≤ B ∨ (A < B ∧ A < B)
◮ A — sequence of immediate subtrees ◮ A ≤ B A ≤ some immediate subtree of B ◮ A < B
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The ordering
A < B ⇔ A ≤ B ∨ (A < B ∧ A < B)
◮ A — sequence of immediate subtrees ◮ A ≤ B A ≤ some immediate subtree of B ◮ A < B all immediate subtrees of A are < B
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The ordering
A < B ⇔ A ≤ B ∨ (A < B ∧ A < B)
◮ A — sequence of immediate subtrees ◮ A ≤ B A ≤ some immediate subtree of B ◮ A < B all immediate subtrees of A are < B ◮ A < B
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The ordering
A < B ⇔ A ≤ B ∨ (A < B ∧ A < B)
◮ A — sequence of immediate subtrees ◮ A ≤ B A ≤ some immediate subtree of B ◮ A < B all immediate subtrees of A are < B ◮ A < B
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The ordering
A < B ⇔ A ≤ B ∨ (A < B ∧ A < B)
◮ A — sequence of immediate subtrees ◮ A ≤ B A ≤ some immediate subtree of B ◮ A < B all immediate subtrees of A are < B ◮ A < B lexicographical ordering
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The ordering
A < B ⇔ A ≤ B ∨ (A < B ∧ A < B)
◮ A — sequence of immediate subtrees ◮ A ≤ B A ≤ some immediate subtree of B ◮ A < B all immediate subtrees of A are < B ◮ A < B lexicographical ordering
◮ Length of sequences
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The ordering
A < B ⇔ A ≤ B ∨ (A < B ∧ A < B)
◮ A — sequence of immediate subtrees ◮ A ≤ B A ≤ some immediate subtree of B ◮ A < B all immediate subtrees of A are < B ◮ A < B lexicographical ordering
◮ Length of sequences ◮ Rightmost element where they differ
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Elementary properties
A < B ⇔ A ≤ B ∨ (A < B ∧ A < B)
◮ Decidable
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Elementary properties
A < B ⇔ A ≤ B ∨ (A < B ∧ A < B)
◮ Decidable ◮ Transitive
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Elementary properties
A < B ⇔ A ≤ B ∨ (A < B ∧ A < B)
◮ Decidable ◮ Transitive ◮ Linear
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Elementary properties
A < B ⇔ A ≤ B ∨ (A < B ∧ A < B)
◮ Decidable ◮ Transitive ◮ Linear ◮ Equality is the usual tree equality
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Some ordinal functions
Zero: · = 0
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Some ordinal functions
Zero: · = 0 Successor: · α = α + 1
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Some ordinal functions
Zero: · = 0 Successor: · α = α + 1 Exponentiation: · · α ∼ ωωα where ∼ means we jump over fix points.
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Some ordinal functions
Zero: · = 0 Successor: · α = α + 1 Exponentiation: · · α ∼ ωωα where ∼ means we jump over fix points. In general we get the fix point free n-ary Veblen functions.
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Approximating from below 1
Γ0 = · · · · · Start with immediate subtrees:
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Approximating from below 1
Γ0 = · · · · · Start with immediate subtrees: 0 = · 0 = · 1 = · ·
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Approximating from below 1
Γ0 = · · · · · Start with immediate subtrees: 0 = · 0 = · 1 = · · Use function with smaller arity:
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Approximating from below 1
Γ0 = · · · · · Start with immediate subtrees: 0 = · 0 = · 1 = · · Use function with smaller arity: · α · β γ
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Approximating from below 1
Γ0 = · · · · · Start with immediate subtrees: 0 = · 0 = · 1 = · · Use function with smaller arity: · α · β γ
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Approximating from below 2
Γ0 = · · · · · Less in lexicographical ordering:
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Approximating from below 2
Γ0 = · · · · · Less in lexicographical ordering: · α β ·
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Approximating from below 2
Γ0 = · · · · · Less in lexicographical ordering: · α β · This gives all trees less than Γ0.
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Approximating from below 2
Γ0 = · · · · · Less in lexicographical ordering: · α β · This gives all trees less than Γ0. To get a cofinal set we only need · · γ ·
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Wellfoundedness
◮ Minimal bad argument
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Wellfoundedness
◮ Minimal bad argument
◮ Minimal height
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Wellfoundedness
◮ Minimal bad argument
◮ Minimal height
◮ Induction over wellfounded trees
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Wellfoundedness
◮ Minimal bad argument
◮ Minimal height
◮ Induction over wellfounded trees
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Wellfoundedness
◮ Minimal bad argument
◮ Minimal height
◮ Induction over wellfounded trees
Both arguments are straightforward.
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Further work
Linear extensions of embeddings
◮ Diana Schmidt
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Further work
Linear extensions of embeddings
◮ Diana Schmidt ◮ Linear extensions of topological embeddings of trees
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Further work
Linear extensions of embeddings
◮ Diana Schmidt ◮ Linear extensions of topological embeddings of trees ◮ |A| maximal ordertype
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Further work
Linear extensions of embeddings
◮ Diana Schmidt ◮ Linear extensions of topological embeddings of trees ◮ |A| maximal ordertype
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Further work
Linear extensions of embeddings
◮ Diana Schmidt ◮ Linear extensions of topological embeddings of trees ◮ |A| maximal ordertype
- ·
A B
- ≤
· |A| |B| ⊕ · |B| |A| This gives Higmans lemma. Further work gives Kruskals theorem.
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Further work
Finite trees with labels
◮ Wellordered set of labels
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Further work
Finite trees with labels
◮ Wellordered set of labels ◮ Each node has a label
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Further work
Finite trees with labels
◮ Wellordered set of labels ◮ Each node has a label ◮ Ai – sequence of i-subtrees
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Further work
Finite trees with labels
◮ Wellordered set of labels ◮ Each node has a label ◮ Ai – sequence of i-subtrees ◮ Defines <i and <∞
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Further work
Finite trees with labels
◮ Wellordered set of labels ◮ Each node has a label ◮ Ai – sequence of i-subtrees ◮ Defines <i and <∞ ◮ A <i B ⇔ A ≤i Bi ∨ (Ai < B ∨ A <i+ B)
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Further work
Finite trees with labels
◮ Wellordered set of labels ◮ Each node has a label ◮ Ai – sequence of i-subtrees ◮ Defines <i and <∞ ◮ A <i B ⇔ A ≤i Bi ∨ (Ai < B ∨ A <i+ B) ◮ A <∞ B — lexicographical ordering
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Further work
Finite trees with labels
◮ Wellordered set of labels ◮ Each node has a label ◮ Ai – sequence of i-subtrees ◮ Defines <i and <∞ ◮ A <i B ⇔ A ≤i Bi ∨ (Ai < B ∨ A <i+ B) ◮ A <∞ B — lexicographical ordering ◮ Linear wellfounded preorderings
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