[PPT] - Harvey Friedmans Finite Phase Transitions L. Gordeev Uni-T PowerPoint Presentation

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Harvey Friedman’s Finite Phase Transitions

L. Gordeev

Uni-T¨ ubingen, Uni-Ghent, PUC-Rio de Janeiro

Schloss Dagstuhl, January 17 - 22, 2016

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Abstract

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Abstract

Definition (H. Friedman)

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Abstract

Definition (H. Friedman) The proof theoretic integer of formal system T (abbreviation: PTI (T) ) is the least integer n such that every Σ0

m sentence

∃x1 · · · ∃xmA (x1, · · · , xm) that has a proof in T with at most 10, 000 symbols, has witnesses x1, · · · , xm < n.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Abstract

Definition (H. Friedman) The proof theoretic integer of formal system T (abbreviation: PTI (T) ) is the least integer n such that every Σ0

m sentence

∃x1 · · · ∃xmA (x1, · · · , xm) that has a proof in T with at most 10, 000 symbols, has witnesses x1, · · · , xm < n. (Actually m = 1 would suffice.)

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Abstract

Definition (H. Friedman) The proof theoretic integer of formal system T (abbreviation: PTI (T) ) is the least integer n such that every Σ0

m sentence

∃x1 · · · ∃xmA (x1, · · · , xm) that has a proof in T with at most 10, 000 symbols, has witnesses x1, · · · , xm < n. (Actually m = 1 would suffice.) A good source of examples is in the area surrounding Kruskal’s theorem.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Abstract

Definition (H. Friedman) The proof theoretic integer of formal system T (abbreviation: PTI (T) ) is the least integer n such that every Σ0

m sentence

∃x1 · · · ∃xmA (x1, · · · , xm) that has a proof in T with at most 10, 000 symbols, has witnesses x1, · · · , xm < n. (Actually m = 1 would suffice.) A good source of examples is in the area surrounding Kruskal’s theorem. This talk is devoted to PTI (T)’s basic properties, examples, comparisons and related phase transitions.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Introduction by Harvey Friedman

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Introduction by Harvey Friedman

Suppose we have a Π0

2 theorem ∀k∃nA (k, n). We then get a

recursive function F (k) = the least n such that A (k, n) holds.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Introduction by Harvey Friedman

Suppose we have a Π0

2 theorem ∀k∃nA (k, n). We then get a

recursive function F (k) = the least n such that A (k, n) holds. In the intended cases, we have F (0) < F (1) < ... . We want to look at F (0) , F (1) , ... and determine where there is a “qualitative jump” in size. I.e., a “phase transition”.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Introduction by Harvey Friedman

Suppose we have a Π0

2 theorem ∀k∃nA (k, n). We then get a

recursive function F (k) = the least n such that A (k, n) holds. In the intended cases, we have F (0) < F (1) < ... . We want to look at F (0) , F (1) , ... and determine where there is a “qualitative jump” in size. I.e., a “phase transition”. In the cases we focus on, after the first few terms - say about 16 or less - we simply get qualitatively indistinguishable very large integers.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Introduction by Harvey Friedman

Suppose we have a Π0

2 theorem ∀k∃nA (k, n). We then get a

recursive function F (k) = the least n such that A (k, n) holds. In the intended cases, we have F (0) < F (1) < ... . We want to look at F (0) , F (1) , ... and determine where there is a “qualitative jump” in size. I.e., a “phase transition”. In the cases we focus on, after the first few terms - say about 16 or less - we simply get qualitatively indistinguishable very large integers. There are a number of forms that such results can take. Some are more descriptive than quantitative and others are more quantitative than descriptive.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -1- by Harvey Friedman

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -1- by Harvey Friedman

In the Quantitative Approach, we simply provide upper and lower bounds on some of the terms in F (0) , F (1) , ... using a notation system for integers.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -1- by Harvey Friedman

In the Quantitative Approach, we simply provide upper and lower bounds on some of the terms in F (0) , F (1) , ... using a notation system for integers. We try to make exact calculations, if possible. But what notation system to use for the integers?

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -1- by Harvey Friedman

In the Quantitative Approach, we simply provide upper and lower bounds on some of the terms in F (0) , F (1) , ... using a notation system for integers. We try to make exact calculations, if possible. But what notation system to use for the integers? In case the numbers involved are less than, say, 10100, the usual base 10 notation is the clear choice.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -1- by Harvey Friedman

In the Quantitative Approach, we simply provide upper and lower bounds on some of the terms in F (0) , F (1) , ... using a notation system for integers. We try to make exact calculations, if possible. But what notation system to use for the integers? In case the numbers involved are less than, say, 10100, the usual base 10 notation is the clear choice. But if the numbers involved are greater than, say, 1010100, base 10 notation is generally of no use whatsoever since it cannot even be presented.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -1- by Harvey Friedman

In the Quantitative Approach, we simply provide upper and lower bounds on some of the terms in F (0) , F (1) , ... using a notation system for integers. We try to make exact calculations, if possible. But what notation system to use for the integers? In case the numbers involved are less than, say, 10100, the usual base 10 notation is the clear choice. But if the numbers involved are greater than, say, 1010100, base 10 notation is generally of no use whatsoever since it cannot even be presented. There are many special approaches that may be particularly illuminating.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -1- by Harvey Friedman

In the Quantitative Approach, we simply provide upper and lower bounds on some of the terms in F (0) , F (1) , ... using a notation system for integers. We try to make exact calculations, if possible. But what notation system to use for the integers? In case the numbers involved are less than, say, 10100, the usual base 10 notation is the clear choice. But if the numbers involved are greater than, say, 1010100, base 10 notation is generally of no use whatsoever since it cannot even be presented. There are many special approaches that may be particularly illuminating. However, we are led to the following general approach.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -2- by Harvey Friedman

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -2- by Harvey Friedman

Let us now focus on a very natural system.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -2- by Harvey Friedman

Let us now focus on a very natural system. We use constants 0, 1, addition, multiplication, and exponentiation to any positive base.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -2- by Harvey Friedman

Let us now focus on a very natural system. We use constants 0, 1, addition, multiplication, and exponentiation to any positive base. We can choose to use terms in these basics with at most, say, 100 symbols, or even say, just 16 symbols, before wishing to use a “more powerful” system.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -2- by Harvey Friedman

Let us now focus on a very natural system. We use constants 0, 1, addition, multiplication, and exponentiation to any positive base. We can choose to use terms in these basics with at most, say, 100 symbols, or even say, just 16 symbols, before wishing to use a “more powerful” system. The obvious motivation for moving to a “more powerful system” is because the number in question is larger than anything given by a term of, say, 100 or maybe 16 symbols.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -2- by Harvey Friedman

Let us now focus on a very natural system. We use constants 0, 1, addition, multiplication, and exponentiation to any positive base. We can choose to use terms in these basics with at most, say, 100 symbols, or even say, just 16 symbols, before wishing to use a “more powerful” system. The obvious motivation for moving to a “more powerful system” is because the number in question is larger than anything given by a term of, say, 100 or maybe 16 symbols. The above system - or some more convenient variant - is a good system for investigating the finite Ramsey numbers, or numbers in Adjacent Ramsey Theory.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -3- by Harvey Friedman

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -3- by Harvey Friedman

We think of the above system as an integer notation system.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -3- by Harvey Friedman

We think of the above system as an integer notation system. I used the integer notation system based on the Ackermann hierarchy of functions, in my work on Long Finite Sequences.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -3- by Harvey Friedman

We think of the above system as an integer notation system. I used the integer notation system based on the Ackermann hierarchy of functions, in my work on Long Finite Sequences. Definition Long Finite Sequences.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -3- by Harvey Friedman

We think of the above system as an integer notation system. I used the integer notation system based on the Ackermann hierarchy of functions, in my work on Long Finite Sequences. Definition Long Finite Sequences. Suppose we have a sequence a1, a2, a3, · · ·

ver {1, 2, 3}. (i.e. each ai ∈ {1, 2, 3}). We select a sequence of

parts from this sequence:

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -3- by Harvey Friedman

We think of the above system as an integer notation system. I used the integer notation system based on the Ackermann hierarchy of functions, in my work on Long Finite Sequences. Definition Long Finite Sequences. Suppose we have a sequence a1, a2, a3, · · ·

ver {1, 2, 3}. (i.e. each ai ∈ {1, 2, 3}). We select a sequence of

parts from this sequence: (a1, a2) , (a2, a3, a4) , (a3, a4, a5, a6) , · · · , (ak, ak+1, · · · , a2k) , · · ·

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -3- by Harvey Friedman

We think of the above system as an integer notation system. I used the integer notation system based on the Ackermann hierarchy of functions, in my work on Long Finite Sequences. Definition Long Finite Sequences. Suppose we have a sequence a1, a2, a3, · · ·

ver {1, 2, 3}. (i.e. each ai ∈ {1, 2, 3}). We select a sequence of

parts from this sequence: (a1, a2) , (a2, a3, a4) , (a3, a4, a5, a6) , · · · , (ak, ak+1, · · · , a2k) , · · · If this sequence of parts does not contain two elements of which the first is a subsequence of the second, then we say that the sequence a1, a2, a3, · · · has property F.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -3- by Harvey Friedman

We think of the above system as an integer notation system. I used the integer notation system based on the Ackermann hierarchy of functions, in my work on Long Finite Sequences. Definition Long Finite Sequences. Suppose we have a sequence a1, a2, a3, · · ·

ver {1, 2, 3}. (i.e. each ai ∈ {1, 2, 3}). We select a sequence of

parts from this sequence: (a1, a2) , (a2, a3, a4) , (a3, a4, a5, a6) , · · · , (ak, ak+1, · · · , a2k) , · · · If this sequence of parts does not contain two elements of which the first is a subsequence of the second, then we say that the sequence a1, a2, a3, · · · has property F. Let n (3) be the longest length of a sequence over {1, 2, 3} with property F.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -4- by Harvey Friedman

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -4- by Harvey Friedman

The Ackermann hierarchy of functions is encapsulated in terms of An (m), which is the n-th Ackermann function at m.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -4- by Harvey Friedman

The Ackermann hierarchy of functions is encapsulated in terms of An (m), which is the n-th Ackermann function at m. An (m) :=    m + 1 if n = 0 An−1 (1) if m = 0 An−1 (An (m − 1))

thw.
L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -4- by Harvey Friedman

The Ackermann hierarchy of functions is encapsulated in terms of An (m), which is the n-th Ackermann function at m. An (m) :=    m + 1 if n = 0 An−1 (1) if m = 0 An−1 (An (m − 1))

thw.

There I gave the lower bound A7198 (158386) < n (3).

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -4- by Harvey Friedman

The Ackermann hierarchy of functions is encapsulated in terms of An (m), which is the n-th Ackermann function at m. An (m) :=    m + 1 if n = 0 An−1 (1) if m = 0 An−1 (An (m − 1))

thw.

There I gave the lower bound A7198 (158386) < n (3). In terms of the binary Ackermann function A (n, m) = An (m), this lower bound reads A (7198, 158386) < n (3).

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -4- by Harvey Friedman

The Ackermann hierarchy of functions is encapsulated in terms of An (m), which is the n-th Ackermann function at m. An (m) :=    m + 1 if n = 0 An−1 (1) if m = 0 An−1 (An (m − 1))

thw.

There I gave the lower bound A7198 (158386) < n (3). In terms of the binary Ackermann function A (n, m) = An (m), this lower bound reads A (7198, 158386) < n (3). I didn’t give an upper bound there, but I later conjectured n (3) < A (A (5, 5) , A (5, 5)).

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -4- by Harvey Friedman

The Ackermann hierarchy of functions is encapsulated in terms of An (m), which is the n-th Ackermann function at m. An (m) :=    m + 1 if n = 0 An−1 (1) if m = 0 An−1 (An (m − 1))

thw.

There I gave the lower bound A7198 (158386) < n (3). In terms of the binary Ackermann function A (n, m) = An (m), this lower bound reads A (7198, 158386) < n (3). I didn’t give an upper bound there, but I later conjectured n (3) < A (A (5, 5) , A (5, 5)). So base 10 notation and the binary Ackermann function can serve as a reasonable notation system for integers.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -5-

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -5-

For a unified approach to integers too large to be bounded by a reasonable sized term in the above notation system for integers, we can use ε0 with its usual system of fundamental sequences.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -5-

For a unified approach to integers too large to be bounded by a reasonable sized term in the above notation system for integers, we can use ε0 with its usual system of fundamental sequences. Definition (Hardy functions)

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -5-

For a unified approach to integers too large to be bounded by a reasonable sized term in the above notation system for integers, we can use ε0 with its usual system of fundamental sequences. Definition (Hardy functions) For any ordinal α, let Hα be the corresponding Hardy function that is defined by transfinite recursion: Hα (x) := x if α = 0 Hα[x] (x + 1) if α > 0 where α [−] : N ∋ x → α [x] < α is the correlated canonical monotone increasing fundamental sequence.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -5-

For a unified approach to integers too large to be bounded by a reasonable sized term in the above notation system for integers, we can use ε0 with its usual system of fundamental sequences. Definition (Hardy functions) For any ordinal α, let Hα be the corresponding Hardy function that is defined by transfinite recursion: Hα (x) := x if α = 0 Hα[x] (x + 1) if α > 0 where α [−] : N ∋ x → α [x] < α is the correlated canonical monotone increasing fundamental sequence. The Ackermann hierarchy appears at very low Hardy ordinal levels (< ωω).

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -7- by Harvey Friedman

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -7- by Harvey Friedman

This provides a notation system for integers, using 0 and the binary function corresponding to the above Hardy hierarchy (or Wainer hierarchy).

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -7- by Harvey Friedman

This provides a notation system for integers, using 0 and the binary function corresponding to the above Hardy hierarchy (or Wainer hierarchy). We may want to sugar it with base 10 notation.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -7- by Harvey Friedman

This provides a notation system for integers, using 0 and the binary function corresponding to the above Hardy hierarchy (or Wainer hierarchy). We may want to sugar it with base 10 notation. For integers too large to be bounded by a reasonable sized term in this notation system for integers, we would use the obvious extension of this for larger proof theoretic ordinals.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -7- by Harvey Friedman

This provides a notation system for integers, using 0 and the binary function corresponding to the above Hardy hierarchy (or Wainer hierarchy). We may want to sugar it with base 10 notation. For integers too large to be bounded by a reasonable sized term in this notation system for integers, we would use the obvious extension of this for larger proof theoretic ordinals. But there remains the question: what do we mean by a qualitative jump in size? What is a phase transition in this context?

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -7- by Harvey Friedman

This provides a notation system for integers, using 0 and the binary function corresponding to the above Hardy hierarchy (or Wainer hierarchy). We may want to sugar it with base 10 notation. For integers too large to be bounded by a reasonable sized term in this notation system for integers, we would use the obvious extension of this for larger proof theoretic ordinals. But there remains the question: what do we mean by a qualitative jump in size? What is a phase transition in this context? We can, of course, let the estimates speak for themselves.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Quantitative approach -7- by Harvey Friedman

This provides a notation system for integers, using 0 and the binary function corresponding to the above Hardy hierarchy (or Wainer hierarchy). We may want to sugar it with base 10 notation. For integers too large to be bounded by a reasonable sized term in this notation system for integers, we would use the obvious extension of this for larger proof theoretic ordinals. But there remains the question: what do we mean by a qualitative jump in size? What is a phase transition in this context? We can, of course, let the estimates speak for themselves. However, we may demand a more principled answer.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -1- by Harvey Friedman

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -1- by Harvey Friedman

We offer the following Qualitative Approach.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -1- by Harvey Friedman

We offer the following Qualitative Approach. In the qualitative approach, we look to formal systems for the more principled answer.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -1- by Harvey Friedman

We offer the following Qualitative Approach. In the qualitative approach, we look to formal systems for the more principled answer. In particular, we associate an integer to every formal system.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -1- by Harvey Friedman

We offer the following Qualitative Approach. In the qualitative approach, we look to formal systems for the more principled answer. In particular, we associate an integer to every formal system. Now, there will be some ad hoc features involved in the associated integer.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -1- by Harvey Friedman

We offer the following Qualitative Approach. In the qualitative approach, we look to formal systems for the more principled answer. In particular, we associate an integer to every formal system. Now, there will be some ad hoc features involved in the associated integer. However, we Conjecture that there is a great deal of robustness here.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -1- by Harvey Friedman

We offer the following Qualitative Approach. In the qualitative approach, we look to formal systems for the more principled answer. In particular, we associate an integer to every formal system. Now, there will be some ad hoc features involved in the associated integer. However, we Conjecture that there is a great deal of robustness here. We call this associated integer - defined below - the PROOF THEORETIC INTEGER OF T.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -1- by Harvey Friedman

We offer the following Qualitative Approach. In the qualitative approach, we look to formal systems for the more principled answer. In particular, we associate an integer to every formal system. Now, there will be some ad hoc features involved in the associated integer. However, we Conjecture that there is a great deal of robustness here. We call this associated integer - defined below - the PROOF THEORETIC INTEGER OF T. We assume that T is #) a formal system in a finite relational type in many sorted predicate calculus with equality, containing a sort for natural numbers, with 0, S, +, ·, exp, <.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -1- by Harvey Friedman

We offer the following Qualitative Approach. In the qualitative approach, we look to formal systems for the more principled answer. In particular, we associate an integer to every formal system. Now, there will be some ad hoc features involved in the associated integer. However, we Conjecture that there is a great deal of robustness here. We call this associated integer - defined below - the PROOF THEORETIC INTEGER OF T. We assume that T is #) a formal system in a finite relational type in many sorted predicate calculus with equality, containing a sort for natural numbers, with 0, S, +, ·, exp, <. The ∆0 formulas are defined as usual, using bounded quantifiers.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -1- by Harvey Friedman

We offer the following Qualitative Approach. In the qualitative approach, we look to formal systems for the more principled answer. In particular, we associate an integer to every formal system. Now, there will be some ad hoc features involved in the associated integer. However, we Conjecture that there is a great deal of robustness here. We call this associated integer - defined below - the PROOF THEORETIC INTEGER OF T. We assume that T is #) a formal system in a finite relational type in many sorted predicate calculus with equality, containing a sort for natural numbers, with 0, S, +, ·, exp, <. The ∆0 formulas are defined as usual, using bounded quantifiers. The Σ0

1 formulas are obtained from the ∆0 formulas by putting

zero or more existential quantifiers in front of ∆0 formulas.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -2- by Harvey Friedman

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -2- by Harvey Friedman

Definition The proof theoretic integer of T is the least integer n such that every Σ0

1 sentence that has a proof in T with at most 10, 000

symbols, has witnesses less than n.

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Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -2- by Harvey Friedman

Definition The proof theoretic integer of T is the least integer n such that every Σ0

1 sentence that has a proof in T with at most 10, 000

symbols, has witnesses less than n. Of course, this definition needs some exact spelling out – e.g., what exact proof system is to be used, and what exactly counts as a symbol (what about parentheses), etcetera?

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -2- by Harvey Friedman

Definition The proof theoretic integer of T is the least integer n such that every Σ0

1 sentence that has a proof in T with at most 10, 000

symbols, has witnesses less than n. Of course, this definition needs some exact spelling out – e.g., what exact proof system is to be used, and what exactly counts as a symbol (what about parentheses), etcetera? However, it is expected that there is a lot of robustness. Of course, not robustness in the form of the exact number being unchanged. But robustness in a more subtle sense.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -2- by Harvey Friedman

Definition The proof theoretic integer of T is the least integer n such that every Σ0

1 sentence that has a proof in T with at most 10, 000

symbols, has witnesses less than n. Of course, this definition needs some exact spelling out – e.g., what exact proof system is to be used, and what exactly counts as a symbol (what about parentheses), etcetera? However, it is expected that there is a lot of robustness. Of course, not robustness in the form of the exact number being unchanged. But robustness in a more subtle sense. In particular, we make the following robustness conjecture.

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Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -3- by Harvey Friedman

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Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -3- by Harvey Friedman

ROBUSTNESS CONJECTURE. Let S, T be two naturally

ccurring formal systems obeying #), which prove EFA

(exponential function arithmetic). Suppose S proves the 1-consistency of T. Then the proof theoretic integer of S is at least a double exponential of the proof theoretic integer of T.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -3- by Harvey Friedman

ROBUSTNESS CONJECTURE. Let S, T be two naturally

ccurring formal systems obeying #), which prove EFA

(exponential function arithmetic). Suppose S proves the 1-consistency of T. Then the proof theoretic integer of S is at least a double exponential of the proof theoretic integer of T. QUESTION: Can we use a significantly smaller number than 10, 000 in the definition of the proof theoretic integer, and still have the robustness conjecture?

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -3- by Harvey Friedman

ROBUSTNESS CONJECTURE. Let S, T be two naturally

ccurring formal systems obeying #), which prove EFA

(exponential function arithmetic). Suppose S proves the 1-consistency of T. Then the proof theoretic integer of S is at least a double exponential of the proof theoretic integer of T. QUESTION: Can we use a significantly smaller number than 10, 000 in the definition of the proof theoretic integer, and still have the robustness conjecture? I used 10, 000 because I want to accommodate some technically neat but entirely crude Hilbert style system, without any sugar.

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Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -3- by Harvey Friedman

ROBUSTNESS CONJECTURE. Let S, T be two naturally

ccurring formal systems obeying #), which prove EFA

(exponential function arithmetic). Suppose S proves the 1-consistency of T. Then the proof theoretic integer of S is at least a double exponential of the proof theoretic integer of T. QUESTION: Can we use a significantly smaller number than 10, 000 in the definition of the proof theoretic integer, and still have the robustness conjecture? I used 10, 000 because I want to accommodate some technically neat but entirely crude Hilbert style system, without any sugar. Because this Conjecture has “naturally occurring”, it takes on an experimental character.

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Qualitative approach -3- by Harvey Friedman

ROBUSTNESS CONJECTURE. Let S, T be two naturally

ccurring formal systems obeying #), which prove EFA

(exponential function arithmetic). Suppose S proves the 1-consistency of T. Then the proof theoretic integer of S is at least a double exponential of the proof theoretic integer of T. QUESTION: Can we use a significantly smaller number than 10, 000 in the definition of the proof theoretic integer, and still have the robustness conjecture? I used 10, 000 because I want to accommodate some technically neat but entirely crude Hilbert style system, without any sugar. Because this Conjecture has “naturally occurring”, it takes on an experimental character. We Conjecture that there is a form of the Conjecture that can be proved, where we assume instead that the complexity of S, T is low, and the size of the proof of 1-consistency of T in S is also low.

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Qualitative approach -4-

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Harvey Friedman’s Finite Phase Transitions

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Theorem (L. G.: Upper bounds)

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Theorem (L. G.: Upper bounds) For all basic formal systems T (such as EFA, 1 quantifier induction, 2 quantifier induction, PA, ACA0, ACA, ATR0, ATR, Π1

1-CA0), the following holds.

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Qualitative approach -4-

Theorem (L. G.: Upper bounds) For all basic formal systems T (such as EFA, 1 quantifier induction, 2 quantifier induction, PA, ACA0, ACA, ATR0, ATR, Π1

1-CA0), the following holds.

The proof theoretic integer of T is smaller than Ho(T) (100, 000), where o (T) is the canonical proof theoretic ordinal of T.

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Qualitative approach -4-

Theorem (L. G.: Upper bounds) For all basic formal systems T (such as EFA, 1 quantifier induction, 2 quantifier induction, PA, ACA0, ACA, ATR0, ATR, Π1

1-CA0), the following holds.

The proof theoretic integer of T is smaller than Ho(T) (100, 000), where o (T) is the canonical proof theoretic ordinal of T. Proof.

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Qualitative approach -4-

Theorem (L. G.: Upper bounds) For all basic formal systems T (such as EFA, 1 quantifier induction, 2 quantifier induction, PA, ACA0, ACA, ATR0, ATR, Π1

1-CA0), the following holds.

The proof theoretic integer of T is smaller than Ho(T) (100, 000), where o (T) is the canonical proof theoretic ordinal of T. Proof. Buchholz-Weiermann style cut elimination (100, 000 is entirely crude upper bound).

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Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -4-

Theorem (L. G.: Upper bounds) For all basic formal systems T (such as EFA, 1 quantifier induction, 2 quantifier induction, PA, ACA0, ACA, ATR0, ATR, Π1

1-CA0), the following holds.

The proof theoretic integer of T is smaller than Ho(T) (100, 000), where o (T) is the canonical proof theoretic ordinal of T. Proof. Buchholz-Weiermann style cut elimination (100, 000 is entirely crude upper bound). This result provides basic quantitative upper bounds for the qualitative evaluations.

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Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -5- by Harvey Friedman

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -5- by Harvey Friedman

We propose using proof theoretic integers of basic formal systems T [as above].

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -5- by Harvey Friedman

We propose using proof theoretic integers of basic formal systems T [as above]. AN IMPORTANT EXAMPLE.

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Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -5- by Harvey Friedman

We propose using proof theoretic integers of basic formal systems T [as above]. AN IMPORTANT EXAMPLE. A good source of examples is in the area surrounding Kruskal’s theorem (starting with k = 0). We will not allow an empty tree. Here is my original finite form of Kruskal’s theorem.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -5- by Harvey Friedman

We propose using proof theoretic integers of basic formal systems T [as above]. AN IMPORTANT EXAMPLE. A good source of examples is in the area surrounding Kruskal’s theorem (starting with k = 0). We will not allow an empty tree. Here is my original finite form of Kruskal’s theorem. Theorem

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -5- by Harvey Friedman

We propose using proof theoretic integers of basic formal systems T [as above]. AN IMPORTANT EXAMPLE. A good source of examples is in the area surrounding Kruskal’s theorem (starting with k = 0). We will not allow an empty tree. Here is my original finite form of Kruskal’s theorem. Theorem For all k ≥ 0 there exists n ≥ 0 such that the following holds. For all structured finite trees T1, · · · , Tn, where each Ti has at most i + k vertices, there exists i < j such that Ti is inf and structure preserving embeddable into Tj.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -5- by Harvey Friedman

We propose using proof theoretic integers of basic formal systems T [as above]. AN IMPORTANT EXAMPLE. A good source of examples is in the area surrounding Kruskal’s theorem (starting with k = 0). We will not allow an empty tree. Here is my original finite form of Kruskal’s theorem. Theorem For all k ≥ 0 there exists n ≥ 0 such that the following holds. For all structured finite trees T1, · · · , Tn, where each Ti has at most i + k vertices, there exists i < j such that Ti is inf and structure preserving embeddable into Tj. Let F s (k) be the least n such that the Theorem holds.

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Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -6-

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Qualitative approach -6-

F s (0) = 2.

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Qualitative approach -6-

F s (0) = 2. F s (1) = 3.

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Qualitative approach -6-

F s (0) = 2. F s (1) = 3. F s (2) = 6.

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Qualitative approach -6-

F s (0) = 2. F s (1) = 3. F s (2) = 6. F s (3) =?

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Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -7-

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Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -7-

Theorem (R. Peng)

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Qualitative approach -7-

Theorem (R. Peng) F s (3) >> Hω2

Hω2
Hω·2
Hω·2
Hω·2
Hω
1010100

.

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Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -7-

Theorem (R. Peng) F s (3) >> Hω2

Hω2
Hω·2
Hω·2
Hω·2
Hω
1010100

. Clearly phase transition!

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Qualitative approach -7-

Theorem (R. Peng) F s (3) >> Hω2

Hω2
Hω·2
Hω·2
Hω·2
Hω
1010100

. Clearly phase transition! Corollary (: Lower bounds)

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Qualitative approach -7-

Theorem (R. Peng) F s (3) >> Hω2

Hω2
Hω·2
Hω·2
Hω·2
Hω
1010100

. Clearly phase transition! Corollary (: Lower bounds) F s (3) > proof theoretic integer of 1 quantifier induction.a

aaccording to the basic quantitative upper bound theorem, above.

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Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -8-

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Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -8-

Theorem (Unstructured Kruskal-Friedman (UKFT) )

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Qualitative approach -8-

Theorem (Unstructured Kruskal-Friedman (UKFT) ) For all k ≥ 0 there exists n ≥ 0 such that the following holds. For all finite trees T1, · · · , Tn, where each Ti has at most i + k vertices, there exists i < j such that Ti is inf preserving (i.e. just homomorphically) embeddable into Tj.

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Theorem (Unstructured Kruskal-Friedman (UKFT) ) For all k ≥ 0 there exists n ≥ 0 such that the following holds. For all finite trees T1, · · · , Tn, where each Ti has at most i + k vertices, there exists i < j such that Ti is inf preserving (i.e. just homomorphically) embeddable into Tj. Let F (k) be the least n such that UKFT holds.

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Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -8-

Theorem (Unstructured Kruskal-Friedman (UKFT) ) For all k ≥ 0 there exists n ≥ 0 such that the following holds. For all finite trees T1, · · · , Tn, where each Ti has at most i + k vertices, there exists i < j such that Ti is inf preserving (i.e. just homomorphically) embeddable into Tj. Let F (k) be the least n such that UKFT holds. Theorem (L. G.: Lower bounds)

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Qualitative approach -8-

Theorem (Unstructured Kruskal-Friedman (UKFT) ) For all k ≥ 0 there exists n ≥ 0 such that the following holds. For all finite trees T1, · · · , Tn, where each Ti has at most i + k vertices, there exists i < j such that Ti is inf preserving (i.e. just homomorphically) embeddable into Tj. Let F (k) be the least n such that UKFT holds. Theorem (L. G.: Lower bounds) F (0) = 2.

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Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -8-

Theorem (Unstructured Kruskal-Friedman (UKFT) ) For all k ≥ 0 there exists n ≥ 0 such that the following holds. For all finite trees T1, · · · , Tn, where each Ti has at most i + k vertices, there exists i < j such that Ti is inf preserving (i.e. just homomorphically) embeddable into Tj. Let F (k) be the least n such that UKFT holds. Theorem (L. G.: Lower bounds) F (0) = 2. F (1) = 3.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -8-

Theorem (Unstructured Kruskal-Friedman (UKFT) ) For all k ≥ 0 there exists n ≥ 0 such that the following holds. For all finite trees T1, · · · , Tn, where each Ti has at most i + k vertices, there exists i < j such that Ti is inf preserving (i.e. just homomorphically) embeddable into Tj. Let F (k) be the least n such that UKFT holds. Theorem (L. G.: Lower bounds) F (0) = 2. F (1) = 3. F (2) = 6.

L. Gordeev

Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -8-

Theorem (Unstructured Kruskal-Friedman (UKFT) ) For all k ≥ 0 there exists n ≥ 0 such that the following holds. For all finite trees T1, · · · , Tn, where each Ti has at most i + k vertices, there exists i < j such that Ti is inf preserving (i.e. just homomorphically) embeddable into Tj. Let F (k) be the least n such that UKFT holds. Theorem (L. G.: Lower bounds) F (0) = 2. F (1) = 3. F (2) = 6. F (3) = 125.

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Harvey Friedman’s Finite Phase Transitions

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Qualitative approach -8-

Theorem (Unstructured Kruskal-Friedman (UKFT) ) For all k ≥ 0 there exists n ≥ 0 such that the following holds. For all finite trees T1, · · · , Tn, where each Ti has at most i + k vertices, there exists i < j such that Ti is inf preserving (i.e. just homomorphically) embeddable into Tj. Let F (k) be the least n such that UKFT holds. Theorem (L. G.: Lower bounds) F (0) = 2. F (1) = 3. F (2) = 6. F (3) = 125. F (4) > Hε0

1010100

> proof theoretic integer of PA. a

aaccording to the basic quantitative upper bound theorem.

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Summary

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Harvey Friedman’s Finite Phase Transitions

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Summary

Abbreviation: PTI (T) stands for “proof theoretic integer of T”.

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Harvey Friedman’s Finite Phase Transitions

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Summary

Abbreviation: PTI (T) stands for “proof theoretic integer of T”. Theorem

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Harvey Friedman’s Finite Phase Transitions

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Summary

Abbreviation: PTI (T) stands for “proof theoretic integer of T”. Theorem

1 F (0) = F s (0) = 2.

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Harvey Friedman’s Finite Phase Transitions

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Summary

Abbreviation: PTI (T) stands for “proof theoretic integer of T”. Theorem

1 F (0) = F s (0) = 2. 2 F (1) = F s (1) = 3.

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Harvey Friedman’s Finite Phase Transitions

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Summary

Abbreviation: PTI (T) stands for “proof theoretic integer of T”. Theorem

1 F (0) = F s (0) = 2. 2 F (1) = F s (1) = 3. 3 F (2) = F s (2) = 6.

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Harvey Friedman’s Finite Phase Transitions

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Summary

Abbreviation: PTI (T) stands for “proof theoretic integer of T”. Theorem

1 F (0) = F s (0) = 2. 2 F (1) = F s (1) = 3. 3 F (2) = F s (2) = 6. 4 F (3) = 125 << F s (3) << F (4).

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Harvey Friedman’s Finite Phase Transitions

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Summary

Abbreviation: PTI (T) stands for “proof theoretic integer of T”. Theorem

1 F (0) = F s (0) = 2. 2 F (1) = F s (1) = 3. 3 F (2) = F s (2) = 6. 4 F (3) = 125 << F s (3) << F (4). 5 PTI (PA) < F (4) < F s (4) ≤ PTI (PA+Consis (PA)).

Moreover PA proves the existence of F (4) (and F s (4)). a

aHence the length of any PA-proof of ∃x (x = F (4)) must exceed 10, 000.

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Harvey Friedman’s Finite Phase Transitions

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Summary

Abbreviation: PTI (T) stands for “proof theoretic integer of T”. Theorem

1 F (0) = F s (0) = 2. 2 F (1) = F s (1) = 3. 3 F (2) = F s (2) = 6. 4 F (3) = 125 << F s (3) << F (4). 5 PTI (PA) < F (4) < F s (4) ≤ PTI (PA+Consis (PA)).

Moreover PA proves the existence of F (4) (and F s (4)). a

aHence the length of any PA-proof of ∃x (x = F (4)) must exceed 10, 000.

Proof.

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Summary

Abbreviation: PTI (T) stands for “proof theoretic integer of T”. Theorem

1 F (0) = F s (0) = 2. 2 F (1) = F s (1) = 3. 3 F (2) = F s (2) = 6. 4 F (3) = 125 << F s (3) << F (4). 5 PTI (PA) < F (4) < F s (4) ≤ PTI (PA+Consis (PA)).

Moreover PA proves the existence of F (4) (and F s (4)). a

aHence the length of any PA-proof of ∃x (x = F (4)) must exceed 10, 000.

Proof. 5: Careful ordinal analysis of the descending tree sequences involved plus quantitative upper bound theorem followed by the formalization thereof in PA+Consis (PA) .

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More on F (3) = 125

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Harvey Friedman’s Finite Phase Transitions

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More on F (3) = 125

Maximal bad sequence is T1, · · · , T124, and hence F (3) = 125, where Ti are as follows.

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Harvey Friedman’s Finite Phase Transitions

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More on F (3) = 125

Maximal bad sequence is T1, · · · , T124, and hence F (3) = 125, where Ti are as follows. T1 = (• • •) , |T1| = 4 ρ (T1) = ωω·2

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Harvey Friedman’s Finite Phase Transitions

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More on F (3) = 125

Maximal bad sequence is T1, · · · , T124, and hence F (3) = 125, where Ti are as follows. T1 = (• • •) , |T1| = 4 ρ (T1) = ωω·2 T2 = ((••) •) , |T2| = 5 ρ (T2) = ωω + ω

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Harvey Friedman’s Finite Phase Transitions

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More on F (3) = 125

Maximal bad sequence is T1, · · · , T124, and hence F (3) = 125, where Ti are as follows. T1 = (• • •) , |T1| = 4 ρ (T1) = ωω·2 T2 = ((••) •) , |T2| = 5 ρ (T2) = ωω + ω T3 = ((((•))) •) , |T3| = 6 ρ (T3) = ω4 + ω

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Harvey Friedman’s Finite Phase Transitions

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More on F (3) = 125

Maximal bad sequence is T1, · · · , T124, and hence F (3) = 125, where Ti are as follows. T1 = (• • •) , |T1| = 4 ρ (T1) = ωω·2 T2 = ((••) •) , |T2| = 5 ρ (T2) = ωω + ω T3 = ((((•))) •) , |T3| = 6 ρ (T3) = ω4 + ω T4 = (((•)) ((•))) , |T4| = 7 ρ (T4) = ω3 · 2

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Harvey Friedman’s Finite Phase Transitions

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More on F (3) = 125

Maximal bad sequence is T1, · · · , T124, and hence F (3) = 125, where Ti are as follows. T1 = (• • •) , |T1| = 4 ρ (T1) = ωω·2 T2 = ((••) •) , |T2| = 5 ρ (T2) = ωω + ω T3 = ((((•))) •) , |T3| = 6 ρ (T3) = ω4 + ω T4 = (((•)) ((•))) , |T4| = 7 ρ (T4) = ω3 · 2 T5 = (((((•)) (•)))) , |T5| = 8 ρ (T5) = ω3 + 2

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Harvey Friedman’s Finite Phase Transitions

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More on F (3) = 125

Maximal bad sequence is T1, · · · , T124, and hence F (3) = 125, where Ti are as follows. T1 = (• • •) , |T1| = 4 ρ (T1) = ωω·2 T2 = ((••) •) , |T2| = 5 ρ (T2) = ωω + ω T3 = ((((•))) •) , |T3| = 6 ρ (T3) = ω4 + ω T4 = (((•)) ((•))) , |T4| = 7 ρ (T4) = ω3 · 2 T5 = (((((•)) (•)))) , |T5| = 8 ρ (T5) = ω3 + 2 T6 = ((((•)) (•))) , |T6| = 7 ρ (T6) = ω3 + 1

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Harvey Friedman’s Finite Phase Transitions

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More on F (3) = 125

Maximal bad sequence is T1, · · · , T124, and hence F (3) = 125, where Ti are as follows. T1 = (• • •) , |T1| = 4 ρ (T1) = ωω·2 T2 = ((••) •) , |T2| = 5 ρ (T2) = ωω + ω T3 = ((((•))) •) , |T3| = 6 ρ (T3) = ω4 + ω T4 = (((•)) ((•))) , |T4| = 7 ρ (T4) = ω3 · 2 T5 = (((((•)) (•)))) , |T5| = 8 ρ (T5) = ω3 + 2 T6 = ((((•)) (•))) , |T6| = 7 ρ (T6) = ω3 + 1 T7 = (((•)) (•)) , |T7| = 6 ρ (T7) = ω3

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Harvey Friedman’s Finite Phase Transitions

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More on F (3) = 125

Maximal bad sequence is T1, · · · , T124, and hence F (3) = 125, where Ti are as follows. T1 = (• • •) , |T1| = 4 ρ (T1) = ωω·2 T2 = ((••) •) , |T2| = 5 ρ (T2) = ωω + ω T3 = ((((•))) •) , |T3| = 6 ρ (T3) = ω4 + ω T4 = (((•)) ((•))) , |T4| = 7 ρ (T4) = ω3 · 2 T5 = (((((•)) (•)))) , |T5| = 8 ρ (T5) = ω3 + 2 T6 = ((((•)) (•))) , |T6| = 7 ρ (T6) = ω3 + 1 T7 = (((•)) (•)) , |T7| = 6 ρ (T7) = ω3 T8 = ((((((((•) (•)))))))) = (

×6

((•) (•)) )

×6

, |T8| = 11 ρ (T8) = ω2 + 6

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More on F (3) = 125

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More on F (3) = 125

T9 = (

×5

((•) (•)) )

×5

, |T9| = 10 ρ (T9) = ω2 + 5

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More on F (3) = 125

T9 = (

×5

((•) (•)) )

×5

, |T9| = 10 ρ (T9) = ω2 + 5 T10 = (

×4

((•) (•)) )

×4

, |T10| = 9 ρ (T10) = ω2 + 4

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More on F (3) = 125

T9 = (

×5

((•) (•)) )

×5

, |T9| = 10 ρ (T9) = ω2 + 5 T10 = (

×4

((•) (•)) )

×4

, |T10| = 9 ρ (T10) = ω2 + 4 T11 = (

×3

((•) (•)) )

×3

, |T11| = 8 ρ (T11) = ω2 + 3

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Harvey Friedman’s Finite Phase Transitions

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More on F (3) = 125

T9 = (

×5

((•) (•)) )

×5

, |T9| = 10 ρ (T9) = ω2 + 5 T10 = (

×4

((•) (•)) )

×4

, |T10| = 9 ρ (T10) = ω2 + 4 T11 = (

×3

((•) (•)) )

×3

, |T11| = 8 ρ (T11) = ω2 + 3 T12 = ((((•) (•)))) , |T12| = 7 ρ (T12) = ω2 + 2

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Harvey Friedman’s Finite Phase Transitions

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More on F (3) = 125

T9 = (

×5

((•) (•)) )

×5

, |T9| = 10 ρ (T9) = ω2 + 5 T10 = (

×4

((•) (•)) )

×4

, |T10| = 9 ρ (T10) = ω2 + 4 T11 = (

×3

((•) (•)) )

×3

, |T11| = 8 ρ (T11) = ω2 + 3 T12 = ((((•) (•)))) , |T12| = 7 ρ (T12) = ω2 + 2 T13 = (((•) (•))) , |T13| = 6 ρ (T13) = ω2 + 1

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Harvey Friedman’s Finite Phase Transitions

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More on F (3) = 125

T9 = (

×5

((•) (•)) )

×5

, |T9| = 10 ρ (T9) = ω2 + 5 T10 = (

×4

((•) (•)) )

×4

, |T10| = 9 ρ (T10) = ω2 + 4 T11 = (

×3

((•) (•)) )

×3

, |T11| = 8 ρ (T11) = ω2 + 3 T12 = ((((•) (•)))) , |T12| = 7 ρ (T12) = ω2 + 2 T13 = (((•) (•))) , |T13| = 6 ρ (T13) = ω2 + 1 T14 = ((•) (•)) , |T14| = 5 ρ (T14) = ω2

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More on F (3) = 125

T9 = (

×5

((•) (•)) )

×5

, |T9| = 10 ρ (T9) = ω2 + 5 T10 = (

×4

((•) (•)) )

×4

, |T10| = 9 ρ (T10) = ω2 + 4 T11 = (

×3

((•) (•)) )

×3

, |T11| = 8 ρ (T11) = ω2 + 3 T12 = ((((•) (•)))) , |T12| = 7 ρ (T12) = ω2 + 2 T13 = (((•) (•))) , |T13| = 6 ρ (T13) = ω2 + 1 T14 = ((•) (•)) , |T14| = 5 ρ (T14) = ω2 T15 = (

×14

((•) •) )

×14

, |T15| = 18 ρ (T15) = ω · 2 + 14

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Harvey Friedman’s Finite Phase Transitions

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More on F (3) = 125

T9 = (

×5

((•) (•)) )

×5

, |T9| = 10 ρ (T9) = ω2 + 5 T10 = (

×4

((•) (•)) )

×4

, |T10| = 9 ρ (T10) = ω2 + 4 T11 = (

×3

((•) (•)) )

×3

, |T11| = 8 ρ (T11) = ω2 + 3 T12 = ((((•) (•)))) , |T12| = 7 ρ (T12) = ω2 + 2 T13 = (((•) (•))) , |T13| = 6 ρ (T13) = ω2 + 1 T14 = ((•) (•)) , |T14| = 5 ρ (T14) = ω2 T15 = (

×14

((•) •) )

×14

, |T15| = 18 ρ (T15) = ω · 2 + 14 T16 = (

×13

((•) •) )

×13

, |T16| = 17 ρ (T16) = ω · 2 + 13

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More on F (3) = 125

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More on F (3) = 125

. . .

12
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More on F (3) = 125

. . .

12

T28 = (((•) •)) , |T28| = 5 ρ (T28) = ω · 2 + 1

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More on F (3) = 125

. . .

12

T28 = (((•) •)) , |T28| = 5 ρ (T28) = ω · 2 + 1 T29 = ((•) •) , |T29| = 4 ρ (T29) = ω · 2

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More on F (3) = 125

. . .

12

T28 = (((•) •)) , |T28| = 5 ρ (T28) = ω · 2 + 1 T29 = ((•) •) , |T29| = 4 ρ (T29) = ω · 2 T30 = (

×30

(••) )

×30

, |T30| = 33 ρ (T30) = ω + 30

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More on F (3) = 125

. . .

12

T28 = (((•) •)) , |T28| = 5 ρ (T28) = ω · 2 + 1 T29 = ((•) •) , |T29| = 4 ρ (T29) = ω · 2 T30 = (

×30

(••) )

×30

, |T30| = 33 ρ (T30) = ω + 30 T31 = (

×29

(••) )

×29

, |T31| = 32 ρ (T31) = ω + 29

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More on F (3) = 125

. . .

12

T28 = (((•) •)) , |T28| = 5 ρ (T28) = ω · 2 + 1 T29 = ((•) •) , |T29| = 4 ρ (T29) = ω · 2 T30 = (

×30

(••) )

×30

, |T30| = 33 ρ (T30) = ω + 30 T31 = (

×29

(••) )

×29

, |T31| = 32 ρ (T31) = ω + 29 . . .

28
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More on F (3) = 125

. . .

12

T28 = (((•) •)) , |T28| = 5 ρ (T28) = ω · 2 + 1 T29 = ((•) •) , |T29| = 4 ρ (T29) = ω · 2 T30 = (

×30

(••) )

×30

, |T30| = 33 ρ (T30) = ω + 30 T31 = (

×29

(••) )

×29

, |T31| = 32 ρ (T31) = ω + 29 . . .

28

T60 = (••) , |T60| = 3 ρ (T60) = ω

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More on F (3) = 125

. . .

12

T28 = (((•) •)) , |T28| = 5 ρ (T28) = ω · 2 + 1 T29 = ((•) •) , |T29| = 4 ρ (T29) = ω · 2 T30 = (

×30

(••) )

×30

, |T30| = 33 ρ (T30) = ω + 30 T31 = (

×29

(••) )

×29

, |T31| = 32 ρ (T31) = ω + 29 . . .

28

T60 = (••) , |T60| = 3 ρ (T60) = ω T61 = (

×63
)
×63

, |T61| = 64 ρ (T61) = 63

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More on F (3) = 125

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More on F (3) = 125

T62 = (

×62
)
×62

, |T62| = 63 ρ (T62) = 62

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More on F (3) = 125

T62 = (

×62
)
×62

, |T62| = 63 ρ (T62) = 62 . . .

61
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More on F (3) = 125

T62 = (

×62
)
×62

, |T62| = 63 ρ (T62) = 62 . . .

61

T124 = • , |T124| = 1 ρ (T124) = 1

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More on F (3) = 125

T62 = (

×62
)
×62

, |T62| = 63 ρ (T62) = 62 . . .

61

T124 = • , |T124| = 1 ρ (T124) = 1 This actually yields F (3) = 125

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More on F (3) = 125

T62 = (

×62
)
×62

, |T62| = 63 ρ (T62) = 62 . . .

61

T124 = • , |T124| = 1 ρ (T124) = 1 This actually yields F (3) = 125 Summing up: We extend the 4-vertex ternary tree T1 by a suitable bad sequence of at most binary trees T2 − T60 followed by the (uniquely determined) maximal bad sequence

f unary trees, i.e. just linear paths, T61 − T124. The whole

sequence is bad, since no ternary (resp. binary) tree is embeddable into any binary or unary (resp. unary) tree.

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Ordinal approximations -1-

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Ordinal approximations -1-

Definition

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Ordinal approximations -1-

Definition For any T ⊆ Tree call ρ : T 1−1 → O an ordinal evaluation iff T T ′ ⇒ ρ (T) ≤ ρ

T ′
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Ordinal approximations -1-

Definition For any T ⊆ Tree call ρ : T 1−1 → O an ordinal evaluation iff T T ′ ⇒ ρ (T) ≤ ρ

T ′

Define Φρ,T : T × N ∋ T → Φρ,T (T) ∈ N by transfinite recursion

n ρ (T):
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Ordinal approximations -1-

Definition For any T ⊆ Tree call ρ : T 1−1 → O an ordinal evaluation iff T T ′ ⇒ ρ (T) ≤ ρ

T ′

Define Φρ,T : T × N ∋ T → Φρ,T (T) ∈ N by transfinite recursion

n ρ (T):

Φρ,T (T, x) =

1

, if ρ (T) = 0 1 + Φρ,T

T, x + 1
, if

ρ (T) > 0 ,

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Ordinal approximations -1-

Definition For any T ⊆ Tree call ρ : T 1−1 → O an ordinal evaluation iff T T ′ ⇒ ρ (T) ≤ ρ

T ′

Define Φρ,T : T × N ∋ T → Φρ,T (T) ∈ N by transfinite recursion

n ρ (T):

Φρ,T (T, x) =

1

, if ρ (T) = 0 1 + Φρ,T

T, x + 1
, if

ρ (T) > 0 , where

T ∈ T is uniquely determined by a following condition

ρ

T
= max {ρ (T ′) < ρ (T) : T ′ ∈ T ∧ |T ′| ≤ x + 1}.
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Ordinal approximations -2-

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Ordinal approximations -2-

Definition

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Ordinal approximations -2-

Definition For any given S ∈ Tree let BadEi (S) := T ∈ Tree : ∃ S = T1, · · · , Tn = T ∈ Bad ∧ (∀1 < i ≤ n) (|Ti| ≤ |T1| + i − 1)

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Ordinal approximations -2-

Definition For any given S ∈ Tree let BadEi (S) := T ∈ Tree : ∃ S = T1, · · · , Tn = T ∈ Bad ∧ (∀1 < i ≤ n) (|Ti| ≤ |T1| + i − 1)

Bad := {T1, · · · , Tn ∈ Tree* : (∀1 ≤ i < j ≤ n) (Ti Tj)}
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Ordinal approximations -2-

Definition For any given S ∈ Tree let BadEi (S) := T ∈ Tree : ∃ S = T1, · · · , Tn = T ∈ Bad ∧ (∀1 < i ≤ n) (|Ti| ≤ |T1| + i − 1)

Bad := {T1, · · · , Tn ∈ Tree* : (∀1 ≤ i < j ≤ n) (Ti Tj)}

Now for any k > 0 let Tk := BadEi (Sk) where Sk =  • · · · •

k

  .

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Ordinal approximations -2-

Definition For any given S ∈ Tree let BadEi (S) := T ∈ Tree : ∃ S = T1, · · · , Tn = T ∈ Bad ∧ (∀1 < i ≤ n) (|Ti| ≤ |T1| + i − 1)

Bad := {T1, · · · , Tn ∈ Tree* : (∀1 ≤ i < j ≤ n) (Ti Tj)}

Now for any k > 0 let Tk := BadEi (Sk) where Sk =  • · · · •

k

  . Corollary

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