Very large finite numbers

Andres E. Caicedo

Department of MathematicsBoise State University

Graduate Student Seminar, October 15, 2008

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Introduction

I would like to present two sets of results in finite combinatoricsmotivated by problems in mathematical logic. The commontheme is that the results produce sequences of numbers thatgrow very fast.I will be discussing regressive functions on dimension 2. Thistopic is part of the branch of combinatorics known as Ramseytheory.I will also talk about Goodstein’s function, an easy example of acomputable function that grows faster than any function onecan reasonably understand.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Introduction

I would like to present two sets of results in finite combinatoricsmotivated by problems in mathematical logic. The commontheme is that the results produce sequences of numbers thatgrow very fast.I will be discussing regressive functions on dimension 2. Thistopic is part of the branch of combinatorics known as Ramseytheory.I will also talk about Goodstein’s function, an easy example of acomputable function that grows faster than any function onecan reasonably understand.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Introduction

I would like to present two sets of results in finite combinatoricsmotivated by problems in mathematical logic. The commontheme is that the results produce sequences of numbers thatgrow very fast.I will be discussing regressive functions on dimension 2. Thistopic is part of the branch of combinatorics known as Ramseytheory.I will also talk about Goodstein’s function, an easy example of acomputable function that grows faster than any function onecan reasonably understand.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

Ramsey theory

Given a collection V of vertices, the complete graph on Vconsists of all the possible edges between these vertices.

����������@

@@@@@@@@@

4 3

1 2

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

Ramsey theory

Given a collection V of vertices, the complete graph on Vconsists of all the possible edges between these vertices.

����������@

@@@@@@@@@

4 3

1 2

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

Ramsey theory is concerned with substructures and partitions:You color each edge of the complete graph on V and look for asubset W of the set of vertices such that the completesubgraph on W is monochromatic.Classical Ramsey theory fixes the number of colors in advance,and asks how large V needs to be given the size of the desiredset W .For example: Let’s use only two colors. Then V needs to havesize 6 to ensure that we can find a monochromatic triangle. Insymbols, r3 = 6.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

Ramsey theory is concerned with substructures and partitions:You color each edge of the complete graph on V and look for asubset W of the set of vertices such that the completesubgraph on W is monochromatic.Classical Ramsey theory fixes the number of colors in advance,and asks how large V needs to be given the size of the desiredset W .For example: Let’s use only two colors. Then V needs to havesize 6 to ensure that we can find a monochromatic triangle. Insymbols, r3 = 6.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

Ramsey theory is concerned with substructures and partitions:You color each edge of the complete graph on V and look for asubset W of the set of vertices such that the completesubgraph on W is monochromatic.Classical Ramsey theory fixes the number of colors in advance,and asks how large V needs to be given the size of the desiredset W .For example: Let’s use only two colors. Then V needs to havesize 6 to ensure that we can find a monochromatic triangle. Insymbols, r3 = 6.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

This is usually presented in the following way:

In any party of six people, there are always three whoknow each other, or three who do not know oneanother.

Five does not suffice: Color the lines of a pentagon “dark” or“transparent” as below, and notice no monochromatic trianglesappear.

1

2 HHHHH3

vvvvv

4)))))

5

�����

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

This is usually presented in the following way:

In any party of six people, there are always three whoknow each other, or three who do not know oneanother.

Five does not suffice: Color the lines of a pentagon “dark” or“transparent” as below, and notice no monochromatic trianglesappear.

1

2 HHHHH3

vvvvv

4)))))

5

�����

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

This is usually presented in the following way:

In any party of six people, there are always three whoknow each other, or three who do not know oneanother.

Five does not suffice: Color the lines of a pentagon “dark” or“transparent” as below, and notice no monochromatic trianglesappear.

1

2 HHHHH3

vvvvv

4)))))

5

�����

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

This is usually presented in the following way:

In any party of six people, there are always three whoknow each other, or three who do not know oneanother.

Five does not suffice: Color the lines of a pentagon “dark” or“transparent” as below, and notice no monochromatic trianglesappear.

1

2 HHHHH3

vvvvv

4)))))

5

�����

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

To see that six works, fix one of the vertices, call it A. There arefive more vertices, B–F . Necessarily, three of the linesconnecting A to these vertices must have the same color.

�������������

BBBBBBBBBBBBB

JJ

JJ

JJ

JJ

JJ

JJJ

B C D E F

A

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

To see that six works, fix one of the vertices, call it A. There arefive more vertices, B–F . Necessarily, three of the linesconnecting A to these vertices must have the same color.

�������������

BBBBBBBBBBBBB

JJ

JJ

JJ

JJ

JJ

JJJ

B C D E F

A

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

�������������

BBBBBBBBBBBBB

JJ

JJ

JJ

JJ

JJ

JJJ

B C D E F

A

Now notice that no line between B, C, E can be “dark”, or weobtain a monochromatic dark triangle. But then, if these threelines are light, we obtain a monochromatic light triangle.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

�������������

BBBBBBBBBBBBB

JJ

JJ

JJ

JJ

JJ

JJJ

B C D E F

A

Now notice that no line between B, C, E can be “dark”, or weobtain a monochromatic dark triangle. But then, if these threelines are light, we obtain a monochromatic light triangle.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

Similarly, using only two colors, V needs to have size 18 toensure that we can find W ⊆ V of size 4, with all the edgesamong vertices in W of the same color. In symbols, r4 = 18.The exact value of r5 is not known. We know that 43 ≤ r5 ≤ 49.

On the other hand, although exact values are not known, wehave a good understanding of the rate of growth of rn, the nth

Ramsey number, which turns out to be exponential in n:

2n/2 ≤ rn ≤ 22n.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

Similarly, using only two colors, V needs to have size 18 toensure that we can find W ⊆ V of size 4, with all the edgesamong vertices in W of the same color. In symbols, r4 = 18.The exact value of r5 is not known. We know that 43 ≤ r5 ≤ 49.

On the other hand, although exact values are not known, wehave a good understanding of the rate of growth of rn, the nth

Ramsey number, which turns out to be exponential in n:

2n/2 ≤ rn ≤ 22n.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

Similarly, using only two colors, V needs to have size 18 toensure that we can find W ⊆ V of size 4, with all the edgesamong vertices in W of the same color. In symbols, r4 = 18.The exact value of r5 is not known. We know that 43 ≤ r5 ≤ 49.

On the other hand, although exact values are not known, wehave a good understanding of the rate of growth of rn, the nth

Ramsey number, which turns out to be exponential in n:

2n/2 ≤ rn ≤ 22n.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

Regressive Ramsey numbers

The work I want to concentrate on studies a similar function Rn,but now we allow the number of colors to increase. To describethe numbers Rn, it is convenient to imagine the vertices in Vnumbered 1,2, . . . ,n. I write ij for the edge between vertices iand j .Now we use whole numbers as colors. The color of an edge 1iis always 0. The color of an edge 2i (with 2 < i) can be 0 or 1.The color of an edge 3i (with 3 < i) can be 0, 1, or 2. And soon. We call these colorings regressive.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

Regressive Ramsey numbers

The work I want to concentrate on studies a similar function Rn,but now we allow the number of colors to increase. To describethe numbers Rn, it is convenient to imagine the vertices in Vnumbered 1,2, . . . ,n. I write ij for the edge between vertices iand j .Now we use whole numbers as colors. The color of an edge 1iis always 0. The color of an edge 2i (with 2 < i) can be 0 or 1.The color of an edge 3i (with 3 < i) can be 0, 1, or 2. And soon. We call these colorings regressive.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

Now, instead of a mono-chromatic “polygon,” we look atmin-homogeneous sets. (Homogeneous sets of more than twoelements might not exist.)

Let W ⊂ V , say W = {i1, i2, . . . , ik} where i1 < · · · < ik . We saythat W is min-homogeneous for a given regressive coloring ifall edges i1i2, i1i3, . . . , i1ik receive the same color c < i1; all theedges i2i3, i2i4, . . . , i2ik receive the same color d < i2 (that maybe different from c); and so on.

Let Rn be the smallest number of vertices that V needs to haveto guarantee that for any regressive coloring there is W ⊂ V ofsize n and min-homogeneous.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

Now, instead of a mono-chromatic “polygon,” we look atmin-homogeneous sets. (Homogeneous sets of more than twoelements might not exist.)

Let W ⊂ V , say W = {i1, i2, . . . , ik} where i1 < · · · < ik . We saythat W is min-homogeneous for a given regressive coloring ifall edges i1i2, i1i3, . . . , i1ik receive the same color c < i1; all theedges i2i3, i2i4, . . . , i2ik receive the same color d < i2 (that maybe different from c); and so on.

Let Rn be the smallest number of vertices that V needs to haveto guarantee that for any regressive coloring there is W ⊂ V ofsize n and min-homogeneous.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

Now, instead of a mono-chromatic “polygon,” we look atmin-homogeneous sets. (Homogeneous sets of more than twoelements might not exist.)

Let W ⊂ V , say W = {i1, i2, . . . , ik} where i1 < · · · < ik . We saythat W is min-homogeneous for a given regressive coloring ifall edges i1i2, i1i3, . . . , i1ik receive the same color c < i1; all theedges i2i3, i2i4, . . . , i2ik receive the same color d < i2 (that maybe different from c); and so on.

Let Rn be the smallest number of vertices that V needs to haveto guarantee that for any regressive coloring there is W ⊂ V ofsize n and min-homogeneous.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

For example, R3 = 3, because 12 and 13 always receive color0, so 23 can receive any color, and {1,2,3} ismin-homogeneous.

Similarly, R4 = 5 and we can always find a min-homogeneousset of the form {1,2,a,b} with 3 ≤ a < b ≤ 5.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

For example, R3 = 3, because 12 and 13 always receive color0, so 23 can receive any color, and {1,2,3} ismin-homogeneous.

Similarly, R4 = 5 and we can always find a min-homogeneousset of the form {1,2,a,b} with 3 ≤ a < b ≤ 5.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

The numbers Rn were originally studied by Kanamori andMcAloon. Their result has two parts.

Theorem (Kanamori-McAloon)For any n ∈ N, Rn exists.

To state the second part of their theorem, we need the notion ofprimitive recursive function.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

The numbers Rn were originally studied by Kanamori andMcAloon. Their result has two parts.

Theorem (Kanamori-McAloon)For any n ∈ N, Rn exists.

To state the second part of their theorem, we need the notion ofprimitive recursive function.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

The numbers Rn were originally studied by Kanamori andMcAloon. Their result has two parts.

Theorem (Kanamori-McAloon)For any n ∈ N, Rn exists.

To state the second part of their theorem, we need the notion ofprimitive recursive function.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

We define the primitive recursive functions by starting with thebasic functions f : Nk → N consisting of constant functions, thesuccessor function S(n) = n + 1, and projectionsf (n1, . . . ,nk ) = ni .We close these functions under composition, and recursion: Ifg : Nk → N and h : Nk+2 → N are primitive recursive, so isf : Nk+1 → N given by

f (~x , y) =

{g(~x) if y = 0,h(~x ,n, f (~x ,n)) if y = n + 1.

To understand this definition, it is perhaps useful to see a fewexamples.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

We define the primitive recursive functions by starting with thebasic functions f : Nk → N consisting of constant functions, thesuccessor function S(n) = n + 1, and projectionsf (n1, . . . ,nk ) = ni .We close these functions under composition, and recursion: Ifg : Nk → N and h : Nk+2 → N are primitive recursive, so isf : Nk+1 → N given by

f (~x , y) =

{g(~x) if y = 0,h(~x ,n, f (~x ,n)) if y = n + 1.

To understand this definition, it is perhaps useful to see a fewexamples.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

We define the primitive recursive functions by starting with thebasic functions f : Nk → N consisting of constant functions, thesuccessor function S(n) = n + 1, and projectionsf (n1, . . . ,nk ) = ni .We close these functions under composition, and recursion: Ifg : Nk → N and h : Nk+2 → N are primitive recursive, so isf : Nk+1 → N given by

f (~x , y) =

{g(~x) if y = 0,h(~x ,n, f (~x ,n)) if y = n + 1.

To understand this definition, it is perhaps useful to see a fewexamples.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

The following functions are primitive recursive:

1 f (x , y) = x + y . We can describe it in the format abovenoting that

f (x , y) =

{x if y = 0,(x + n) + 1 if y = n + 1.

2 f (x , y) = xy .3 f (x , y) = xy .

4 f (x , y) = xxx

...x

, where the tower has height y .5 Etc.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

The following functions are primitive recursive:

1 f (x , y) = x + y . We can describe it in the format abovenoting that

f (x , y) =

{x if y = 0,(x + n) + 1 if y = n + 1.

2 f (x , y) = xy .3 f (x , y) = xy .

4 f (x , y) = xxx

...x

, where the tower has height y .5 Etc.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

The following functions are primitive recursive:

1 f (x , y) = x + y . We can describe it in the format abovenoting that

f (x , y) =

{x if y = 0,(x + n) + 1 if y = n + 1.

2 f (x , y) = xy .3 f (x , y) = xy .

4 f (x , y) = xxx

...x

, where the tower has height y .5 Etc.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

The following functions are primitive recursive:

1 f (x , y) = x + y . We can describe it in the format abovenoting that

f (x , y) =

{x if y = 0,(x + n) + 1 if y = n + 1.

2 f (x , y) = xy .3 f (x , y) = xy .

4 f (x , y) = xxx

...x

, where the tower has height y .5 Etc.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

The following functions are primitive recursive:

1 f (x , y) = x + y . We can describe it in the format abovenoting that

f (x , y) =

{x if y = 0,(x + n) + 1 if y = n + 1.

2 f (x , y) = xy .3 f (x , y) = xy .

4 f (x , y) = xxx

...x

, where the tower has height y .5 Etc.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

The following functions are primitive recursive:

1 f (x , y) = x + y . We can describe it in the format abovenoting that

f (x , y) =

{x if y = 0,(x + n) + 1 if y = n + 1.

2 f (x , y) = xy .3 f (x , y) = xy .

4 f (x , y) = xxx

...x

, where the tower has height y .5 Etc.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

We can now state the second part of the Kanamori-McAloonresult. Say that f : N→ N grows faster than g : N→ N or that feventually dominates g, iff f (n) > g(n) for all but finitely manyvalues of n.

Theorem (Kanamori-McAloon)

The function f (n) = Rn grows faster than any primitive recursivefunction.

Their proof used tools of mathematical logic and did notproduce any explicit bounds.

Several mathematicians tried to provide explicit proofs of theKanamori-McAloon result; among others, Kojman, Shelah,Blanchard, and Weiermann.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

We can now state the second part of the Kanamori-McAloonresult. Say that f : N→ N grows faster than g : N→ N or that feventually dominates g, iff f (n) > g(n) for all but finitely manyvalues of n.

Theorem (Kanamori-McAloon)

The function f (n) = Rn grows faster than any primitive recursivefunction.

Their proof used tools of mathematical logic and did notproduce any explicit bounds.

Several mathematicians tried to provide explicit proofs of theKanamori-McAloon result; among others, Kojman, Shelah,Blanchard, and Weiermann.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

We can now state the second part of the Kanamori-McAloonresult. Say that f : N→ N grows faster than g : N→ N or that feventually dominates g, iff f (n) > g(n) for all but finitely manyvalues of n.

Theorem (Kanamori-McAloon)

The function f (n) = Rn grows faster than any primitive recursivefunction.

Their proof used tools of mathematical logic and did notproduce any explicit bounds.

Several mathematicians tried to provide explicit proofs of theKanamori-McAloon result; among others, Kojman, Shelah,Blanchard, and Weiermann.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

We can now state the second part of the Kanamori-McAloonresult. Say that f : N→ N grows faster than g : N→ N or that feventually dominates g, iff f (n) > g(n) for all but finitely manyvalues of n.

Theorem (Kanamori-McAloon)

The function f (n) = Rn grows faster than any primitive recursivefunction.

Their proof used tools of mathematical logic and did notproduce any explicit bounds.

Several mathematicians tried to provide explicit proofs of theKanamori-McAloon result; among others, Kojman, Shelah,Blanchard, and Weiermann.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

DefinitionAckermann’s function A : N× N→ N is defined “by doublerecursion” as follows:

A(0,m) = m + 1.A(n,0) = A(n − 1,1) for n > 0.A(n,m) = A(n − 1,A(n,m − 1)) for n,m > 0.

Let An = A(n, ·). For example, A0(m) = m + 1, A1(m) = m + 2,A2(m) = 2m + 3, A3 grows like 2m, A4 grows like a tower oftwo’s of length m, etc.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

DefinitionAckermann’s function A : N× N→ N is defined “by doublerecursion” as follows:

A(0,m) = m + 1.A(n,0) = A(n − 1,1) for n > 0.A(n,m) = A(n − 1,A(n,m − 1)) for n,m > 0.

Let An = A(n, ·). For example, A0(m) = m + 1, A1(m) = m + 2,A2(m) = 2m + 3, A3 grows like 2m, A4 grows like a tower oftwo’s of length m, etc.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

DefinitionAckermann’s function A : N× N→ N is defined “by doublerecursion” as follows:

A(0,m) = m + 1.A(n,0) = A(n − 1,1) for n > 0.A(n,m) = A(n − 1,A(n,m − 1)) for n,m > 0.

Let An = A(n, ·). For example, A0(m) = m + 1, A1(m) = m + 2,A2(m) = 2m + 3, A3 grows like 2m, A4 grows like a tower oftwo’s of length m, etc.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

DefinitionAckermann’s function A : N× N→ N is defined “by doublerecursion” as follows:

A(0,m) = m + 1.A(n,0) = A(n − 1,1) for n > 0.A(n,m) = A(n − 1,A(n,m − 1)) for n,m > 0.

Let An = A(n, ·). For example, A0(m) = m + 1, A1(m) = m + 2,A2(m) = 2m + 3, A3 grows like 2m, A4 grows like a tower oftwo’s of length m, etc.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

DefinitionAckermann’s function A : N× N→ N is defined “by doublerecursion” as follows:

A(0,m) = m + 1.A(n,0) = A(n − 1,1) for n > 0.A(n,m) = A(n − 1,A(n,m − 1)) for n,m > 0.

Let An = A(n, ·). For example, A0(m) = m + 1, A1(m) = m + 2,A2(m) = 2m + 3, A3 grows like 2m, A4 grows like a tower oftwo’s of length m, etc.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

TheoremAckermann’s function is not primitive recursive. In fact, An(n)grows faster than any primitive recursive function.

Ackermann’s original definition used three variables. Theversion given above was introduced by Rafael Robinson andRozsa Peter, and has become the standard example of acomputable function that is not primitive recursive.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

TheoremAckermann’s function is not primitive recursive. In fact, An(n)grows faster than any primitive recursive function.

Ackermann’s original definition used three variables. Theversion given above was introduced by Rafael Robinson andRozsa Peter, and has become the standard example of acomputable function that is not primitive recursive.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

Let g(n,m) be the least l such that if V = {m,m + 1, . . . , l} thenany regressive coloring of V admits a min-homogeneous W ofsize n. For example, g(2,m) = m + 1 and g(3,m) = 2m + 1.The nth regressive Ramsey number Rn is g(n,1) = g(n − 1,2).

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

Let g(n,m) be the least l such that if V = {m,m + 1, . . . , l} thenany regressive coloring of V admits a min-homogeneous W ofsize n. For example, g(2,m) = m + 1 and g(3,m) = 2m + 1.The nth regressive Ramsey number Rn is g(n,1) = g(n − 1,2).

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

Set g0(n,m) = m and gk+1(n,m) = g(n,gk (n,m)). We thenhave:

Theoremg(n + 1,m + 1) ≥ g(n,g(n + 1,m) + 1). In particular, for n ≥ 2and m ≥ 1, g(n,m) ≥ An−1(m − 1).

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

Set g0(n,m) = m and gk+1(n,m) = g(n,gk (n,m)). We thenhave:

Theoremg(n + 1,m + 1) ≥ g(n,g(n + 1,m) + 1). In particular, for n ≥ 2and m ≥ 1, g(n,m) ≥ An−1(m − 1).

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

Theorem1 For m ≥ 2, g(4,m) ≤ 2m(m + 2)− 2m−1 + 1.2 For all n there is a constant cn such that

g(n,m) < An−1(cnm) for all m.

Thus g(n, ·) grows like An−1 and g(·,m) grows like Ackermann’sfunction.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

Theorem1 For m ≥ 2, g(4,m) ≤ 2m(m + 2)− 2m−1 + 1.2 For all n there is a constant cn such that

g(n,m) < An−1(cnm) for all m.

Thus g(n, ·) grows like An−1 and g(·,m) grows like Ackermann’sfunction.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

Theorem1 For m ≥ 2, g(4,m) ≤ 2m(m + 2)− 2m−1 + 1.2 For all n there is a constant cn such that

g(n,m) < An−1(cnm) for all m.

Thus g(n, ·) grows like An−1 and g(·,m) grows like Ackermann’sfunction.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

The proofs I have obtained are explicit, considerably improvethe previously known bounds, and identify the right rate ofgrowth of the numbers Rn. They provide us with a combinatorialproof of a result formerly obtained by non-constructive means.They also identify the first example of a naturally occurringfunction whose rate of growth is Ackermannian.

Similar problems can be studied in dimension 3 or larger, andmany interesting questions related to the computation of explicitbounds remain.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

The Kanamori-McAloon theorem, part 1The Kanamori-McAloon theorem, part 2Bounds

The proofs I have obtained are explicit, considerably improvethe previously known bounds, and identify the right rate ofgrowth of the numbers Rn. They provide us with a combinatorialproof of a result formerly obtained by non-constructive means.They also identify the first example of a naturally occurringfunction whose rate of growth is Ackermannian.

Similar problems can be studied in dimension 3 or larger, andmany interesting questions related to the computation of explicitbounds remain.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

Goodstein’s function

DefinitionThe depth-1 base b representation of n ∈ N is just the usualbase b representation of n:

n = bm1n1 + · · ·+ bmk nk

where m1 > · · · > mk ≥ 0 and 1 ≤ ni < b for each i .The depth-(m + 1) representation is obtained by replacing eachmi with their depth-m base b representation.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

Goodstein’s function

DefinitionThe depth-1 base b representation of n ∈ N is just the usualbase b representation of n:

n = bm1n1 + · · ·+ bmk nk

where m1 > · · · > mk ≥ 0 and 1 ≤ ni < b for each i .The depth-(m + 1) representation is obtained by replacing eachmi with their depth-m base b representation.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

For example, the depth-1 base 2 representation of 266 is28 + 23 + 21, its depth-2 base 2 representation is223

+ 221+1 + 21 and so its depth-3 (or higher) base 2representation is

266 = 222+1+ 22+1 + 2.

For any n and b, as m increases, the depth-(m + 1) base brepresentations of n eventually stabilize.

DefinitionWe call this stable representation the complete base brepresentation of n ∈ N.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

For example, the depth-1 base 2 representation of 266 is28 + 23 + 21, its depth-2 base 2 representation is223

+ 221+1 + 21 and so its depth-3 (or higher) base 2representation is

266 = 222+1+ 22+1 + 2.

For any n and b, as m increases, the depth-(m + 1) base brepresentations of n eventually stabilize.

DefinitionWe call this stable representation the complete base brepresentation of n ∈ N.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

For example, the depth-1 base 2 representation of 266 is28 + 23 + 21, its depth-2 base 2 representation is223

+ 221+1 + 21 and so its depth-3 (or higher) base 2representation is

266 = 222+1+ 22+1 + 2.

For any n and b, as m increases, the depth-(m + 1) base brepresentations of n eventually stabilize.

DefinitionWe call this stable representation the complete base brepresentation of n ∈ N.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

DefinitionThe change of base function Rb : N→ N takes a naturalnumber n, and then replaces every b with b + 1 in the completebase b representation of n.

For example,

R2(266) = 333+1+ 33+1 + 3

= 443426488243037769948249630619149892887.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

DefinitionThe change of base function Rb : N→ N takes a naturalnumber n, and then replaces every b with b + 1 in the completebase b representation of n.

For example,

R2(266) = 333+1+ 33+1 + 3

= 443426488243037769948249630619149892887.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

DefinitionThe Goodstein Sequence beginning with n, (n)k , is defined by(n)1 = n and for k ≥ 1,

(n)k+1 =

{Rk+1((n)k )− 1 if (n)k > 00 if (n)k = 0.

For example, the sequence for n = 3 is 3,3,3,2,1,0,0, . . .

DefinitionThe Goodstein Function G : N→ N is defined to be the smallestnumber k for which (n)k = 0.

Clearly, G is computable.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

DefinitionThe Goodstein Sequence beginning with n, (n)k , is defined by(n)1 = n and for k ≥ 1,

(n)k+1 =

{Rk+1((n)k )− 1 if (n)k > 00 if (n)k = 0.

For example, the sequence for n = 3 is 3,3,3,2,1,0,0, . . .

DefinitionThe Goodstein Function G : N→ N is defined to be the smallestnumber k for which (n)k = 0.

Clearly, G is computable.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

DefinitionThe Goodstein Sequence beginning with n, (n)k , is defined by(n)1 = n and for k ≥ 1,

(n)k+1 =

{Rk+1((n)k )− 1 if (n)k > 00 if (n)k = 0.

For example, the sequence for n = 3 is 3,3,3,2,1,0,0, . . .

DefinitionThe Goodstein Function G : N→ N is defined to be the smallestnumber k for which (n)k = 0.

Clearly, G is computable.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

DefinitionThe Goodstein Sequence beginning with n, (n)k , is defined by(n)1 = n and for k ≥ 1,

(n)k+1 =

{Rk+1((n)k )− 1 if (n)k > 00 if (n)k = 0.

For example, the sequence for n = 3 is 3,3,3,2,1,0,0, . . .

DefinitionThe Goodstein Function G : N→ N is defined to be the smallestnumber k for which (n)k = 0.

Clearly, G is computable.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

Theorem (Goodstein)

The function G is well defined, i.e., G(n) exists for all n.

Here are the first few values of G:

G(0) = 1

G(1) = 2

G(2) = 4

G(3) = 6

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

Theorem (Goodstein)

The function G is well defined, i.e., G(n) exists for all n.

Here are the first few values of G:

G(0) = 1

G(1) = 2

G(2) = 4

G(3) = 6

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

Theorem (Goodstein)

The function G is well defined, i.e., G(n) exists for all n.

Here are the first few values of G:

G(0) = 1

G(1) = 2

G(2) = 4

G(3) = 6

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

Theorem (Goodstein)

The function G is well defined, i.e., G(n) exists for all n.

Here are the first few values of G:

G(0) = 1

G(1) = 2

G(2) = 4

G(3) = 6

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

Theorem (Goodstein)

The function G is well defined, i.e., G(n) exists for all n.

Here are the first few values of G:

G(0) = 1

G(1) = 2

G(2) = 4

G(3) = 6

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

G(4) = 3 · 2402653211 − 2 ≈ 6.895× 10121210694.

The number of digits of G(5) is much larger than G(4). Thenumber of elementary particles in the universe is estimated tobe (significantly) below 1090.To analyze this function one is required to step out of the naturalnumbers and consider the more general notion of ordinals.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

G(4) = 3 · 2402653211 − 2 ≈ 6.895× 10121210694.

The number of digits of G(5) is much larger than G(4). Thenumber of elementary particles in the universe is estimated tobe (significantly) below 1090.To analyze this function one is required to step out of the naturalnumbers and consider the more general notion of ordinals.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

G(4) = 3 · 2402653211 − 2 ≈ 6.895× 10121210694.

The number of digits of G(5) is much larger than G(4). Thenumber of elementary particles in the universe is estimated tobe (significantly) below 1090.To analyze this function one is required to step out of the naturalnumbers and consider the more general notion of ordinals.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

The ordinals are obtained by continuing the natural numbersinto the transfinite. We have two operations:

Given any ordinal α, one can add 1 to it to obtain itssuccessor ordinal, α+ 1.Or we can look at a collection of ordinals without amaximum, and add its least upper bound to it. These arecalled limit ordinals.

It is customary to call ω the first infinite ordinal. The first fewordinals are 0,1,2, . . . , and then:

ω, ω + 1, . . . , ω · 2, ω · 2 + 1, . . . , ω · 3, . . . , ω2, . . . , ω3, . . . , ωω, . . .

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

The ordinals are obtained by continuing the natural numbersinto the transfinite. We have two operations:

Given any ordinal α, one can add 1 to it to obtain itssuccessor ordinal, α+ 1.Or we can look at a collection of ordinals without amaximum, and add its least upper bound to it. These arecalled limit ordinals.

It is customary to call ω the first infinite ordinal. The first fewordinals are 0,1,2, . . . , and then:

ω, ω + 1, . . . , ω · 2, ω · 2 + 1, . . . , ω · 3, . . . , ω2, . . . , ω3, . . . , ωω, . . .

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

The ordinals are obtained by continuing the natural numbersinto the transfinite. We have two operations:

Given any ordinal α, one can add 1 to it to obtain itssuccessor ordinal, α+ 1.Or we can look at a collection of ordinals without amaximum, and add its least upper bound to it. These arecalled limit ordinals.

It is customary to call ω the first infinite ordinal. The first fewordinals are 0,1,2, . . . , and then:

ω, ω + 1, . . . , ω · 2, ω · 2 + 1, . . . , ω · 3, . . . , ω2, . . . , ω3, . . . , ωω, . . .

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

The ordinals are obtained by continuing the natural numbersinto the transfinite. We have two operations:

Given any ordinal α, one can add 1 to it to obtain itssuccessor ordinal, α+ 1.Or we can look at a collection of ordinals without amaximum, and add its least upper bound to it. These arecalled limit ordinals.

It is customary to call ω the first infinite ordinal. The first fewordinals are 0,1,2, . . . , and then:

ω, ω + 1, . . . , ω · 2, ω · 2 + 1, . . . , ω · 3, . . . , ω2, . . . , ω3, . . . , ωω, . . .

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

The ordinals are obtained by continuing the natural numbersinto the transfinite. We have two operations:

Given any ordinal α, one can add 1 to it to obtain itssuccessor ordinal, α+ 1.Or we can look at a collection of ordinals without amaximum, and add its least upper bound to it. These arecalled limit ordinals.

It is customary to call ω the first infinite ordinal. The first fewordinals are 0,1,2, . . . , and then:

ω, ω + 1, . . . , ω · 2, ω · 2 + 1, . . . , ω · 3, . . . , ω2, . . . , ω3, . . . , ωω, . . .

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

We denote by ε0 the first ordinal α such that α = ωα.Any ordinal α < ε0 can be written in a unique way asα = ωβ(γ + 1) where β < α. Define for limit α < ε0 anincreasing sequence d(α,n) that converges to α by setting

d(α,n) = ωβγ +

{ωδn if β = δ + 1,ωd(β,n) if β is limit.

For example, d(ω,n) = n, d(ω3,n) = ω2 · n, d(ωωω,n) = ωω

n.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

We denote by ε0 the first ordinal α such that α = ωα.Any ordinal α < ε0 can be written in a unique way asα = ωβ(γ + 1) where β < α. Define for limit α < ε0 anincreasing sequence d(α,n) that converges to α by setting

d(α,n) = ωβγ +

{ωδn if β = δ + 1,ωd(β,n) if β is limit.

For example, d(ω,n) = n, d(ω3,n) = ω2 · n, d(ωωω,n) = ωω

n.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

We denote by ε0 the first ordinal α such that α = ωα.Any ordinal α < ε0 can be written in a unique way asα = ωβ(γ + 1) where β < α. Define for limit α < ε0 anincreasing sequence d(α,n) that converges to α by setting

d(α,n) = ωβγ +

{ωδn if β = δ + 1,ωd(β,n) if β is limit.

For example, d(ω,n) = n, d(ω3,n) = ω2 · n, d(ωωω,n) = ωω

n.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

We denote by ε0 the first ordinal α such that α = ωα.Any ordinal α < ε0 can be written in a unique way asα = ωβ(γ + 1) where β < α. Define for limit α < ε0 anincreasing sequence d(α,n) that converges to α by setting

d(α,n) = ωβγ +

{ωδn if β = δ + 1,ωd(β,n) if β is limit.

For example, d(ω,n) = n, d(ω3,n) = ω2 · n, d(ωωω,n) = ωω

n.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

The Lob-Wainer fast growing hierarchy (fα)α<ε0 of functionsf : N→ N is:

Definition1 f0(n) = n + 1.2 For α < ε0, fα+1(n) = f n

α(n), where the superindexindicates that fα is iterated n times.

3 For limit α < ε0, fα(n) = fd(α,n)(n).

For example: f1(n) = 2n, f2(n) = n2n and f3(n) is (significantly)larger than a stack of powers of two of length n; fω(n) = fn(n)grows like Ackermann’s function An(n).

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

The Lob-Wainer fast growing hierarchy (fα)α<ε0 of functionsf : N→ N is:

Definition1 f0(n) = n + 1.2 For α < ε0, fα+1(n) = f n

α(n), where the superindexindicates that fα is iterated n times.

3 For limit α < ε0, fα(n) = fd(α,n)(n).

For example: f1(n) = 2n, f2(n) = n2n and f3(n) is (significantly)larger than a stack of powers of two of length n; fω(n) = fn(n)grows like Ackermann’s function An(n).

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

The Lob-Wainer fast growing hierarchy (fα)α<ε0 of functionsf : N→ N is:

Definition1 f0(n) = n + 1.2 For α < ε0, fα+1(n) = f n

α(n), where the superindexindicates that fα is iterated n times.

3 For limit α < ε0, fα(n) = fd(α,n)(n).

For example: f1(n) = 2n, f2(n) = n2n and f3(n) is (significantly)larger than a stack of powers of two of length n; fω(n) = fn(n)grows like Ackermann’s function An(n).

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

The Lob-Wainer fast growing hierarchy (fα)α<ε0 of functionsf : N→ N is:

Definition1 f0(n) = n + 1.2 For α < ε0, fα+1(n) = f n

α(n), where the superindexindicates that fα is iterated n times.

3 For limit α < ε0, fα(n) = fd(α,n)(n).

For example: f1(n) = 2n, f2(n) = n2n and f3(n) is (significantly)larger than a stack of powers of two of length n; fω(n) = fn(n)grows like Ackermann’s function An(n).

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

The Lob-Wainer fast growing hierarchy (fα)α<ε0 of functionsf : N→ N is:

Definition1 f0(n) = n + 1.2 For α < ε0, fα+1(n) = f n

α(n), where the superindexindicates that fα is iterated n times.

3 For limit α < ε0, fα(n) = fd(α,n)(n).

For example: f1(n) = 2n, f2(n) = n2n and f3(n) is (significantly)larger than a stack of powers of two of length n; fω(n) = fn(n)grows like Ackermann’s function An(n).

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

Peano Arithmetic PA is the formal theory that captures theintuitive notion of “finite mathematics.” If a result is not provablein PA, it must use infinite objects in an essential way.

Theorem (Wainer)1 Each fα is strictly increasing.2 If α < β < ε0 then fα is eventually dominated by fβ.3 Each fα is computable, and provably total in PA.4 Any computable f provably total in PA is eventually

dominated by some fα, α < ε0.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

Peano Arithmetic PA is the formal theory that captures theintuitive notion of “finite mathematics.” If a result is not provablein PA, it must use infinite objects in an essential way.

Theorem (Wainer)1 Each fα is strictly increasing.2 If α < β < ε0 then fα is eventually dominated by fβ.3 Each fα is computable, and provably total in PA.4 Any computable f provably total in PA is eventually

dominated by some fα, α < ε0.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

Peano Arithmetic PA is the formal theory that captures theintuitive notion of “finite mathematics.” If a result is not provablein PA, it must use infinite objects in an essential way.

Theorem (Wainer)1 Each fα is strictly increasing.2 If α < β < ε0 then fα is eventually dominated by fβ.3 Each fα is computable, and provably total in PA.4 Any computable f provably total in PA is eventually

dominated by some fα, α < ε0.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

Peano Arithmetic PA is the formal theory that captures theintuitive notion of “finite mathematics.” If a result is not provablein PA, it must use infinite objects in an essential way.

Theorem (Wainer)1 Each fα is strictly increasing.2 If α < β < ε0 then fα is eventually dominated by fβ.3 Each fα is computable, and provably total in PA.4 Any computable f provably total in PA is eventually

dominated by some fα, α < ε0.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

Peano Arithmetic PA is the formal theory that captures theintuitive notion of “finite mathematics.” If a result is not provablein PA, it must use infinite objects in an essential way.

Theorem (Wainer)1 Each fα is strictly increasing.2 If α < β < ε0 then fα is eventually dominated by fβ.3 Each fα is computable, and provably total in PA.4 Any computable f provably total in PA is eventually

dominated by some fα, α < ε0.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

DefinitionLet Rω

n (m) be the change of base function replacing each n byω in the complete base n representation of m.

For example, since 266 = 33+2 + 32 · 2 + 3 + 2, then

Rω3 (266) = ωω+2 + ω22 + ω + 2.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

DefinitionLet Rω

n (m) be the change of base function replacing each n byω in the complete base n representation of m.

For example, since 266 = 33+2 + 32 · 2 + 3 + 2, then

Rω3 (266) = ωω+2 + ω22 + ω + 2.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

TheoremLet

n = 2m1 + 2m2 + · · ·+ 2mk

where m1 > m2 > · · · > mk . Let αi = Rω2 (mi). Then

G(n) = fα1(fα2(. . . (fαk (3)) . . . ))− 2.

For example, G(266) = fωω+1(fω+1(6))− 2, because266 = 222+1

+ 22+1 + 21 and f1(3) = 6.Similarly,

G(4) = fω(3)− 2 = f3(3)− 2 = 3 · 23 · 23·23 · 23·23·23·23

− 2.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

TheoremLet

n = 2m1 + 2m2 + · · ·+ 2mk

where m1 > m2 > · · · > mk . Let αi = Rω2 (mi). Then

G(n) = fα1(fα2(. . . (fαk (3)) . . . ))− 2.

For example, G(266) = fωω+1(fω+1(6))− 2, because266 = 222+1

+ 22+1 + 21 and f1(3) = 6.Similarly,

G(4) = fω(3)− 2 = f3(3)− 2 = 3 · 23 · 23·23 · 23·23·23·23

− 2.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

TheoremLet

n = 2m1 + 2m2 + · · ·+ 2mk

where m1 > m2 > · · · > mk . Let αi = Rω2 (mi). Then

G(n) = fα1(fα2(. . . (fαk (3)) . . . ))− 2.

For example, G(266) = fωω+1(fω+1(6))− 2, because266 = 222+1

+ 22+1 + 21 and f1(3) = 6.Similarly,

G(4) = fω(3)− 2 = f3(3)− 2 = 3 · 23 · 23·23 · 23·23·23·23

− 2.

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

We obtain at once:

Corollary1 (Goodstein) G is total.2 (Kirby-Paris) Item (1) is not a theorem of PA.

THE END

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

We obtain at once:

Corollary1 (Goodstein) G is total.2 (Kirby-Paris) Item (1) is not a theorem of PA.

THE END

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

We obtain at once:

Corollary1 (Goodstein) G is total.2 (Kirby-Paris) Item (1) is not a theorem of PA.

THE END

Caicedo Very large numbers

IntroductionRegressive Ramsey numbers

Goodstein’s function

Goodstein’s functionThe Kirby-Paris theorem

We obtain at once:

Corollary1 (Goodstein) G is total.2 (Kirby-Paris) Item (1) is not a theorem of PA.

THE END

Caicedo Very large numbers