by Louis H. Kauffman

Mathematics Department
University of Illinois at Chicago
851 South Morgan Street
Chicago, Illinois 60607-7045

In 1961 Warren McCulloch wrote an essay entitled "What is a Number, that a Man May Know It, and a Man that He May Know a Number?" [M] . McCulloch's essay indicates a birthdate for second order cybernetics that is somewhat earlier than what is usually assumed. In fact, on careful reading, it indicates that second order cybernetics is neccessarily born at once with the concept of cybernetics and that there is no such thing as a separate field of second order cybernetics. Such is the power of arithmetic, or rather, such is the power of the non-numerical basis of numerical arithmetic!

It is through the non-numerical mathematics of the observer - the mathematics of form, reference, self-reference and logic, that arithmetic emerges into the world, and that world learns to count and do science, even to do first order cybernetics. This paper will discuss the foundations of arithmetic in the non-numerical worlds of perception, action and distinction. This paper will discuss the construction of the peculiar smokescreen that is taken to be an eternal world of distinct forms, a world that we have made, a world that we are making and can never be completed.

I take up the theme of number in three sections. Each section orbits the source at a somewhat different distance from the center. The common theme is the relation of simple recursion, awareness of awareness and the apparent generation of multiplicity. Section 2 is a non-technical discussion in the light of Russell's definition of number and Russell's paradox of the set of all sets that are not members of themselves. Section 3 is a constructive account of numbers from the empty set. Section 4 is a brief direct discussion of number and self-reference. Section 5, in the form of a dialogue, shows how Cantor's transfinite numbers become a metalanguage for the the finite numbers, a field of self-reference where the metalanguage and the language are one.

What is a number?

Language betrays us even as we speak. For the existence of a number is unlike any existence that we call physical or mental. To obtain the isness of number you need to call up the Platonic realm of eternal forms. Where is the number of the Platonic realm in the cosmic telephone directory? You cannot even get a call through to the Platonic realm without the existence of number and you cannot get the existence of number without the Platonic realm.

But I can count! I do count! So you say. Everybody counts, but who knows the nature of number. Well we all do know it. It?

How can a number exist?

This

consists in three stars. But three stars is not the number three. Where is the number three? Could it be that the number three is not a thing?

Lets try to be quite down to earth about this. We use the concept of the number three with some ease. It is easy to agree that concepts are not things in the sense of physical objects. In fact concepts often have a precision that is never found in the physical. But numbers are peculiar and central concepts indeed, and we need to get a look at them from a new angle.

Lets begin by recalling the famous and remarkable definition of number due to Bertrand Russell: " A number is the class of all classes similar to a given class." Here the numbers under discussion are zero and the positive integers. In Russell's definition, a concept (number) has been identified with a class. This Russell class is a set of classes and each class in this set has the same "number" of members (that is what similarity of classes means). Now we see that underlying Russell's definition is the notion of when two classes "have the same number" of members. And this notion of having the same number of members (called similarity) does not depend upon the definition of or the existence of any given number. Similarity of classes is utterly non-numerical.

Let's recall the meaning of similarity: Two classes are said to be similar if there is a one-to-one correspondence between the members of the one class and the members of the other class. That is, A and B are similar if and only if there is a way to match the members of A with the members of B so that each member of A is paired with a unique member of B and each member of B is paired with a unique member of A. Similarity can be checked without the ability to count. An illiterate shepherd can obtain the similarity between the set of notches on a stick and the sheep in his flock by herding them past and marking the stick.

Similarity is fundamental and non-numerical. You do not have to be able to count to see that {*#@} and {!&^} are both instances of three. But you do have to be able to read out ! as one and not two. How do you do that? How do you know that an exclamation point, with its two parts, represents a single form and is taken as a single instance of that form? For that matter, how do you know that the form {*#@} is not five rather than three? Certainly, its parts can be put into one to one correspondence with the row of letters ABCDE.

How do you know to take a given form as foreground against which to make the comparisons? How does the shepherd recognize his sheep and how does he know where and when to notch the stick?

How does the shepherd know when to start working the stick and when to stop? How does he know when the sheep are starting through the gate? How does he know when the field is empty and all the sheep have passed?

None of these questions can be addressed by a formal system alone. The ability perform the relations indicated by these questions is a prerequisite to being able to make any formal system at all. The non-numerical capabilities that we are wondering about here can all be summarized in the injunction: Make a distinction!

It is not the intent of this essay to worry about the question - how do distinctions get made? In fact, we had best worry first whether there are any such "things" as distinctions in the first place. Again language gets in the way, for a distinction is not a thing. A distinction is a process or relationship between an observer and his/her world. A distinction arises in the condition of observation. The condition of observation is itself a distinction of a self and a world observed by that self. To the extent that the self understands that the world observed is not distinct from the self, the world and self cohere into one whole, but the full coherence cannot be seen by any individuated self. "It" cannot be seen because "it", the coherence, is not observable directly. The act of observation puts the coherence into the blind spot of the eye of the beholder.

A system that observes itself cannot see all of itself, for (paraphrasing Spencer-Brown [SB]) in order to see at all, it must evidently divide itself into a part that sees and a part that is seen. The part that is seen will not be all. Perhaps the "system" has the capability of absorbing the distinction of seer and seen into a whole and moving to a new level of observation. In this new level the fundamental limitations will hold once again, but a wider vision has been obtained.

Imagine an observing system so subtle as to always be able, in the presence of any distinction to recognise the process of distinction as such and to rise to the level of comment on the condition of the distinction itself. This subtle observing system may, from the point of an outside observer, make many distinctions of which it is unaware. (I am unaware of the direct activity of the neurons in my brain.) But whenever our observing system recognises distinction, it also sees a unity and can look again at the two sides and their join.

I am aware of my distinction of self and world and that they are intertwined. I can indicate the unity of self and world in language. I can make maps and models of the unity of self and world. This unity will never fully come into view. Every attempt to map out the unity of self and world becomes a map of unknown territory that is enfolded once again into the fundamental ground of the original self/world distinction.

Language acts to simplify and speaks of the original self/world distinction, as though there were only one such and that it existed in eternity. Language must be respected for its invocation of ideals, but is this not an illusion generated by the very words themselves? The words would have us imagine that they are independent of their temporal contexts. They would have us believe in the possibility of immutable forms. They rest their case on the stability of typography and the repeatability of sounds and thoughts. They rest their case on our ability to distinguish them. We are able to distinguish langauge through our ability to distinguish anything at all. Language and world live in a circular round. The self does not manage to lift out of that round and look upon it from above. The self occurs only in relation to a given world through a given language.

What then of the Russell definition of number? The set of all classes similar to a given class is the number of that class. To the extent that the observer can distinguish and match the elements of a given class with a reference class, he/she can determine the numerosity of the given class. The Russell definition assumes the ability to compare and distinguish, assumes the existence of all the possible classes similar to the given class, assumes all this and achieves logical closure. Is this closure not an image of our confidence in our own ability to count? We arrive at a concept of number though our ability to compare and distinguish. We can compare and we know it. The observing system has become aware of itself.

Russell circumvents talk of self and reference. If one can speak of all sets similar to a given set, if one speak of all and give to it eternity, then why not speak of all sets? And if we speak of all sets why not speak of all sets that are not members of themselves. And so Russell did speak - of all sets not members of themselves. And this act of speech brought forth an endless creation of new sets, not to be allowed in the static eternity. And it was forbidden.

It was forbidden. It could not be avoided. The observing system looked at any construction from outside and Remade it anew. The building was never completed. Eternity did not submit to construction. There was no exit except in silence.

A set is a special sort of distinction. Sets are concepts held by an observing system. The empty set { } is the bare concept of collecting.

Two sets are identical if they have the same members.

The identity of a set is derived solely from its members. Any set is identical to itself, and there is only one empty set (See the proof below).

(Perhaps you do not like the idea that A=A. After all A may have changed by the time you reached the second version. But in set theory it is assumed that the members of a set do not change. Since nothing in the observed world has such a property, we see that set theory is not a perfect model of the temporal world. Well, nothing is perfect.)

By defining identity in terms of membership, set theory makes a notion of identity possible in a shifting world of representations. When we say A=B, for sets A and B, it means that they share the property of having the same members. There is no difference between two sets that have the same members.

Theorem Zero. There is only one empty set.

Proof. Suppose that S and T are both empty sets. Then S and T have exactly the same members, namely none. Therefore S=T. Hence the empty set is unique. //

Theorem One. The empty set exists.

Proof. Suppose that there is no empty set. Let S denote the collection of all sets that have no members. Then S is neccessarily empty. But we said that empty sets do not exist! S is empty. This is a contradiction. Therefore the empty set exists.//

The uniqueness of the empty set heralds the uniqueness of its successors. There is only one set whose sole member is the empty set: { { } }. There is only one set whose members are the empty set and the set whose member is the empty set: { { } { { } } }.

(I use {{A} {B} {C}} as notation for the set whose members are {A}, {B} and {C}. The locations of outermost pairs of closed brackets delineates the members of a set. No commas are required in this notation.)

Let the successor A' of a set A consist in the set whose members are the members of A plus a new member consisting in A itself.

Thus

{ }' = { { } }

{ { } }' = { { } { { } } }

{ { } { { } } }' = { { } { { } } { { } { { } } } }.

If E denotes the empty set, then the successor sequence

provides a unique sequence of sets each having one more member than the previous set. No two of these sets are equal to one another (because they have different members). Each set in this successor sequence becomes a reference set for the corresponding number. Thus the empty set is a reference for 0, E' is a reference for 1, E'' is a reference for 2 and so on. The successors of the empty set provide a specific construction of multiplicities and thereby bring the numbers into mathematical existence.

In the modern version of Russell's definition, two sets A and B are said to have the same cardinality if there is a one-to-one correspondence between A and B. 0 is the cardinality of E. 1 is the cardinality of E'. 2 is the cardinality of E''. If N is the cardinality of X , then N+1 is the cardinality of X' ( as long as X is not a member of itself).

(Notice how this classical point of view wants to avoid self membership at every turn.)

No longer do we say that a number "is" the collection of all sets of a given cardinality. In fact, a number is not a noun in modern mathematics, it is a relationship. We have returned to the position of the shepherd and the notched stick. The notched stick is now the successor sequence of the empty set. It is a stick that all practicioners of set theoretic shepherding have agreed upon.

Do you have three sheep? Well let me see. Three according to my Von Neumann calculator (John Von Neumann discovered the successor sequence of the empty set.) is the cardinality of { { } { { } } { { } { { } } } }. As your sheep pass the gate I shall erase successive members of my reference set. Now I erase { { } { { } } } , leaving { { } { { } } }. Another sheep passes by and I erase { { } }, leaving { { } }. One more sheep goes through the gate and I erase { } leaving { }. This last set is empty and there are no more sheep. Indeed, you have exactly three sheep.

Three is an adjective, not a noun.

But wait! Lets look at the last erasure operation you did in that sheep count. You had the set { { } }. You erased the member { } and obtained { }. You erased the empty set as a member of { { } } and you obtained as a result the empty set { }. You erased the empty set and were immediately handed the empty set! If there is only one empty set and I erase it, how can that result in the creation of the empty set?

The standard answer: E={ } is the empty set and E is distinct from E' = { { } } = { E }. If I erase E from E' it does not mean that I erase E. It means that I delete E from membership in E'. This changes E' to E. We did not erase E, we changed E'.

But in order to handle the standard answer you have to be an observer who is capable of looking both inside and outside of E'. You have to be aware of E' as a system with a boundary. You have to understand that when the present instance of E disappears it does not mean that E has gone. You have to be an observer that can reconstruct and recognise the empty set.

It would be a bit irreverent to say that numbers arise at that level of consciousness that is just beyond peek-a-boo. Just beyond that place where things exist only in the eye of the observer. Out of sight out of mind.

How is this trick established? It is established through our second theorem. If there is no empty set in sight, the observer immediately finds one in the act of set formation. This is the pure act of distinction. It is beyond peek-a-boo. The observer has become aware not only of awareness, but also of awareness of awareness. This is awareness of awareness. That is the prerequisite for number.

The counting process is self-referential. It uses its production to produce itself. Counting is as simple as one, two three

or | . ||. |||

or { {} }, { {} { {} } }, { {} { {} } { {} { {} } } }

In the case of stick counting we see the essential self-reference as soon as we get a row too large to count.

If R = |||||||..., then R = | R.

In the successor sequence described in the last section X = X' means that X is a member of itself.

Time arises by taking circularity and unfolding it into sequence. It is no wonder that the construction of number is fraught with the avoidance of self-reference. Nevertheless it is time (time!) to see that what is based on awareness of awareness need not avoid awareness.

Numbers do not exist eternally. We have made them. The universe brings forth number just as It becomes aware of itself.

The learned professors Jeremy and Lou decided to discuss the nature of arithmetic. The motivation for this dialogue was a sleepless night spent trying to prove that A+B = B+A for any natural numbers A and B. Lou had decided to represent a number as a row of stars. The number 1 is * ,2 is ** and 3 is *** in Lou's world [K5]. In this world, A+B is denoted by AB, the left-to-right juxtaposition of the strings for A and for B. Lou had decided to assume nothing, start at the very beginning and derive the well-known properties of numbers from his rows of stars. And he had difficulty proving that A+B=B+A. This looks obvious, and yet it baffled him for a long time. Then it dawned on him that he could begin with AB and slowly move B around A until it was on the other side, transforming AB into BA.

This proof that AB = BA was perfectly convincing to Lou. In fact, he realized that he had deliberately forgotten a key concept about numbers during his attempt to prove AB = BA. This concept was the notion that a number is a number of some specific collection, and that the number assigned to this collection does not change when the collection is moved about.

Then Burt came along and said "Why don't you turn the row AB around by 180 degrees? Surely the number of stars does not change while you rotate the row, and just as surely, the row AB will become the row BA under this turning!" {Burt Voorhees at the June 1995, Brussels Conference -"Einstein Meets Magritte".}

Burt's solution was startling to Lou even though it had essentially the same structure as his own proof. In Burt's viewpoint it was possible to let the "number" AB retain its form and yet be transformed into BA by a rotation that reversed the order of its parts. This was rather like turning the number inside out! In both cases the key concept of a number as an invariant assigned to a collection or a form was the vantage point from which the proof became comprehensible.

Jeremy and Lou continued their discussion in the direction of recursions and infinity.

Lou. Consider a theory of natural numbers (1,2,3,...) that begins by representing each number by a row of stars. Thus

1=*

2=**

3=***

4=****

and of course

N+1 = N*.

In this system, we assume that every number N has a successor N* and we define the sum of two star rows N and M by their juxtaposition: N+M = NM. Thus 2+2 = **** = 4.

Jeremy. But what is the meaning of equality in your system? It would beg the question to say that two rows are equal if and only if they have the same number of stars.

Lou. Indeed, you are right . What we must do is give a procedure whereby the equality of rows can be determined. Let us assume that it is possible to recognise if a row is empty. Given a non-empty row N, let N-1 be the row obtained by removing a star from the left of N. Thus *** -1 = **. Let N-k denote the result of subtracting 1 successively from N, k times.

Let 0 denote the empty row. There is only one empty row.

We shall say that N=M if N-k = M-k =0 for some k. This definition does not require us to count, it only requires that we make successive subtractions from N and from M and that we find they both reach zero after the same number of subtractions. For example, if we wish to determine whether

we apply successive subtraction from each row:

***** ?=? ****

**** ?=? ***

*** ?=? **

** ?=? *

* ?=?

and conclude that the original numbers are not equal. Note that it is not neccessary to have a name for a number or to know how to count up to this number to apply the criterion of equality.

Jeremy. Well, I see that you do have a notion of equality, but what is this business about recognising an empty row? An empty row is an empty row. Since the only distinguishing characteristic of a row is its members, I grant you that there is only one empty row. And somehow, all the distinctions among the numbers arise from this ability to recognise emptiness!

Lou. Jeremy, you have put your finger on it! A number is not just its stars. It is a row of stars and even the empty row is a kind of number (called zero in these late days). The empty row is far from empty of concept, it just has no contents! In fact, the empty row embodies the very concept of number and so it becomes our reference point in the act of counting. The idea of a row (even an empty one) is a piece of virtual logic in our construction of number.

Jeremy. Well, if you can make such a fuss about zero, I shall counter with infinity! Consider an "infinite" number w = *****... . An endless number of stars appear at the right of w. Hence w-1 = w and w-k is never zero for any finite k. On the other hand

w+1 = w* = *****...*

is obviously different from w. You will never be able to prove this difference by your method since w+1 is also inexhaustible as far as removing stars from the left is concerned. I say that you should call two "numbers" equal if you can match them up structurally. From that point of view *****... and *****...* and *****...** are all different even though you cannot take any of them to zero in a finite number of steps! In fact, I challenge you to consider the following beginnings in a vast sea of infinite numbers:

w= *****...

w+1 = [*****...]*

w+2 = [*****...]**

w+3 = [*****...]***

...

2w = w+w = [*****...][*****...]

3w = [*****...][*****...][*****...]

... w2 = [*****...][*****...][*****...][*****...] ...

w2 + w +1 = ([*****...][*****...][*****...][*****... ] ... )[*****...]*

The pictures begin to fail me, but it is clear that once you allow that first infinite number w, and the idea of infinite repetition that underlies it, then we can make a great medley of infinite numbers heading on out beyond all the ordinary numbers of our experience.

Lou. The infinite numbers you are describing are called ordinals. They are the discovery of Georg Cantor, the famous proponent of infinity from the last century. I am glad we are fully agreed on the commutativity of the finite ordinals because that commutativity can no longer hold for the infinite ordinals. For example,

1+w = *[*****...] = ******... = w

while w+1 = [*****...]*, and this is not equal to w.

Jeremy. Our old arguments about commutativity do not apply to these structures. The order in an ordinal is important. Even w herself cannot be turned around by 180 degrees as I was so fond of doing with the finite numbers. We have decided that the indeterminate part of the infinity continues off to the right in these representations. Commutativity goes out the window, but we get the self embedding or self-reference of the equation *w=w coupled with the fact that we can say of any two ordinals whether one is larger or smaller than the other!

Lou. How do you do that?

Jeremy. It is very easy. All ordinals are constructed from previous ordinals. If you truncate an ordinal, taking everything from the left only up to a certain point, then you get another ordinal. This ordinal is called a truncate of the first ordinal.
I shall write X=T(Y) to indicate that X is a truncate of Y.
Thus *** = T(****) and *** = T(***...) and www...=T(www...w*).
I say that X

Lou. We can speak of products of ordinals. If X#Y denotes the product of X and Y, then X#Y is obtained by replacing every * in Y by a copy of X. Thus X#2 = X#** = XX. I cannot specify 2#Y until you tell me the structure of Y. Lets do an example w#2= w#[**] = [***...][***...] 2#w = **#[***...] = [(**)(**)(**) ...] = ******... = w. Multiplication of ordinals is not commutative either!

Jeremy. I have always disliked the notations that send exponents running up the page along incomprehensible diagonals. Do you mind if we use A{B} to denote A raised to the B-th power? Then A{B{C}} denotes A raised to the (B raised to the C) power and A{B}{C} denotes (A raised to the B) raised to the C power.

We can go ahead and construct powers. w{2} = w#w= www...=(**...)(**...)(**...)... w{3} = w#(w#w) =(ww...)(ww...)(ww...)... =([(**...)(**...)...][(**...)(**...)...]...)([(**...)(**...)...][(**...)(**...)...]...) .... I am tempted to find w{w}. What does this look like?

Lou. Well, it is clear that for ordinals we should have the rule a{(b+c)} = a{b} #a{c}, so w{w} = w{1+w} = w{1}#w{w } = w#w{w}. w{w} is a solution of the equation X = w#X.

Jeremy. Yes, but what does w{w} look like?! It is the limit of the sequence

*

**...

[**...][**...]...

[[**...][**...]...][[**...][**...]...]...

[[[**...][**...]...][[**...][**...]...]...][[[**...][**...]...][[**...][**...]...]...]...

...

s

s' = s#w=ss...

...

The parentheses are present to indicate the self-embeddings and repetitions that are present at each stage of the recursion. If s denotes the sequence at a given stage, then s'= s#w = ss... is the next stage. Each stage in the series is a truncate of the next stage and so ww is simply the union of all the stages in this infinite series. Indeed, this limit will satisfy w#(w{w}) = w{w} and so we get a specific model for this infinite product.

Lou. Now you have convinced me that we can go on building these ordinals forever. Are they of any real use in thinking about numbers?

Jeremy. As a matter of fact they are useful, but we need a few more facts about ordinals. Did you know that there are no infinite descending sequences of ordinals?

Lou. Lets see, I'll try to make an infinite descending sequence. Lets start with w{w}. Now I must choose something smaller than w{w}. I guess I could take w{137}. This is a big jump down, but ordinary integers are the only ordinals strictly less than w. Now I have to choose something less than w{137} and of course I could take w{136}, w{135}, w{134},..., w{3},w{2},w, but this is a finite sequence. Then I need to take an ordinal less than w and this causes another big jump down to say, 13. And so I get the descending sequence w{w}, w{137},w{136},...,w{2},w,13,12,11,...,1. It is finite, even though I did my best to create an infinite descending sequence. The jumps down are too big!

Jeremy. You can try higher exponentiations. We can consider large ordinals such as L = w{w{w{w{...}}}}. L is exponentially self-referential with L = w{L}. I could try to make a descending sequence starting with L and I will again find that it will stop after a finite number of steps.

Lou. You are giving me enough time to puzzle this thing out. Any ordinal is either obtained from a previous ordinal by adding 1, or it is the limit of a sequence of ordinals as we have seen in our examples. An ordinal less than a limit ordinal will be a big jump down into the limit sequence with only a finite number of members of the sequence below it. It is clear that the finite descent property is inherited in the process of adding one or taking limits. So all ordinals have the finite descent property.

Jeremy. Good. Now I can show you a use for ordinals. We can use them to analyze the behaviour of certain sequences of natural numbers. First recall how a number can be written uniquely in a given base. For example 87 in base 3 is

Here I am writing the number as a sum of powers of B with each power of B multiplied by one of the numbers 0,1,..., (B-1) where B is the base (B is 3 in this example). We are using . to denote multiplication and X{Y} to denote exponentiation just as we did with ordinals a moment ago. In such an expression, we get exponents in the powers of 3 and these may themselves be expressed in base 3. For example, we can re-write the 4 in 3{4} and get 3{4} = 3{3{1}.1 + 3{0}.1} giving

Every number that occurs in this expression for 87 is expressed in base 3. We say that 3{3{1}.1 + 1}+3{1}.2 is the complete base 3 expression for 87. <\p>

I am going to define a method for producing sequences of natural numbers. These are called Goodstein sequences [LN]. Take a given number N and base B. Let B[N] be the complete base B expression for N. Let B[N]C be the result of replacing all instances of B in B[N] by C. For example, 3[87] = 3{3{1}.1 + 1}+3{1}.2, 3[87]11 = 11{11{1}.1 + 1}+11{1}.2. As long as C is greater than B, we have that B[N]C will be a number written in complete base C. To define the Goodstein sequence, start with B and N and let N' = (B+1)[ B[N](B+1) -1 ].

N' is the next number in the sequence and (B+1) is the next base. Each time we have a number fully expressed in a given base B, we shift it up to the next base (B+1), subtract 1 and re-express this number in base (B+1). A Goodstein sequence is the potentially infinite sequence of numbers obtained by this process from any starting point (N,B).

Lou. These sequences tend to get very large quite rapidly.

3{3} + 3{2}

4{4} + 4{2} -1 = 4{4} + 4{1}.3 + 4{0}.3

5{5} + 5{1}.3 + 5{0}.2

6{6} + 6{1}.3 + 6{0}.1

7{7} + 7{1}.3

8{8} + 8{1}.2 + 8{0}.7

9{9} + 9{1}.2 + 9{0}.6

10{10} + 10{1}.2 + 10{0}.5

In this example I started with 36 = 27 + 9 = 3{3} +3{2} in complete base three. By the time the sequence has gotten to the base 10, the value has skyrocketed to a number larger than 10 to the 10th power. This is astronomical, and only the beginning of the rise.

Jeremy. Yes they do get big, but the surprising result is

Theorem (See [LN]). Every Goodstein sequence terminates at zero in a finite number of steps.

Proof. With the help of the ordinals the proof is very simple. Map each element N (base B) of the Goodstein sequence to the ordinal obtained by replacing each occurrence of B in N (written in complete base B) by the ordinal w. Call this ordinal O(N). Then it is easy to check that O(N') < O(N) as ordinals. Hence if the Goodstein sequence is N, N', N'', N''', ... then we have descending sequence of ordinals O(N) > O(N') > O(N'') > ... . Since there are no infinite descending sequences of ordinals, this descent must terminate. The theorem follows at once from this termination. Q.E.D.

Lou. Can't you prove this Theorem without invoking the ordinals?

Jeremy. As a matter of fact I do need a structure like the ordinals to prove the Theorem. It can be shown [LN] that there is no proof that the Goodstein sequences terminate that is just based on the classical Peano axioms for the natural numbers. One needs key concepts related to higher induction such as our results about the ordinals. There may be other proofs, but the ordinals provide the most beautiful ground for understanding the Goodstein sequence. We see from the proof that ordinals are not just ways to consider infinity. They in fact are a kind of algebra that reflects just the right properties of numbers to see into the problem about the Goodstein sequence. The proof is logical, but it depends upon the concept and logic of the ordinals in order to attain our comprehension.

Lou. The ordinals are a language that talks about natural numbers without getting involved in all their particularities. Since the ordinals are in fact an extension of the natural numbers this is a new arithmetic that speaks about the old arithmetic. Language and meta-language are entwined.

Jeremy. In the whole discourse, the language is the meta-language.

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