How We Got From There To Here: A Story of Real Analysis

Interlocuters:Salviati, Sagredo, and Simplicio; Three Friends of Galileo Galilei

Setting: Three friends meet in a garden for lunch in Renassaince Italy. Prior to their meal they discuss the book How We Got From There to Here: A Story of Real Analysis. How they obtained a copy is not clear.

Salviati: My good sirs. I have read this very strange volume as I hope you have?

Sagredo: I have and I also found it very strange.

Simplicio: Very strange indeed; at once silly and mystifying.

Salviati: Silly? How so?

Simplicio: These authors begin their tome with the question, “What is a number?” This is an unusually silly question, don’t you think? Numbers are numbers. Everyone knows what they are.

Sagredo: I thought so as well until I reached the last chapter. But now I am not so certain. What about this quantity \(\aleph_0?\) If this counts the positive integers, isn’t it a number? If not, then how can it count anything? If so, then what number is it? These questions plague me ’til I scarcely believe I know anything anymore.

Simplicio: Of course \(\aleph_0\) is not a number! It is simply a new name for the infinite, and infinity is not a number.

Sagredo: But isn’t \(\aleph_0\) the cardinality of the set of natural numbers, \(\NN\text{,}\) in just the same way that the cardinality of the set \(S=\left\{Salviati, Sagredo, Simplicio\right\}\) is \(3?\) If \(3\) is a number, then why isn’t \(\aleph_0?\)

Simplicio: Ah, my friend, like our authors you are simply playing with words. You count the elements in the set \(S=\left\{Salviati, Sagredo, Simplicio\right\};\) you see plainly that the number of elements it contains is \(3\) and then you change your language. Rather than saying that the number of elements in \(S\) is \(3\) you say that the cardinality is \(3\text{.}\) But clearly “cardinality” and “number of elements” mean the same thing.

Similarly you use the symbol \(\NN\) to denote the set of positive integers. With your new word and symbol you make the statement “the cardinality (number of elements) of \(\NN\) is \(\aleph_0\text{.}\)” This statement has the same grammatical form as the statement “the number of elements (cardinality) of \(S\) is three.” Since three is a number you conclude that \(\aleph_0\) is also a number.

But this is simply nonsense dressed up to sound sensible. If we unwind our notation and language, your statement is simply, “The number of positive integers is infinite.” This is obviously nonsense because infinity is not a number.

Even if we take infinity as an undefined term and try to define it by your statement this is still nonsense since you are using the word “number” to define a new “number” called infinity. This definition is circular. Thus it is no definition at all. It is nonsense.

Salviati: Your reasoning on this certainly seems sound.

Simplicio: Thank you.

Salviati: However, there are a couple of small points I would like to examine more closely if you will indulge me?

Simplicio: Of course. What troubles you?

Salviati: You’ve said that we cannot use the word “number” to define numbers because this would be circular reasoning. I entirely agree, but I am not sure this is what our authors are doing.

Consider the set \(\left\{1, 2, 3\right\}\text{.}\) Do you agree that it contains three elements?

Simplicio: Obviously.

Sagredo: Ah! I see your point! That there are three elements does not depend on what those elements are. Any set with three elements has three elements regardless of the nature of the elements. Thus saying that the set \(\left\{1, 2, 3\right\}\) contains three elements does not define the word “number” in a circular manner because it is irrelevant that the number 3 is one of the elements of the set. Thus to say that three is the cardinality of the set \(\left\{1, 2, 3\right\}\) has the same meaning as saying that there are three elements in the set \(\left\{Salviati, Sagredo, Simplicio\right\}\text{.}\) In both cases the number “\(3\)” is the name that we give to the totality of the elements of each set.

Salviati: Precisely. In exactly the same way \(\aleph_0\) is the symbol we use to denote the totality of the set of positive integers.

Thus \(\aleph_0\) is a number in the same sense that ’\(3\)’ is a number, is it not?

Simplicio: I see that we can say in a meaningful way that three is the cardinality of any set with . . . well, . . . with three elements (it becomes very difficult to talk about these things) but this is simply a tautology! It is a way of saying that a set which has three elements has three elements!

This means only that we have counted them and we had to stop at three. In order to do this we must have numbers first. Which, of course, we do. As I said, everyone knows what numbers are.

Sagredo: I must confess, my friend, that I become more confused as we speak. I am no longer certain that I really know what a number is. Since you seem to have retained your certainty can you clear this up for me? Can you tell me what a number is?

Simplicio: Certainly. A number is what we have just been discussing. It is what you have when you stop counting. For example, three is the totality (to use your phrase) of the elements of the sets \(\left\{Salviati, Sagredo, Simplicio\right\}\) or \(\left\{1, 2, 3\right\}\) because when I count the elements in either set I have to stop at three. Nothing less, nothing more. Thus three is a number.

Salviati: But this definition only confuses me! Surely you will allow that fractions are numbers? What is counted when we end with, say \(4/5\) or \(1/5?\)

Simplicio: This is simplicity itself. \(4/5\) is the number we get when we have divided something into \(5\) equal pieces and we have counted four of these fifths. This is four-fifths. You see? Even the language we use naturally bends itself to our purpose.

Salviati: But what of one-fifth? In order to count one fifth we must first divide something into fifths. To do this we must know what one-fifth is, musn’t we? We seem to be using the word “number” to define itself again. Have we not come full circle and gotten nowhere?

Simplicio: I confess this had not occurred to me before. But your objection is easily answered. To count one-fifth we simply divide our “something” into tenths. Then we count two of them. Since two-tenths is the same as one-fifth the problem is solved. Do you see?

Sagredo: I see your point but it will not suffice at all! It merely replaces the question, “What is one-fifth?” with, “What is one-tenth?” Nor will it do to say that one-tenth is merely two-twentieths. This simply shifts the question back another level.

Archimedes said, “Give me a place to stand and a lever long enough and I will move the earth.” But of course he never moved the earth because he had nowhere to stand. We seem to find ourselves in Archimedes’ predicament: We have no place to stand.

Simplicio: I confess I don’t see a way to answer this right now. However I’m sure an answer can be found if we only think hard enough. In the meantime I cannot accept that \(\aleph_0\) is a number. It is, as I said before, infinity and infinity is not a number! We may as well believe in fairies and leprechauns if we call infinity a number.

Sagredo: But again we’ve come full circle. We cannot say definitively that \(\aleph_0\) is or is not a number until we can state with confidence what a number is. And even if we could find solid ground on which to solve the problem of fractions, what of \(\sqrt{2}?\) Or \(\pi?\) Certainly these are numbers but I see no way to count to either of them.

Simplicio: Alas! I am beset by demons! I am bewitched! I no longer believe what I know to be true!

Salviati: Perhaps things are not quite as bad as that. Let us consider further. You said earlier that we all know what numbers are, and I agree. But perhaps your statement needs to be more precisely formulated. Suppose we say instead that we all know what numbers need to be? Or that we know what we want numbers to be?

Even if we cannot say with certainly what numbers are surely we can say what we want and need for them to be. Do you agree?

Sagredo: I do.

Simplicio: And so do I.

Salviati: Then let us invent numbers anew, as if we’ve never seen them before, always keeping in mind those properties we need for numbers to have. If we take this as a starting point then the question we need to address is, “What do we need numbers to be?”

Sagredo: This is obvious! We need to be able to add them and we need to be able to multiply them together, and the result should also be a number.

Simplicio: And subtract and divide too, of course.

Sagredo: I am not so sure we actually need these. Could we not define “subtract two from three” to be “add negative two to three” and thus dispense with subtraction and division?

Simplicio: I suppose we can but I see no advantage in doing so. Why not simply have subtraction and division as we’ve always known them?

Sagredo: The advantage is parsimony. Two arithmetic operations are easier to keep track of than four. I suggest we go forward with only addition and multiplication for now. If we find we need subtraction or division we can consider them later.

Simplicio: Yes! We have it! Any objects which combine in these precise ways can be called numbers.

Salviati: Certainly these properties are necessary, but I don’t think they are yet sufficient to our purpose. For example, the number \(1\) is unique in that it is the only number which, when multiplying another number leaves it unchanged.

For example: \(1\cdot3=3\text{.}\) Or, in general, if \(x\) is a number then \(1\cdot x =x\text{.}\)

Sagredo: Yes. Indeed. It occurs to me that the number zero plays a similar role for addition: \(0+x=x\text{.}\)

Salviati: It does not seem to me that addition and multiplication, as we have defined them, force \(1\) or \(0\) into existence so I believe we will have to postulate their existence independently.

Sagredo: Is this everything then? Is this all we require of numbers?

Simplicio: I don’t think we are quite done yet. How shall we get division?

Sagredo: In the same way that we defined subtraction to be the addition of a negative number, can we not define division to be multiplication by a reciprocal? For example, \(3\) divided by \(2\) can be considered \(3\) multiplied by \(1/2\text{,}\) can it not?

Salviati: I think it can. But observe that every number will need to have a corresponding negative so that we can subtract any amount. And again nothing we’ve discussed so far forces these negative numbers into existence so we will have to postulate their existence separately.

Simplicio: And in the same way every number will need a reciprocal so that we can divide by any amount.

Sagredo: Every number that is, except zero.

Simplicio: Yes, this is true. Strange is it not, that of them all only this one number needs no reciprocal? Shall we also postulate that zero has no reciprocal?

Salviati: I don’t see why we should. Possibly \(\aleph_0\) is the reciprocal of zero. Or possibly not. But I see no need to concern ourselves with things we do not need.

Simplicio: Is this everything then? Have we discovered all that we need for numbers to be?

Salviati: I believe there is only one property missing. We have postulated addition and we have postulated multiplication and we have described the numbers zero and one which play similar roles for addition and multiplication respectively. But we have not described how addition and multiplication work together.

With this in place I believe we have everything we need.

Simplicio: Indeed. We can also see from this that \(\aleph_0\) cannot be a number since, in the first place, it cannot be added to another number and in the second, even if it could be added to a number the result is surely not also a number.

Salviati: My dear Simplicio, I fear you have missed the point entirely! Our axioms do not declare what a number is, only how it behaves with respect to addition and multiplication with other numbers. Thus it is a mistake to presume that “numbers” are only those objects that we have always believed them to be. In fact, it now occurs to me that “addition” and “multiplication” also needn’t be seen as the operations we have always believed them to be.

I submit that our set along with these definitions satisfy all of our axioms and thus \(a\text{,}\) \(b\) and \(c\) qualify to be called “numbers.”

Simplicio: This cannot be! There is no zero, no one!

Sagredo: But there is. Do you not see that \(a\) plays the role of zero — if you add it to any number you get that number back. Similarly \(b\) plays the role of one.

Simplicio: This is astonishing! If \(a, b\) and \(c\) can be numbers then I am less sure than ever that I know what numbers are! Why, if we replace \(a, b\text{,}\) and \(c\) with Simplicio Sagredo, and Salviati then we become numbers ourselves!

Salviati: Perhaps we will have to be content with knowing how numbers behave rather than knowing what they are.

However I confess that I have a certain affection for the numbers I grew up with. Let us call those the “real” numbers. Any other set of numbers, such as our \(\left\{a,b,c\right\}\) above we will call a field of numbers, since they seem to provide us with new ground to explore. Or perhaps just a number field?

As we have been discussing this I have been writing down our axioms. They are these. Numbers are any objects which satisfy all of the following properties:

Sagredo: My friend, this is a thing of surpassing beauty! All seems clear to me now. Numbers are any group of objects which satisfy our axioms. That is, a number is anything that acts like a number.

Salviati: Yes this seems to be true.

Simplicio: But wait! We have not settled the question: Is \(\aleph_0\) a number or not?

Salviati: If everything we have just done is valid then \(\aleph_0\) could be a number. And so could \(\aleph_1\text{,}\)\(\aleph_2 \ldots\) if we can find a way to define addition and multiplication on the set \(\left\{\aleph_0, \aleph_1, \aleph_2 \ldots\right\}\) in a manner that agrees with our axioms.

Sagredo: An arithmetic of infinities! This is a very strange idea. Can such a thing be made sensible?

Simplicio: Not, I think, before lunch. Shall we retire to our meal?