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The Effort to Reduce Numbers to Sets I




An odd campaign emerged in the 19th century – the effort to reduce numbers to sets. This brings us back, though in a rather different light, to the question with which we opened: what is a number?

The simplest possible kinds of numbers are the counting numbers, those we tick off from one to ten on our fingers. There can be no doubt that they are numbers, they are the paradigm, they are at heart what we mean by “numbers.” Thus, whether anything else is a “number” is, if you will, a question about the family resemblance between the counting numbers and that other sort.

Whatever else is a number: three is a number!

“Is pi a number?” means, “does pi have a close family relationship to three?”

“Is i a number?” means the same.  

But in the 19th century, for the first time, mathematicians and logicians started wondering, not idly but in all workaday seriousness, what it means to call three a number.

They decided that numbers were special sorts of set. The number zero is the null set, sometimes indicated by this notation:∅.  The number one is the set of all sets that contain only the null set. In other words, {∅}.

Spelling that out, the set of nothing is different from the set consisting only of that set;  is different from {∅}. And so we have gone from 0 to 1. We can keep going in regular counting order. Two is the set consisting of both of the above sets. Three is the set consisting of all of the sets thus far named. And so forth for as long as we have patience.  And we have built the natural numbers, a set of these strange formulations, building it out of … nothing.

It is possible to re-define the other sorts of numbers we’ve been discussing using set theoretic conceptions, and a good deal of ingenuity has been expended on this. But it was with his venture into infinity that one of the founders of set theory, Georg Cantor, blew everybody’s mind. [By the way, Cantor’s doctoral adviser was Karl Weierstrass. That’s a name we’ve encountered here before. He’s the fellow who figured out how to re-work calculus without reference to those dubious infinitesimals.]  

 Cantor worked on the concept of the equivalence of sets, or the one-to-one relationship between the items in one set and the items in another.  It seems obvious, for example, that the set of odd positive integers is equivalent to the set of even positive integers. We can line them up in a one-to-one correspondence this way:

2   -   1

4   -   3

6   -  5 

and so forth. We can keep pairing them off forever, without embarrassing themselves or the mathematicians.

What about the set of all odd positive integers’ numbers and the set of all positive integers? This seems different. Surely, we say at first thought, the set of odd positive integers is only one-half as large as the set of all positive integers!
But no, our mind is accustomed to dealing with finite groupings, and here the difference between infinite and finite groupings leads us astray. 
If we understand the equivalence of sets as the possibility of this sort of one-to-one correspondence, we’ll soon have to agree that the odd integers and all integers are equivalent sets. Line them up and see!
1   -   1
2   -   3
3   -   5
4   -   7.
At what point will we run out of odds that we can put on the right side of that column to match the newest integer, odd or even, on the left?  Never. Thus, the two infinite sets are equivalent.
Does this mean that every infinite set is equivalent to every other? No.  In particular, by the same standard the set of all integers is decidedly not equivalent to the set of all points on a line segment: any line segment.
Let’s prove this. Suppose we’re discussing a line segment one unit long. It doesn’t matter of course what the unit is: a foot, a meter, a mile. We will represent the point on the leftward most end of this segment as 0, the midpoint as 0.5, and the rightward most point as 1.
Now what? Can we establish a one-to-one relationship between the infinity of integers and the infinity of these points, expressed in decimal form? It seems unlikely. After all, whatever point we associate with the number 1, and whatever point we associate with the number 2, we will have missed an infinite number of other points in between them.
But that isn’t a proof. Cantor offered a rigorous and elegant proof on this point.  That will hold until tomorrow.

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