Photo by Arnold L. Hirsch
On December 15 I went to a holiday party in Hyde Park, where I had a conversation with Mr. Anastaplo on this topic: it was the kind of party where talking about orders of infinity was appropriate. Here are my further thoughts, which he invited me to post here.
On one hand, his intuition is correct–it IS somehow nonsensical that the infinity of primes should not be less than the infinity of counting numbers. We all somehow understand that the part should be less than the whole, and in this case, it is a rather ‘sparse’ part. It was established long ago that for any counting number P, the number of primes less than P is approximately log(P); furthermore, log(P) is certainly less, and generally much less, than P, for P > 100. These are points that are almost intuitive. So even if there are an infinite number of primes, then that infinity OUGHT to be less than the intinite number of counting numbers.
Our intuition is based on things we actually encounter, which are never infinite and seldom, actually, of any really great size (which is why Einstein’s relativity runs counter to our intuition, too). When we try to make infinity consistent, we end up with some surprising, and maybe counter-intuitive results.
We know (maybe) what it means to count a finite set: we put into one-to-one correspondence the counting numbers and the things in that set. When we run out of things to put in one-to-one correspondence, then the last number we used is the count of that set. But when we generalize it’s a bit different (or at least in one way that everyone generalizes, it’s a bit different.)
Rather than saying that we have established a one-to-one correspondence, we say we can find an ordering of that infinite set so that if you give any counting number P, we can find the Pth member of that set, and for any member of the set, we can count at least that far. We never actually count, because we never run out of numbers. Thus maybe we are already into the potential, rather than the actual. Or maybe we are into dispute, rather than into discourse.
If we take this as the way to count infinite collections, then we end up with the apparently counter-intuitive result that the (infinite) number of primes is EQUAL to the (infinite) number of counting numbers, even though for any finite number P the number of primes less than P is much less than P, and the disparity grows larger as P grows larger.
Is there another way of generalizing the way we count? Maybe, but I don’t think so.
Is this consistent? I certainly hope so.
Does this make sense? That depends on how closely one wants to be wedded to the intuitions we have about finite things.
If one wants to remain wedded to that intuition, one might deny the existence of infinity. That would be really hard, I imagine; I don’t know if it would be impossible, because I’ve never tried.
The machines we rely on daily, tho’, can only work with concrete numbers, so we need to work around this. Most digital computers can’t really imagine there are larger numbers than 2^64, or smaller positive numbers than 2^(-64). For the machines we rely on daily, there are actually only a finite number of numbers, and no number is irrational. So the machines we use daily work with a kind of arithmetic that only sort of resembles the arithmetic we use, or think we use.
It is worth noting that there are sets that can’t be put into a one-to-one correspondence with the counting numbers. The easiest example is the set of numbers defined by Dedekind’s cuts, the so-called “real” numbers, or the continuum. There are more than a countable number of such numbers. So there are at least two different kinds of infinity, one greater than the other: the number of countable numbers and the number of continuous numbers.
“For the machines we rely on daily, there are actually only a finite number of numbers, and no number is irrational.”
The machines we use do not interpret their own activity: “For them” there is nothing at all.
But we can interpret the machine’s calculations as calculations with a peculiar kind of arithimetic involving only finitely many number-like entities, or we can interpret the machine’s calculations as calculations with ordinary integers or rational numbers or real numbers. Which interpretation is most useful depends on our purposes in interpreting.
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This blog was created by Joel Rich, and maintained by him until his death in June 2011. The blog is now maintained by John Metz (firstname.lastname@example.org).