From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math.num-analysis Subject: Re: Infinity question... please help solve a bet! Date: June 19, 1997 In article <33A84AA4.6B51@mci.com.bite.me.spammers>, Mike Secorsky wrote: >An individual I work with has made a bet with another co-worker >concerning infinity. His claim is that there are an infinite number of >infinities, and he uses the following example... This is not really fodder for s.m.n-a. I've directed followups to sci.math, where it'll keep 'em busy for weeks. Yes, "infinite" is a concept rather than a quantity; it means "not finite" and may be described, for example, as the ability to pair off the elements of a set with the elements of a proper subset. Your example, for instance, shows that the set of counting numbers is infinite, since it can clearly be put in one-to-one correspondence with a subset of itself (the set of _even_ counting numbers). But if you wish to say more than "not finite", you may. Given two infinite sets, you may ask whether or not it is possible to pair off the elements of one set with the elements of the other. So while the set of counting numbers is infinite, as is the set of even counting numbers, and the set of all even perfect squares, these sets are all "equally large" (i.e. the elements can be paired off) even though each is clearly a proper subset of the one listed before it. Are all inifinite sets "equally large" in this sense? No - for example, one may prove that the set of all real numbers is so large that you can never pair off _all_ its elements with the set of counting numbers. So everyone's a winner here: yes, there are "different infinities" but no, the proffered example does not suffice. By the way, the fun is just beginning here, hence a large field of mathematics known as set theory. The axioms usually used to clarify what makes a set enable us to prove something rather remarkable: it's possible to set up a consistent "ranking" of all sets so that we can call some sets "larger" than others (Here I put this in quotes because I don't mean that either of them has to be a subset of the other; and moreover, as we have already seen, a set is not necessarily "larger" than a proper subset -- the two may be equally large). Even better, there will always be the concept of "the next biggest size". Then we can call the "smallest" infinite size "aleph-0" (this would be for any set which can be paired off with the counting numbers), the next size "aleph-1", and so on. None of this is remarkable if you only deal with finite sets but with infinite sets stranger things happen. For example: * There is not always the concept of "the next smaller size" (e.g., all sets which are "smaller" than the set of counting numbers are in fact finite, and there's no last finite size! To give another example, if you have sets A0, A1, A2, ... of sizes aleph-0, aleph-1, etc., then their union A_w has a brand new size, but there is no "next smaller size" just smaller than A_w.) * The usual axioms for set theory are insufficient to decide whether or not there are "in-between" sizes in general. The Continuum Hypothesis isn't so much a hypothesis but a question: about this first set A_1 which is so big that it can't be paired off with the counting numbers -- is it so big that it can be paired off with the set of all real numbers? Or is there a set of intermediate size? Turns out the question is independent of the other, more believable axioms for set theory. That is, the answer to this question depends on what you mean by "set" ! * There's an infinite number of different sizes of infinite sets. If you're still with me, you should be asking, "OK, so we have this twisty maze of little labels, all different: aleph-0, aleph-1, etc. Sure, the set of these labels is not finite. But can it be paired off with the counting numbers? Or are there so many _labels_ that we can't even pair them off with, say, the set of all real numbers?" Well, your worst fears will be realized: the collection of all labels is so big isn't even really a set... Some folks say you should never attempt to think about set theory while drinking; the theory is too disorienting as it is. But others say that's the _only_ state of mind in which the subject can be approached! dave