Cantor's Infinities and Universal Sets
Date: 07/17/2008 at 03:27:17 From: Martin Subject: Cantor's Infinities In Cantor's set theory, the idea of having a universal set or a set of everything cannot be true, due to the basic contradiction that arises from the nature of set theory. Based on this, when looking at Cantor's Hierarchy of Infinities, does the hierarchy of infinities still hold? Truly, if an absolute infinity existed then it would accommodate everything...contradicting the idea of no universal set.
Date: 07/17/2008 at 08:32:31 From: Doctor Tom Subject: Re: Cantor's Infinities Hi Martin. You're right--you cannot have a set that, say, contains one infinite set of each size or you'll run into the problem you mention. One way around it, for the purposes of discussion of the idea, is to use the idea of a "class" that can be that large. The restriction, of course is that if such a class is too large to be a set (in other words, it can't be built from the allowed operations of set theory), then it is called a "proper class", and cannot be contained in any set or other class. That way I can talk to other mathematicians about "the class containing all the ordinal numbers", but I can't include that in any set or class. - Doctor Tom, The Math Forum http://mathforum.org/dr.math/
Date: 07/18/2008 at 09:23:44 From: Martin Subject: Cantor's Infinities But the idea of Cantor's infinities is based upon looking at bijections. For instance, the set of integers has a lower cardinality than the set of real numbers because a bijection between the two sets does not exist. So Cantor looked at bijections to determine the size of infinities. If so, are we stuck with looking at Sets? In addition, could it be that there does not even exist a hierarchy of Infinities as Cantor has hypothesized? Many thanks. I'm doing a maths essay on Cantor and infinity so I just thought of this...it would be great if you can offer any advice.
Date: 07/18/2008 at 10:35:34 From: Doctor Tom Subject: Re: Cantor's Infinities The problem with naive set theory, where you're sloppy about what you allow to be a "set", is (Bertrand) Russell's paradox: Consider the set of all sets that don't contain themselves as an element. Is that set a member of itself or not? If it isn't, it is, and if it is, it isn't. To get around this what's done is to carefully build up a collection of valid sets, starting from the empty set, so that nothing you make can contain itself. In such a system, you can't have a universal set (that contains all sets) or it would have to contain itself and you're back to Russell's paradox. The most common set of axioms used to build up a formal (and as far as we know, contradiction-free) theory of sets is called the "Zermelo- Frankel" axioms for set theory. Those axioms include the "axiom of infinity" that tell us basically that the natural numbers can be put in a set, and the "power set axiom" that tells us that if S is a set, then the power set of S is also a set. With Cantor's work, we can show that a hierarchy of infinities exists: a sequence of sets, each infinite, and each larger than the last. We cannot put ALL of those into one set, however, since we'd be back at Russell's paradox. But we, as mathematicians, need to talk about "all the sizes of infinity", so when we do want to talk this way, we certainly don't want to say, "the set of all the infinite cardinal numbers" or we'll face Russell again. So when we talk about these larger groups, we just call them "classes", with the understanding that a class cannot be made that contains another class, and classes cannot be members of sets. Actually, the word "class" is a little looser, and a class could be a set, but if it's too large, we just call it a "proper class", and so to be very precise, I should say that no PROPER class can be contained in another class or set. I'm not sure how much you know about formal set theory or the hierarchy of infinities, but I've written a couple of papers aimed at the level of bright high-school students that you might find interesting as background, although they don't treat your topic directly. Take a look at these: Infinity http://www.geometer.org/mathcircles/Infinity.pdf and Nothing http://www.geometer.org/mathcircles/nothing.pdf Good luck with your project! - Doctor Tom, The Math Forum http://mathforum.org/dr.math/
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