From: ags@seaman.cc.purdue.edu (Dave Seaman) Subject: Re: re : Cantor Set ....continued Date: 11 Oct 1999 13:07:04 -0500 Newsgroups: sci.math Keywords: exponentiation behaves differently for cardinals and ordinals In article <7tt2oq$qae$1@bgtnsc02.worldnet.att.net>, Ari wrote: >Hmmm but I read in "Numbers and Infinity" by Sondheim and Rogerson that >2^No=c,where No=cardinality of the integers and c=cardinality of the reals. >Are you saying that lim 2^n as n->No is not =2^No? Yes, that's right. I was distinguishing between the cardinal number aleph_0 and the ordinal number omega. Even though they are often considered to be the same set, there is a difference in how arithmetic is defined on cardinal numbers and on ordinal numbers. We have 2^aleph_0 = c (in cardinal arithmetic), but 2^w = lim{n->w} 2^n = w. Cardinal exponentiation is discontinuous at aleph_0. >Be patient w/ me I'm not a mathematician. >On the other hand I always thought the reals can be represented as an >infinite ( in the No sense) string of digits where each digit can be 0 or >1. Yes, they can, though I would prefer to say that the strings are infinite in the omega sense. Each string is a mapping d: w -> {0, 1}, where w = omega = the smallest transfinite ordinal = the set of finite ordinals = the set of natural numbers. -- Dave Seaman dseaman@purdue.edu Pennsylvania Supreme Court Denies Fair Trial for Mumia Abu-Jamal ============================================================================== From: Fred Galvin Subject: Re: Isn't omega an ordinal number? Date: Thu, 7 Oct 1999 14:28:00 -0500 Newsgroups: sci.math On Thu, 7 Oct 1999, David C. Ullrich wrote: > Because it leads to such interesting "paradoxes": In fact omega = > Aleph_0 > (at least in the standard setup the set used to represent the first > infinite > ordinal is the same set as the set used to represent the first infinite > cardinal), but 2^omega <> 2^Aleph_0 (because 2^omega is the limit > of 2^n for finite n, while 2^Aleph_0 is the cardinality of the power > set of a set of cardinality Aleph_0.) IMHO it would be a damn good idea to do it your way. Unfortunately, the symbol 2^omega is thoroughly ambiguous, and in the contemporary literature it usually does mean 2^aleph_0, i.e., cardinal rather than ordinal exponentiation. To quote from Kunen's _Set Theory_ (1st edition, 1980, p. 32): "Cardinal exponentiation is not the same as ordinal exponentiation. The ordinal 2^omega is omega, but the cardinal 2^omega = |scriptP(omega)| > omega. In this book, ordinal exponentiation is rarely used, and kappa^lambda denotes cardinal exponentiation unless otherwise stated." Of course 2^omega is also used to denote the power set of the set omega, construed either as the set of all subsets of omega, or the set of all mappings from omega to {0,1}. :-( ============================================================================== From: Mike Oliver Subject: Re: omega^omega...???Please Clarify Date: Wed, 20 Oct 1999 13:42:41 -0700 Newsgroups: sci.math Ari wrote: > > Is omega^omega A1;A2;A3;A4;A5;....where each Aj has ordinal omega and the > index set of j's has ordinal omega? > Is 1,3,5; 2,6,10; 4,12,20; 8,24,40; 16,48,80; .....; > 2^n,3*2^n,5*2^n....;....an example of the above?(Don't know if I got the > general term right) Exponentiation has a wealth of different meanings, which are usually distinguishable from context, but your posting has not provided enough context to tell what you mean. The most common use of the expression "omega^omega" involves ordinal exponentiation. The easiest way to think about this omega^omega is to work up to it slowly. Imagine omega objects arranged linearly. Now put another omega objects after them; the order type of the two together is omega*2. Add another omega at the end and you have omega*3. Keep on doing it omega times, and you have omega^2 objects laid out. Now you can put another omega^2 objects after them; that's (omega^2)*2. Now you see how to get to omega^3. So arrange a group of omega objects, followed by omega^2 objects, followed by omega^3 objects, and so on--the whole collection has order type omega^omega. Secondly, when A and B are general sets (not just ordinals), A^B sometimes means the collection of all functions from B into A. So sometimes omega^omega means the set of all functions from the natural numbers to the natural numbers, (usually with a particular topology; add the topology and you get what is called "Baire space", a very important structure in descriptive set theory). Sometimes, to distinguish this from other kinds of exponentiation, this A^B is written with the small B at the upper *left* of the large A instead of the upper right. Finally, "omega" is sometimes used as a synonym for the cardinal Aleph_0, so "omega^omega" can even be used to mean *cardinal* exponentiation. In that case, omega^omega equals the cardinality of the continuum, 2^Aleph_0.