On Mon, 03 Aug 2009, Kip Murray wrote: > Bill, taking your points in order: > > Question 1. In abstract set theory, "set" and "is an element of" are > undefined, > but an implementor of a concrete model for set theory must define which > objects > he is using to represent sets. That is, an answer for Question 1 is required > of > an implementor of a concrete model. > > Question 2. You are correct, it is the relation "is an element of" that must > be > defined. I misstated Question 2.
Undefined doesn't mean rubbish or unknown, you can take it to mean anything you like, ie defined by implementor, so long as it doesn't invalidate other postulates. I guess the test for membership is the most important. Once it is defined, the domain of this test automatically determine what is an element and what is a set. > Question 7. That's exactly right. > > Multiple occurrences of an element. I had a teacher who said one day, "You > are > not going to like this, but we are going to allow a number to be a member more > than once, and we will take into account 'multiple membership' in finding the > sum of the numbers in a finite set. We will then use the idea of sum for a > finite set to define what is meant by a sum for the numbers in an infinite > set. > Not every infinite set of numbers has a sum." So, multiple occurrences of > the > same element are sometimes allowed, but are frowned on! Outside of that one > course, they were never allowed in my training, and I learned later it is more > acceptable to talk about a sum of an infinite sequence (in sequences > repetitions > _are_ allowed), and that the teacher was using "sum for a set" to finesse > absolute convergence. ...more than you wanted to know In brief, I think of > the > names you wrote as different names for the same set of only three numbers. > I think this is a matter of representation. You prefer to use the set of distinct element to represent all other sets in the same equivalent class. Likewise you defined set as an order set. It needs to define an extra relation of 'ordering' which is not always possible in general. For a set of complex numbers, it can have a partial ordering for the subset of real numbers, but a total ordering of the entire complex plane is impossible at least in analysis. Fortunately all arrays in J can be sorted. > By the way, what does iirc mean? "if I recall"? > Yes, iirc (correctly) ;-) -- regards, ==================================================== GPG key 1024D/4434BAB3 2008-08-24 gpg --keyserver subkeys.pgp.net --recv-keys 4434BAB3 ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
