| Subject: Set Theory |
Author:
mi chamocha
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Date Posted: 17:47:35 07/11/01 Wed
In reply to:
Flannel
's message, "Response." on 14:55:10 07/11/01 Wed
I don't even think that this argument was very central to Flanner's argument that initiated this thread. Nevertheless, I'll continue it, because I believe that the answer to most of the questions we ask here (with the exception of the more humanitarian questions, eg, ethics, politics, etc.) will inevitably lie somewhere in between math, physics, and philosophy. Furthermore, set theory turns out to be an important aspect of mathematics. So, to begin, here is a (very) small primer on set theory:
Curly braces typically denote the contents of a set. When {0,1} is written, this denotes a set consisting of two elements: 0 and 1. A "pipe" character, '|', denotes a qualification on the elements of a set. So { x + y | 0 < x < 1, 5 < y < 10 } denotes a set consisting of the sum of any real between 0 and 1, and any real between 5 and 10.
Slightly relevant to the current discussion, as it turns out, some infinities are larger than others. There are 'countable' sets, and 'uncountable' sets. A countable set is any set for which there exists a surjection, or a 1-1 mapping, from the set into the counting numbers, which are all integers greater than zero. The counting numbers are obviously countable, since each number can be mapped to itself. Any finite set is also countable (but not all countable sets are finite, obviously). The real numbers are uncountable. The set of all rational numbers happens to be countable as well, which is kind of odd if you think about it, since it means that the integers and the rationals are actually the *same* size. Odd, because if you choose any two integers, there are an infinite number of rationals between them. Hmm. Oh well.
A subset S of a set T is a set such that all the elements of S are also elements of T. Pretty simple.
Now, to the question at hand. Flanner's original statement with which I took issue was:
"A subset [we'll call the subset S] of infinity [we'll call this T] is still infinite"
Within the language of set theory (which is what he uses here), that statement is false. (Anyone happen to know the truth value of the sentence "This statement is false"? ;-) From what I can tell he means one of two things:
1) The set T to which he refers is the set {infinity}. This is a set consisting of a single element, which is the element 'infinity'. Now I don't really think this is what he means, since the set isn't even infinite itself. Further, the notion of a subset is of an element is not defined. Thus what I think he means is something more like:
2) T is any set with an infinite number of members, e.g., the real numbers. In this case, his statement is still false. As I showed above, the set {0,1} has two elements (hence, is finite) and is a subset of the reals, since 0 and 1 are both real numbers. Therefore this suffices as a counterexample to his claim.
His argument that "a number doesn't exist without its sum" is wrong for (at least) three reasons:
1) It's circular, and removes any basis for the existence of any number.
2) There are many conceivable mathematical structures that have no sum operation defined on them (e.g., the quaternions, the set of mappings from the integers into themselves) which are every bit as real and existent as the numbers - a 'sum' isn't required for these to exist; furthermore, I could just as easily choose any of these sets and my original argument would still proceed unchanged.
3) The argument isn't even specific to the numbers; any set, any collection of things or ideas would suit just as well.
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