Transfinite ArithmeticDate: 10/28/97 at 17:08:19 From: MARK SIMAKOVSKY Subject: Transfinite Arithmetic What is transfinite arithmetic? I pretty much know what it means, but I am having trouble applying it to aleph-null. What is the relation between aleph-null, aleph-one, and aleph-two? How do you put these into sets? Date: 10/28/97 at 17:37:19 From: Doctor Tom Subject: Re: Transfinite Arithmetic Hi Mark, I'll give you a start here, but you've basically asked a question that requires a college course to cover entirely. Find a book on set theory for details. The easiest construction of the ordinal numbers is to define zero as the empty set, and if you've already defined the number n, the number n+1 = n union { n } So (using "U" to indicate "union"): 0 = { } 1 = 0 U { 0 } = { } U { 0 } = { 0 } 2 = 1 U { 1 } = { 0 } U { 1 } = { 0, 1 } 3 = 2 U { 2 } = { 0, 1 } U { 2 } = { 0, 1, 2 } and so on. For the finite ordinals, you basically see that: n = { 0, 1, 2 ,3 ..., (n-1) } The ordinal numbers are the smallest set that: 1) contains the number zero = { }. 2) if it contains n, it contains n+1 (as defined above) 3) if S is a set of ordinal numbers, so is the union of all the elements of S. If you leave out condition 3, you just get the natural numbers, which is the same as the finite ordinals. But the set { 0, 1, 2, 3, ... } is a set of ordinals, so the union of all those sets is also an ordinal, and you'll see that it is { 0, 1, 2, ... } -- the same thing. But notice that every ordinal is a set containing the "right" number of things -- 3 is a set with 3 items in it: 0, 1, and 2. The set above is not any of the finite ordinals -- it's larger than any of them -- it is infinite, the smallest size of infinity, and is called aleph-0, where aleph is the first letter of the Hebrew alphabet. But since aleph-0 is an ordinal, so is aleph-0 + 1: aleph-0 + 1 = aleph-0 U { aleph-0 } = { aleph-0, 0, 1, 2, ... } it is different from aleph-0. Similarly, you can get aleph-0 + 2, aleph-0 + 3, ... and so on. Then you can union all of those and get aleph-0 * 2, and on, and on, and on ... Eventually, you get to a set so large that it can't be matched one-to-one with aleph-0, and the first time that happens, call it aleph-1 -- the second smallest size of infinity. This goes on and on, too. Arithmetic can be defined on the ordinals, and that's what you should look up in a set-theory book. You've probably seen a proof (Cantor's diagonalization proof) that there are more real numbers than integers. That means that C, the size of the real numbers, is larger than aleph-0, Up until about 1965 or so, nobody knew if C was equal to aleph-1 or not. Then Paul Cohen of Stanford proved that the theorem "C = aleph-1" is independent of the other axioms of set theory. In other words, it can be neither proved nor disproved. This is only the first of a large set of amazing facts about the ordinal numbers. But find a set-theory book -- I'm too lazy to type in chapter after chapter. :^) -Doctor Tom, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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