Transfinite NumbersDate: 11/07/97 at 00:45:00 From: John McSwain Subject: Transfinite numbers I have done some research and found that Georg Cantor discovered transfinite numbers, but I don't understand what they are. Could you please give an example or some depiction that might help me? Thanks a lot. Date: 11/07/97 at 10:39:22 From: Doctor Rob Subject: Re: Transfinite numbers To talk about cardinal numbers, including transfinite cardinal numbers, you need to talk about counting the numbers of elements in sets. If a set A is finite, there is a nonnegative integer, denoted #A, which is the number of elements in A. That is one of the finite cardinal numbers. To do arithmetic with cardinal numbers, you use facts about finite sets and the number of elements in them, such as the following: If A and B can be put into one-to-one correspondence, then #A = #B, and conversely. If A is contained in B, then #A <= #B. If A is disjoint from B and C is their union, then #C = #A + #B. If A and B are sets, and C = A x B is the set of all ordered pairs of elements, the first from A and the second from B, then #C = (#A)*(#B). If C is the set of all subsets of A, then #C = 2^(#A). Try some of these with small sets to satisfy yourself that they are so. If the set is infinite, the corresponding cardinal number is not one of the finite cardinal numbers, so it is called a transfinite (or infinite) cardinal number. The first transfinite number is called aleph-sub-zero (or aleph- naught, or aleph_0). (Aleph is the first letter of the Hebrew alphabet.) It is the cardinal number of the set of positive integers. Sets having this cardinal number are called countably infinite sets, or countable sets, because they can be put into one-to-one correspondence with the positive integers, or counting numbers. The above rules for computing with finite cardinal numbers were extended by Cantor to apply to transfinite cardinal numbers. In this way he could talk about doing arithmetic with transfinite cardinal numbers. He discovered and proved that, if n is any finite cardinal number, aleph_0 + n = n + aleph_0 = aleph_0, aleph_0 + aleph_0 = aleph_0, aleph_0*n = n*aleph_0 = aleph_0 (n > 0), aleph_0*aleph_0 = aleph_0, aleph_0^n = aleph_0 (n > 0). This is a pretty strange arithmetic! Note that subtraction and division are not definable operations in this arithmetic. The Associative Laws of Addition and Multiplication hold, and the Commutative Laws of Addition and Multiplication do, too, and the Distributive Law, but there is no zero, no unity, no negatives, and no reciprocals. Cantor also proved that 2^aleph_0 > aleph_0. This means that the set of all subsets of the integers cannot be put in one-to-one correspondence with the integers, that this set is really a different size of infinite set, truly larger. Cantor called 2^aleph_0 = C, which stands for "continuum". C is the cardinality of the set of real numbers. Once again, Cantor showed that n^aleph_0 = C (n > 1), aleph_0^aleph_0 = C, C + n = n + C = C, C + aleph_0 = aleph_0 + C = C, C + C = C,, C*n = n*C = C (n > 0), C*aleph_0 = aleph_0*C = C, C*C = C, C^n = C (n > 0), C^aleph_0 = C, but 2^C > C. This trick of exponentiating one cardinal number to get a larger one works for all transfinite cardinal numbers. In this way, Cantor showed that there are infinitely many different transfinite cardinal numbers. In fact the set of transfinite cardinal numbers itself has a cardinal number, which is transfinite! Let's not think about that - it gets very confusing! But are there any transfinite cardinal numbers between aleph_0 and C? The name aleph_1 has been given to the smallest transfinite cardinal number larger than aleph_0. Then aleph_0 < aleph_1 <= C. The above question can be rephrased as, "Is aleph_1 = C?" Cantor guessed that this is so, but was unable to prove this. It was one of the great unsolved problems of mathematics, called "The Continuum Hypothesis". Finally it was shown that this was not provable from the usual axioms of set theory! It is usually assumed as an additional axiom. I hope that this is helpful. If you need more help, write again. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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