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To: Zeev Hed who wrote (30604)	9/24/1999 10:04:00 PM
From: NHBob	Respond to of 93625

Wow, just when I was beginning to believe that all designs these days are based on worst-on-worst (lets see, Carl would you use devil-on-devil?)and that I should purge all my RSS (root-sum-sq) design guides from my bookshelves, Zeev, you come along and save the day. My technical library budget thanks you. (I hope <VBG>) Bob

To: Zeev Hed who wrote (30604)	9/25/1999 3:57:00 AM
From: Bilow	Read Replies (1) \| Respond to of 93625

You are quite right, Zeev, my infinities are off by one. The count of real numbers is Aleph 1, and Aleph 0 is the, rather small, number of integers, as you said. As far as the other Alephs, any textbook on transfinite arithmetic will prove: Aleph N+1 = 2^(Aleph N) The existence proof is elementary, and I can give an example of it, in the case of the number Aleph 2 being greater than Aleph 1. An example of Aleph 2 is the number of functions mapping the (real) unit interval (i.e. [0,1]) to itself. Proof that the number of such functions is beyond Aleph 1, by counterexample: If the number of such functions were merely Aleph 1, then, choose a mapping from [0,1] to those functions. Suppose that mapping is Phi. Phi maps the unit interval to functions on the unit interval. So Phi(x) is a mapping of the unit interval to itself. In other words, Phi takes a real number, and "turns it into" a function. Note Phi(x) is a function that maps the unit interval to itself. So if y is in the unit interval, then (Phi(x))(y) is also in the unit interval. I hope this makes the notation clear. As an example, one function that maps the unit interval to itself is that defined by y = x^3. This function is supposed to be in the range of Phi, so there must be some real number that maps to this function. Suppose the real number that maps to y = x^3 is 0.36 Then we would have: Phi(0.36)(x) = x^3 Phi(0.3601) would be some other function, presumably. Now define a function Psi. Define Psi as follows. At the point x, choose if Phi(x)(x) > 1/2, then choose Psi(x) = Phi(x)(x) - 1/2 if Phi(x)(x) <= 1/2, then choose Psi(x) = Phi(x)(x) + 1/2 This newly defined function Psi, is a mapping of the unit interval to itself, but it is not equal to any of the functions mapped to by Phi, as it is unequal to Phi(b), for example, at the value b. Therefore Phi could not have mapped the unit interval to the functions on the unit interval. Therefore the number of functions on the unit interval is larger than the number of real numbers in the unit interval. The continuum hypothesis is that there are not infinities between the infinities defined by Aleph N + 1 = 2 ^ (Aleph N) It is a pleasure to recall transfinite math, which I have not been able to find any other use for, for many, many years... -- Carl