X-Message-Number: 21981 From: "michaelprice" <> References: <> Subject: Transfinites Date: Sun, 15 Jun 2003 05:56:35 +0100 Matthew S. Malek writes: > As it turns out, there are actually different sizes of infinity, > but only two of them. Two? There are infinite number of infinities (AKA transfinites) To be more precise there are at least alepth_1 transfinites where alepth_0 = number of natural numbers and alepth_(n+1) = 2 ^^ alepth_n and you can prove that alepth_(n+1) > alepth_n. i.e. 2 to the power of any transfinite is a larger transfinite, which comes from the set theory result that number of all possible subsets of a set is always greater than the number of elements of the original set. This allows us to construct a countable infinity of transfinites. We can then construct a greater transfinite beth_0 = sum (n =0 - infinity) alepth_n which forms the basis of another hierarchy. This process of construction is never ending and generates at least alepth_1 transfinites. (Caveats: I'm ignoring the continuum hypothesis or axiom, which states that there may be more conjectural but unprovable (in the Godelian sense) transfinites lurking "in-between" the ones given by the above construction. In which case the labelling scheme becomes more complex..... Also, I may be underestimating the number of constructible transfinites; there might be alepth_2!) > Which is the larger infinity: The number of > integers alepth_0 >, or the number of [real] numbers between > zero and one? alepth_1 Cheers, Michael C Price ---------------------------------------- http://mcp.longevity-report.com http://www.hedweb.com/manworld.htm Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=21981