X-Message-Number: 21973 From: Date: Sat, 14 Jun 2003 11:36:20 EDT Subject: infinities --part1_12c.2c5f8437.2c1c9af4_boundary Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit I had said that, by Cantor's method of counting, there are the same number of odd integers as integers, but by another method there are twice as many integers. Matthew Malek writes in part: > If there are infinite odd integers, how can there be twice as many even > integers? > > "Twice infinity" doesn't really strike me as being a well defined term. > > As it turns out, there are actually different sizes of infinity, but only > two of them. Which is the larger infinity: The number of integers, or > the number of numbers between zero and one? (First, I didn't say twice as many even integers--I said twice as many integers as odd integers--although the other statement could also be proven by appropriate definition.) No, there are not only two (theoretical) infinities, but an infinite number of them, cardinality aleph-null (the denumerable infinities such as the integers), aleph-one, aleph-two, ad infinitum. The cardinality of the continuum, called "c," may or may not be the same as aleph-one--this proposition is thought to be undecidable. How can there be twice as many integers as odd integers? By choosing your definition of "twice as many." Suppose I choose to say, "Set I of cardinality aleph-null (integers) shall be deemed twice as large as set O of cardinality aleph-null (odd integers) if O is a subset of I and a one-to-one correspondence can be shown between the numbers n in O and the pairs of numbers (n, n + 1) in I. Again, the only importance of this stuff for most of us is the reminder that language is tricky and tradition is often badly mistaken. Of course we remember too that the "infinities" in math are not actual or completed infinities but only potential infinities in the sense of there always being more. Whether there are any physical infinities in the universe, or whether the universe itself is finite, are open questions. Robert Ettinger --part1_12c.2c5f8437.2c1c9af4_boundary Content-Type: text/html; charset="US-ASCII" [ AUTOMATICALLY SKIPPING HTML ENCODING! ] Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=21973