Cantor Confusion
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From: Dave Seaman on 28 Sep 2006 07:45 On 28 Sep 2006 04:06:24 -0700, mueckenh(a)rz.fh-augsburg.de wrote: > Peter Webb schrieb: >> You produce ANY list of all Reals, I can show you a missing real. Therefore >> I can do it for ALL lists, and hence there is no complete list of Reals. >> >> Obviously you have to tell me the list first, as there is no single Real >> which is missing from every list, > No? > But the set of all lists is countable (as is any quantized or > discontinuous set), Wrong. > so is the set of all list entries. Nevertheless, > there is no real number missing in every list? So every real number is > in at least one of the list? So every real number is one element of a > countable set of entries? And there is nothing real really outside of > this countable set? > Regards, WM -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. <http://www.mumia2000.org/>
From: Arturo Magidin on 28 Sep 2006 08:34 In article <1159410937.013643.192240(a)h48g2000cwc.googlegroups.com>, <the_wign(a)yahoo.com> wrote: >Cantor's proof is one of the most popular topics on this NG. It >seems that people are confused or uncomfortable with it, so >I've tried to summarize it to the simplest terms: > >1. Assume there is a list containing all the reals. >2. Show that a real can be defined/constructed from that list. >3. Show why the real from step 2 is not on the list. >4. Conclude that the premise is wrong because of the contradiction. This is hardly the simplest terms. Much simpler is to do a ->direct<- proof instead of a proof by contradiction. 1. Take ANY list of real numbers. 2. Show that a real can be defined/constructed from that list. 3. Show that the real from step 2 is not on the list. 4. Conclude that no list can contain all reals. Why insist on proof by contradiction? It just begs the other person to misidentify what is "the" premise that is false. Maybe the constructed number is not really constructed? Maybe the number is not really a real? Etc. -- ====================================================================== "It's not denial. I'm just very selective about what I accept as reality." --- Calvin ("Calvin and Hobbes" by Bill Watterson) ====================================================================== Arturo Magidin magidin-at-member-ams-org
From: Randy Poe on 28 Sep 2006 11:19 mueckenh(a)rz.fh-augsburg.de wrote: > Peter Webb schrieb: > > > You produce ANY list of all Reals, I can show you a missing real. Therefore > > I can do it for ALL lists, and hence there is no complete list of Reals. > > > > Obviously you have to tell me the list first, as there is no single Real > > which is missing from every list, > > No? No. Let x be any real. Then I can certainly create a list {a_1, a_2, ...} with a_1 = x. > But the set of all lists is countable No it isn't. - Randy
From: Ross A. Finlayson on 28 Sep 2006 11:31 Dave Seaman wrote: > On 28 Sep 2006 04:06:24 -0700, mueckenh(a)rz.fh-augsburg.de wrote: > > > Peter Webb schrieb: > > >> You produce ANY list of all Reals, I can show you a missing real. Therefore > >> I can do it for ALL lists, and hence there is no complete list of Reals. > >> > >> Obviously you have to tell me the list first, as there is no single Real > >> which is missing from every list, > > > No? > > > But the set of all lists is countable (as is any quantized or > > discontinuous set), > > Wrong. > > > so is the set of all list entries. Nevertheless, > > there is no real number missing in every list? So every real number is > > in at least one of the list? So every real number is one element of a > > countable set of entries? And there is nothing real really outside of > > this countable set? > > > Regards, WM > > > > -- > Dave Seaman > U.S. Court of Appeals to review three issues > concerning case of Mumia Abu-Jamal. > <http://www.mumia2000.org/> Hi. There are a variety of actively researched set theories that have representations of numbers in them where the powerset or antidiagonal argument is not said to hold. There are a variety of most types of applied mathematics that ignore transfinite cardinals. Cantor's (Georg Cantor, mathematician's) expectation that there is a universe, is called here the domain principle. In ZF, which is taught to a lot of people, there is no universe. So if you use a universe in a set theory, then it is not ZF. Remove all the (non-logical) axioms from any theory and then it is the null axiom theory. If you're interested in a theory that is designed with the goals of being consistent and complete, I've written some thousands of pages about it to sci.math. Ah, I had some excellent thoughts about definitions the other day, very comforting, in the context as above. Did you know some mathematicians divide by zero? The universe is infinite, infinite sets are equivalent. Ross
From: William Hughes on 28 Sep 2006 11:35
mueckenh(a)rz.fh-augsburg.de wrote: > Peter Webb schrieb: > > > You produce ANY list of all Reals, I can show you a missing real. Therefore > > I can do it for ALL lists, and hence there is no complete list of Reals. > > > > Obviously you have to tell me the list first, as there is no single Real > > which is missing from every list, > > No? > > But the set of all lists is countable Twaddle. Let A ={ (x,0,0,0,...) | x is a real number } Then A is a set of lists and A has as many elements as there are real numbers. >(as is any quantized or > discontinuous set), so is the set of all list entries. Nevertheless, > there is no real number missing in every list? So every real number is > in at least one of the list? So every real number is one element of a > countable set of entries? And there is nothing real really outside of > this countable set? > Note any countable set of entries is a list (possibly with a different ordering). This is staight quantifier dyslexia For every real number x, there exists a list, L_x such that x is a member of L_x There exists a list L, such that every real number x is a member of L. The first is true, the second is false. There is no way to put all the L_x together to get a "countable set of entries" (the list L). - William Hughes |