Cantor Confusion
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From: William Hughes on 10 Oct 2006 17:44 Albrecht wrote: > William Hughes wrote: > > Albrecht wrote: > > > William Hughes schrieb: > > > > > > > Albrecht wrote: > > > > > William Hughes schrieb: > > > > > > > > > > > Albrecht wrote: > > > > > > > > > > > > <...> > > > > > > > > > > > > > I don't controvert the axiomatic methode anymore. But I claim that it > > > > > > > isn't the only and the important one in math. In teaching and in the > > > > > > > mind of the people the axiomatic method appears to be the only right > > > > > > > way to do math. That's not correct. > > > > > > > The nondenumerable infinity of the reals is not the only one truth. > > > > > > > Nobody is wrong who claims only one kind of infinity, the one we only > > > > > > > can know: the endless infinity. > > > > > > > > > > > > > > > > > > > > > > > > > The problem is not that someone who believes > > > > > > in your intuitive "endless infinity" (intuitive because it cannot > > > > > > be put on a mathematical footing) is wrong. > > > > > > > > > > Oh yes, it is the problem. I came to these subject by reading a bunch > > > > > of popular books about math. When I read the diagonal argument the > > > > > third or fourth time I started to wonder. The textes were of differnt > > > > > quality but all of them had a special sort of feeling. And all stated, > > > > > that this proof is so elementary, easy and absolute right that nobody > > > > > had anything to reflect or critizise about it. > > > > > But I found in shortest time a lot of questions about the issue. > > > > > Later I read professional works about set theorie and I found a similar > > > > > feeling in the textes about the diagonal argument. And then I started > > > > > to learn about the role of ZF in the teaching on universities and I had > > > > > a lot of disputes about the matter in newsgroups. > > > > > > > > Don't you find it interesting that of all the places you looked, > > > > the only place where anyone disagreed with the diagonal > > > > argument was the newsgroups? > > > > > > That's really untrue. I had read several books and papers of academics > > > (who do not post in this or the german math newsgroups) in which they > > > formulate (very cautious) criticism about ZF, axiomatic set theory or > > > especially the axiom of infinity. > > > > Did any of them disagree with the diagonal argument? > > > I think, the answer to this question don't enlight the whole problem we > discuss. In other words, no. Now read the rest of the post. This is the problem. Not someone saying, "your arguments lead to conclusions I find counterintuitive, so the axioms you chose need to be modified", but someone saying "your arguments lead to conclusions I find counterintuitive, so there must be something wrong with your arguments" or "your arguments lead to conclusions I find counterintuitive, so your axioms must be inconsistent". - William Hughes
From: Albrecht on 10 Oct 2006 18:55 Arturo Magidin wrote: > 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. > This summarization of the diagonal argument of Cantor seems to be accepted by the most people in sci.math. I like to examine the idea of Russell Easterly - building a kind of diagonal number on lists of natural numbers - in respect to this view. First I review his idea: Let's have an arbitrary list of natural numbers: 1: a 2: b 3: c .... with the numbers of digits of the numbers in the list: 1: m 2: n 3: o .... Now we build the "diagonal number" d of the list as follows: We have a look on the first number of the list a which is build out of m digits. We build a number with m+1 digits with the cipher 1. This number is truely greater than a and therefore different from a. It's the first approach to d. (E.g. a = 765 -> m = 3 -> d = 1111) Now we have a look on the second entry of the list. The number b with n digits. If n <= m we let d unchanged. If n > m we build the new number d with n+1 digits, again with the cipher 1. In this way we go through the list. This construction builds up a number d which is different from any number of the list. Now we want to test this sentences: 1. Take ANY list of natural numbers. 2. Show that a natural number can be defined/constructed from that list. 3. Show that the natural number from step 2 is not on the list. 4. Conclude that no list can contain all natural numbers. 1.: Any lists are any finite and infinite lists of natural numbers (Axiom of infinity). 2.: The number d has a finite, integer difference to any number of the list by construction. A number which could be written as the sum of two natural numbers is a natural number too. 3.: The number d is different to any number of the list by construction. 4.: The natural numbers are nondenumerable. What's wrong with Russell's argument but right with Cantor's? Best regards Albrecht S. Storz > > 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: Albrecht on 10 Oct 2006 19:02 William Hughes wrote: > Albrecht wrote: > > William Hughes wrote: > > > Albrecht wrote: > > > > William Hughes schrieb: > > > > > > > > > Albrecht wrote: > > > > > > William Hughes schrieb: > > > > > > > > > > > > > Albrecht wrote: > > > > > > > > > > > > > > <...> > > > > > > > > > > > > > > > I don't controvert the axiomatic methode anymore. But I claim that it > > > > > > > > isn't the only and the important one in math. In teaching and in the > > > > > > > > mind of the people the axiomatic method appears to be the only right > > > > > > > > way to do math. That's not correct. > > > > > > > > The nondenumerable infinity of the reals is not the only one truth. > > > > > > > > Nobody is wrong who claims only one kind of infinity, the one we only > > > > > > > > can know: the endless infinity. > > > > > > > > > > > > > > > > > > > > > > > > > > > > > The problem is not that someone who believes > > > > > > > in your intuitive "endless infinity" (intuitive because it cannot > > > > > > > be put on a mathematical footing) is wrong. > > > > > > > > > > > > Oh yes, it is the problem. I came to these subject by reading a bunch > > > > > > of popular books about math. When I read the diagonal argument the > > > > > > third or fourth time I started to wonder. The textes were of differnt > > > > > > quality but all of them had a special sort of feeling. And all stated, > > > > > > that this proof is so elementary, easy and absolute right that nobody > > > > > > had anything to reflect or critizise about it. > > > > > > But I found in shortest time a lot of questions about the issue. > > > > > > Later I read professional works about set theorie and I found a similar > > > > > > feeling in the textes about the diagonal argument. And then I started > > > > > > to learn about the role of ZF in the teaching on universities and I had > > > > > > a lot of disputes about the matter in newsgroups. > > > > > > > > > > Don't you find it interesting that of all the places you looked, > > > > > the only place where anyone disagreed with the diagonal > > > > > argument was the newsgroups? > > > > > > > > That's really untrue. I had read several books and papers of academics > > > > (who do not post in this or the german math newsgroups) in which they > > > > formulate (very cautious) criticism about ZF, axiomatic set theory or > > > > especially the axiom of infinity. > > > > > > Did any of them disagree with the diagonal argument? > > > > > > I think, the answer to this question don't enlight the whole problem we > > discuss. > > In other words, no. > > > Now read the rest of the post. > > Since you choose to don't read the rest of my post, why should I do otherwise? Best regards Albrecht S. Storz
From: William Hughes on 10 Oct 2006 19:08 Albrecht wrote: > William Hughes wrote: > > Albrecht wrote: > > > William Hughes wrote: > > > > Albrecht wrote: > > > > > William Hughes schrieb: > > > > > > > > > > > Albrecht wrote: > > > > > > > William Hughes schrieb: > > > > > > > > > > > > > > > Albrecht wrote: > > > > > > > > > > > > > > > > <...> > > > > > > > > > > > > > > > > > I don't controvert the axiomatic methode anymore. But I claim that it > > > > > > > > > isn't the only and the important one in math. In teaching and in the > > > > > > > > > mind of the people the axiomatic method appears to be the only right > > > > > > > > > way to do math. That's not correct. > > > > > > > > > The nondenumerable infinity of the reals is not the only one truth. > > > > > > > > > Nobody is wrong who claims only one kind of infinity, the one we only > > > > > > > > > can know: the endless infinity. > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > The problem is not that someone who believes > > > > > > > > in your intuitive "endless infinity" (intuitive because it cannot > > > > > > > > be put on a mathematical footing) is wrong. > > > > > > > > > > > > > > Oh yes, it is the problem. I came to these subject by reading a bunch > > > > > > > of popular books about math. When I read the diagonal argument the > > > > > > > third or fourth time I started to wonder. The textes were of differnt > > > > > > > quality but all of them had a special sort of feeling. And all stated, > > > > > > > that this proof is so elementary, easy and absolute right that nobody > > > > > > > had anything to reflect or critizise about it. > > > > > > > But I found in shortest time a lot of questions about the issue. > > > > > > > Later I read professional works about set theorie and I found a similar > > > > > > > feeling in the textes about the diagonal argument. And then I started > > > > > > > to learn about the role of ZF in the teaching on universities and I had > > > > > > > a lot of disputes about the matter in newsgroups. > > > > > > > > > > > > Don't you find it interesting that of all the places you looked, > > > > > > the only place where anyone disagreed with the diagonal > > > > > > argument was the newsgroups? > > > > > > > > > > That's really untrue. I had read several books and papers of academics > > > > > (who do not post in this or the german math newsgroups) in which they > > > > > formulate (very cautious) criticism about ZF, axiomatic set theory or > > > > > especially the axiom of infinity. > > > > > > > > Did any of them disagree with the diagonal argument? > > > > > > > > > I think, the answer to this question don't enlight the whole problem we > > > discuss. > > > > In other words, no. > > > > > > Now read the rest of the post. > > > > > > Since you choose to don't read the rest of my post, why should I do > otherwise? Two reasons. You were trying to change the subject, I was not. I posted first. - William Hughes
From: David Marcus on 10 Oct 2006 20:53
David R Tribble wrote: > Dik T. Winter wrote: > >> And you also start with definitions, or a model. I did *not* state that > >> it was difficult to define, or to make a model. But without such a > >> definition or model we are in limbo. I think other (consistent) definitions > >> or models are possible, giving a different outcome. > > > > Virgil wrote: > >> Can you suggest one? One that does not ignore the numbering on the balls > >> as some others have tried to do. > > > > David Marcus wrote: > > Models that ignore the number of the balls are certainly models. Whether > > they are reasonable translations of the original problem into > > Mathematics is a separate question. > > I tried modeling the problem with sets: > B_0 = { } > B_1 = B_0 U {1,2,3,...,10} \ {1} > ... > B_n = B_(n-1) U {10(n-1)+1,10(n-1)+2,...,10(n-1)+10} \ {n} > ... > > The resulting "balls in the vase at noon" would then seem to be B_w. > Which is problematic, since there is no ball labeled "w". So this is > probably a bad model of the problem. (I don't claim to be an expert.) I think you are still using the ball labeling. So, even if your model gave an answer for noon, I don't think it is what Virgil asked for. You could define B_w = {x | exists m such that for all n > m, x in B_n}. Another possibility is B_w = {x | for all m, there is an n > m such that x in B_n}. -- David Marcus |