Cantor Confusion
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From: Tonico on 6 Oct 2006 10:55 Albrecht wrote: ............................. > > 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. I understand, why they are cautious. > They fear the defamation they must be aware of if they would be very > concrete in criticism. I have experienced this defamation (not for me > but other persons). There are a lot of people who do not abhor from > defamation and denuncation. This people react about criticism as their > sanity or life is threatened. It's really not understandable. > > If you want I try to find some of this books and papers again and give > you the information about authors and titels. Some may only be > available in German language. > > Best regards > Albrecht S. Storz ******************************************************** It would REALLY be a refreshing thing to read some serious mathematician's MATHEMATICAL objections to Cantor's Diagonal method, or to ANY other part of mathematics. Maths, just like the other sciences, isn't grounded on dogma, and people forwarding REASONABLE, well-based objections, opinions or ideas on whatever are always welcome. The problem is that so far the only objections to Cantor, infinity, etc. that I have read come from people that usually don't even learn mathematics, leave alone have BSc in maths or stuff. Please Albrecht: if you have some SERIOUS source suporting what you say, in ANY LANGUAGE AT ALL (up to and including Bantoo and Boro-Boro), be gracious enough to post it here. Thanx and regards Tonio
From: MoeBlee on 6 Oct 2006 13:06 Albrecht wrote: > MoeBlee schrieb: > > > Albrecht wrote: > > > > > I will try to explain my claim again: There is only a[n] [anti-]diagonal number > > > which is proveable not in the list if there is a real number which is > > > build up out of all the real numbers in the list. But for an infinite > > > list you can't end the diagonal number. You may have a sequence which > > > converge. But you never have a limit. > > > > Then you're not talking about the reals. We prove that every convergent > > sequence of rationals converges to a unique real number, which is the > > limit of the sequence. If it converges, then it has a limit. > > Who had proved this for arbitrary infinite sequences of decimal digits? The only way a sequence S of DIGITS can converge is if there exists an n such that for all k>n and all j>n, S(k) = S(j). You do understand that that's not what's involved in my remarks, right? > > Thus every > > countable decimal expansion represents a real number. In particular, > > the anti-diagonal of any given countable sequence of denumerable > > decimal expansions is a countable decimal expansion and thus represents > > a real number. And we prove it does not represent any real number in > > the given countable sequence of decimal expansions. > > > > Argue that you don't like the axioms of set theory, if you like. Argue > > that you don't ascribe to the principles of reasoning codified by first > > order logic, if you like. But it is not in any way rationally arguable > > that the uncountability of the reals is not a theorem of the stated > > axioms. And more simply, there is no theory that you have proposed in > > which there exists a function from the counting numbers onto the > > carrier set of a complete ordered field. > > > > > Cantor argues that you must not have the limit. > > > > We must not have the limit of what? And where does Cantor argue this? > > The limit of the antidiagonal. The antidiagonal itself may not have a limit, since the antidiagonal itself is just a sequence of digits. But the sequence of rationals that is represented by the antidiagonal has a limit. That's what matters. Every sequence of digits represents a convergent sequence of rationals, and the limit of that sequence is a real number. You should not conflate the actual sequence of digits with the sequence of rationals that it represents. > > They axiomatize ordinary calculus used for science and engineering. > > Moreover, mathematical logic and set theory were and are very useful in > > the development of the digital computer, as you are using such a > > digital computer to declare that these are ideas are not very useful. > > > > Without axiom of infinity no computers? I didn't say that. But much of the theory that has enabled advances in computing and even the invention of modern digital computers does use mathematical logic and set theory that make use of the set of natural numbers as a set, in particular as the domain of certain functions. Moreover, ordinary calculus also uses the set of natural numbers. MoeBlee
From: georgie on 6 Oct 2006 13:46 Tonico wrote: > Albrecht wrote: > ............................ > > > 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. I understand, why they are cautious. > > They fear the defamation they must be aware of if they would be very > > concrete in criticism. I have experienced this defamation (not for me > > but other persons). There are a lot of people who do not abhor from > > defamation and denuncation. This people react about criticism as their > > sanity or life is threatened. It's really not understandable. > > > > If you want I try to find some of this books and papers again and give > > you the information about authors and titels. Some may only be > > available in German language. > > > > Best regards > > Albrecht S. Storz > ******************************************************** > It would REALLY be a refreshing thing to read some serious > mathematician's MATHEMATICAL objections to Cantor's Diagonal method, or > to ANY other part of mathematics. Why? So you can pick on notation or some other aspect that has nothing to do with serious objections? The only reponses that are in this thread seem to be notational objections or the "You're wrong because lots of mathematicians say so" response. There isn't anything here that shows a flaw in the OP's idea. > Maths, just like the other sciences, isn't grounded on dogma, and > people forwarding REASONABLE, well-based objections, opinions or ideas > on whatever are always welcome. That's what all fanatics say. > The problem is that so far the only objections to Cantor, infinity, > etc. that I have read come from people that usually don't even learn > mathematics, leave alone have BSc in maths or stuff. BS! You don't know that. That's called rationalizing. You haven't seriously considered even trying to understand ANY argument about Cantor's diagonal proof that doesn't align with your own thoughts.
From: Tonico on 6 Oct 2006 15:15 georgie wrote: ............................................................... > BS! You don't know that. That's called rationalizing. You haven't > seriously considered even trying to understand ANY argument about > Cantor's diagonal proof that doesn't align with your own thoughts. ********************************************************************************* Ts,ts,ts...Georgie boy, common! Now you behave like a little kid making a scene because he/she doesn't get her/his way. So I haven't SERIOUSLY considered any argument, uh? Well, well...as an argument THAT is weak, but hey: it's your argument. ;) On the other side, of course, one has to consider: how could anyone having a brain virgin of any mathematical education and without any concern for logic, sound reasoning and intellectual honestity be able to express herself otherwise? Georgie, I see your point. Regards Tonio
From: georgie on 6 Oct 2006 15:25
Tonico wrote: > georgie wrote: > .............................................................. > > BS! You don't know that. That's called rationalizing. You haven't > > seriously considered even trying to understand ANY argument about > > Cantor's diagonal proof that doesn't align with your own thoughts. > ********************************************************************************* > Ts,ts,ts...Georgie boy, common! Now you behave like a little kid making > a scene because he/she doesn't get her/his way. So I haven't SERIOUSLY > considered any argument, uh? > Well, well...as an argument THAT is weak, but hey: it's your argument. > ;) I used your style. So mostly its your argument. > On the other side, of course, one has to consider: how could anyone > having a brain virgin of any mathematical education and without any > concern for logic, sound reasoning and intellectual honestity be able > to express herself otherwise? Georgie, I see your point. Sounds like the establishment to me. I'm sure you don't see any points, however. |