
#37
May912, 03:12 PM

P: 1,781





#38
May912, 03:35 PM

P: 1,583





#39
May912, 04:00 PM

P: 606

Check this: as already noted, by "list" one has to understand "elements of an infinite countable set". Next: nobody adds anything to anything. Upon getting any list, one can construct a number which is not in that specific list. Next: perhaps you think you've arrived to a great insight when you use exclamation marks, but I honestly can't see it. Your example of a finite list with three elements shows nothing as I can construct easily a number not in it without even using the diagonal trick. Next: I don't care, nor any other mathematician would, what's the first element in such a list. It can be 0, 1, 0.5 or whatever. Anyway, after you're done with your list, I can always construct a number not in it. DonAntonio 



#40
May912, 04:06 PM

P: 606

This is what Cantor's theorem states: the set of real numbers is infinite yet uncountable, i.e. it cannot be ennumerated. That's why. Either you're addressing something else or you're completely lost in this matter: neither Cantor nor anyone else trying to prove this theorem "forms" any list at all. We assume such a list exists AND THEN we prove that there's always a real number not belonging to the list. I would never tell you to free your mind, but I' definitely tell you to learn some mathematics. If you want to understand this stuff, that is. DonAntonio 



#41
May912, 04:30 PM

P: 799

Fascinating to see the thread degeneration here. Any discussion of antiCantor cranks draws antiCantor cranks.
Purely from a behavioral point of view, the Cantor deniers and the Cantor denier refutors seem equally obsessive. The fact that one group is mathematically correct and the other not, is irrelevant. Because after all, most people manage to avoid these discussions altogether. And now you can see why. They always, always, always end up in exactly the same place. What I used to like about sci.math on Usenet was that at least there, you could toss in gratuitous personal insults. Here you can't do that. So it's much less fun to tease and torment the deniers. 



#42
May912, 04:32 PM

P: 31

Similarly any decimal expansion 0.a1a2a3... where a1, a2, a3,... are digits 09 is short for the series a1*10^1 + a2*10^2 + ... which converges to a real number between zero and one no matter what the digits are chosen to be. 



#43
May912, 05:28 PM

P: 606

Indeed. For the time being the thread hasn't degenerated that much, imo. I won't get into any crankbashing or crankeducating rant here. For that we have sci.math. So far, though, Antiphon is not a crank but someone with some doubts and some rather deserted areas in his/her mathematical education...for now. If and when he, or anyone else, slip into crankhood I, for one, shall bail out of the thread. DonAntonio 



#44
May1012, 01:04 PM

P: 31





#45
May1012, 01:18 PM

P: 799





#46
May1112, 01:43 AM

P: 1,781

I'm still here. Been traveling coast to coast.
I'm quite sure I'll never become a crank on this issue. As a nonmathematician though I can see how someone would become one. Maybe if this issue can be zoomed in on, many future cranksinwaiting might be saved. Please lead me down the reasoning path. I'll do my best to explain using the language I have available. Proofs by contradiction make sense. You make an assertion or assumption that may or may not be true, then you follow up with some valid deductions based on the assumption. If your subsequent deductions are valid but you arrive at a contradiction or falsehood, then the original assumption was false. This is proof by contradiction as I understand it. For example (and I'm making this up on the fly) lets suppose that division by zero were legitimate arithmetic. I can probably form some simple algebraic expessions which would result in a statement like 1=2. Nobody should have a problem with such a proof. But if you start a proof with 1=2 and then proceed to do valid algebra with it, the contradiction doesn't arise from the proof but is built in at the beginning. I can't speak for any AntiCantor cranks but for me this is an issue. A few posts back MBS says that the proof of the irrationality of sqrt(2) can begin by assuming the existence of two integers m and n such that n^2/m^2=2. You then perform valid reasoning on this and arrive at absurd conclusions. That's great. I don't have trouble with that because the expression above is legitimate algebra, it just so happens that no two integers will satisfy it. But I'd bet it's not legitimate logic to start with the absurd result and reason your way backward to the expression above. What am I missing? 



#47
May1112, 01:56 AM

P: 4,570

This is important not just mathematically but also psychologically because when most people start off disproving something, in the back of their mind they assume that what they are proving is undoubtedly true which ends up screwing up their analysis, proof and way of thinking whereas the above method psychologically says "OK this is what you said, let's go along with this and see what happens" which is a much better approach because mentally you are saying "I'm going to disregard my own prejudices for the moment and I'm going to assume that you are right". It's very subtle, but it's so important as a logical tool and I'm afraid it's not used as much as it could (and should) be. You definitely have the right approach and mindset for analyzing things not only mathematically, but in general situations overall. 



#48
May1112, 02:03 AM

P: 1,623





#49
May1112, 04:00 AM

P: 250

Is this a closed subject in the mathematical world? I ask because I have found this
http://en.wikipedia.org/wiki/Controv...tor%27s_theory And sure everyone is entitled to have an opinion but I'd like to know if experts logicians have reached an agreement on this. 



#50
May1112, 04:22 AM

P: 606

This is one of the instances where Wiki, a very good source of immediate though generally not deep and sometimes even unreliable knowledge, can mislead. This article begins with the following: "In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has found some acceptance in the mathematics community, it has been criticized in several areas by mathematicians and philosophers." The words "some acceptance" are unduly and unjustly misleading: infinite sets, in this or that acception, are widel accepted by an overwhelming majority of mathematicians. Period. Now, the controversy exists within very narrow and, if may I add, unimportant frames and individuals, and it surely isn't something that, as far as we know right now, would affect in some dramatic way neither the development of most of mathematics nor most of its applications to other sciences, technology and/or the "real" world, whatever that is. Back again with cranks: these persons are characterised by an inner and utterly unjustified certainty that they are right and ALL the others are wrong, even when they are NOT mathematicians (99% of the cases) and the others are. DonAntonio 



#51
May1112, 04:51 AM

P: 250

OK, I just thought up this counterargument
Let's imaging it does exist a list of all real numbers such that they have only zeros in the decimal part. The list will look something like: S 1 xxxxxxxxxxxxxxxxxxx.00000000000000000 2 yyyyyyyyyyyyyyyyyyy.00000000000000000 3 xyxyxyxyxyxyxyxyxyx.00000000000000000 4 yxyxyxyxyxyxyxyxyxy.00000000000000000 Then we try to construct S0 using a different digit from the Diagonal yxyy... .00000000000000000 Then S0 cannot possibly be in that list, therefore that list cannot exist... but since the list have only zeros in the decimal places that list is equivalent to the Natural numbers which we know we can count, therefore the diagonal argument makes no sense. What is it that I did wrong? 



#52
May1112, 07:20 AM

P: 95

As an aside, you don't need to phrase the diagonal argument as a contradiction, you can just use it to show that any function from the naturals to the reals must fail to be onto.




#53
May1112, 07:25 AM

P: 95





#54
May1112, 08:34 AM

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