Cantor Confusion
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From: G. Frege on 23 Jan 2007 13:06 On Tue, 23 Jan 2007 12:58:51 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > Why are you reading this thread? Is it sensible to do so? > Well, actually I'm ONLY "reading" some answers to WM. After all, I might learn something. (I'm still a beginner concerning set theory.) F. -- E-mail: info<at>simple-line<dot>de
From: David Marcus on 23 Jan 2007 13:17 Andy Smith wrote: > G. Frege <nomail(a)invalid.?.invalid> writes > >On Tue, 23 Jan 2007 10:45:22 GMT, Andy Smith > ><Andy(a)phoenixsystems.co.uk> wrote: > > > >> But since you cannot have an actually infinite natural number, you > >> cannot have an actually infinite list ... > >> > >Huh? Where did you get that nonsense from. (Did you read one of > >M�ckenheim's papers? :-? > > > No, who he? M�ckenheim is mueckenh(a)rz.fh-augsburg.de. You may have noticed one or two posts from him. -- David Marcus
From: Dave Seaman on 23 Jan 2007 13:27 On Tue, 23 Jan 2007 17:15:52 GMT, Andy Smith wrote: > stephen(a)nomail.com writes >>Cantor's argument simply says that given a list of real numbers, finite >>or infinite, there exists a real not on the list. That is all. >>An immediate consequence is that there does not exist a list >>that contains all the reals. >> >>> OK, well I do see the argument better now, but if that was an argument >>> that I had suggested for the first time, you would be on me like wolves >>> ... >> >>You still seem to be missing it. > Maybe. The issue wasn't with a finite list, it was whether you could > have an infinite list when all the indices of the rows i.e. all natural > numbers, must be finite ... resolved by considering the list as "an > infinite set" just as the set of "all natural numbers" can be considered > as "an infinite set", even though no member of the natural numbers are > infinite; "an infinite set" is an abstract mental concept. By definition, a set is "finite" if it has the size of some natural number. If a set isn't finite, then it's called "infinite". It's obvious that the set of all natural numbers can't be finite, since that would imply the existence of a largest natural number. > If you see this as straightforward it is because your mindset has been > conditioned by your education to see this as normal. I can safely say > that if your concepts of infinite sets was placed in front of the > population at large 99 % would think that this is barking mad > doublethink ... > Cantor provided a perspective to view infinity, and his insight > underpins, as I understand it, modern set theory. It may be consistent, > but I don't see that the philosophical rational is trivial - and, as I > understand it, in Cantor's day there were many eminent and far from > stupid mathematicians who couldn't get a handle on it. From posts on > this site I can see that their descendants are still here and active ... > If I was asked to sum it up, at present I would say that my > understanding is that you can't have an actually infinite integer, but > reals can be defined as having an actually infinite binary > representation .. (with apologies for the adjective "actually"). So no > surprise that the reals are "uncountable". You can't have an actually infinite integer for exactly the same reason that you can't have an actually 200-cm. meter or an actually 4-sided triangle. It's not that we can't imagine an actually infinite cardinal, an actually 200-cm. object or an actually 4-sided polygon, it's just that such things are by definition not actually examples of the previously defined word. -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. <http://www.mumia2000.org/>
From: imaginatorium on 23 Jan 2007 13:43 Andy Smith wrote: > Fuckwit <nomail(a)invalid.?.invalid> writes <snip> > Well I understand what you are saying, but, sorry to be dim, I still > don't see it. No need to apologise. Dogged, but genuine persistance will get you everywhere. > There is no greatest natural number. Right. A very good start! > So I can construct a finite ordered > list of natural numbers, labelled 1 to n, and identify a number not in > the list, and label that n+1. I can do that for all n. Careful! You can do that FOR ANY n. (Any list including all of the natural numbers would never end, so could not be labelled from 1 to anything.) > Cantor's hypothetical numbered list of the reals is also finite ? Not necessarily. Cantor's diagonal proof applies to _any_ list of reals, ending or unending. > So Cantor's > construction for a list with n elements just generates another real, > which he can insert as the n+1 th row, and he can do that for all n? Not exactly. What is 'n' here? Mostly we are concerned (since _obviously_ the reals can't be put in an ending list) with an unending list - the proof just shows that whichever list you choose, it is always incomplete. > As I had previously understood it Cantor's argument relied on a > hypothetically complete set of reals, with an actually infinite number > of rows, and then showing that there was a real not included in the > actually infinite list. But since you cannot have an actually infinite > natural number, you cannot have an actually infinite list, so the > argument is invalid? (I am sure that it isn't, but that is what I am > trying to understand). The expression "actually infinite" only occurs in sci.math threads as part of crank babbling (like Mueckenheim's) or genuine confusion (like yours). Avoid it, and you may find you've understood something. No, no natural number is "infinite", more or less by definition. (You would have to explain exactly what you meant by an "infinite natural number", and explain how the set of natural numbers - all successors of each other - came to be divided into two parts, 'finite' and 'infinite'. ( A challenge that still awaits an answer from a number of cranks hereabouts.)) What is an "actually infinite list"? Suppose you defined it as a list including an "actually infinite integer", then no, I don't think you could have such a list. But this has nothing to do with the natural numbers as mathematicians define them. They can all be put on a list: start with 1 (or 0 if that's your preference), followed by 2, 3, 4, ... 56, 57, ... 13646745344, ... and so on. The obvious feature of this list is that it does not end (another fairly obvious feature is that it does indeed include every natural number). This unendingness (unendlichkeit in German, if I'm not mistaken) is usually called "infinite" in English, because we like to make our words from hashed up bits of other people's languages instead of our own, the sensible German or Chinese way, but all "infinite" means is in- [negative prefix] finite [finished, ended]. So this list includes only perfectly finite, natural numbers, yet it goes on without end. So if you really grok what you said yourself - "There is no greatest natural number." - then you should agree that your later thought: "But since you cannot have an actually infinite natural number, you cannot have an actually infinite list" is, to be blunt, nonsense (not really wrong, just confused). I've been watching various cranks here for some time now - in a gruesome sort of way, it's actually fascinating. I think one of the commonest themes running through what they say is a (mis)conception that "infinity" is a sort of number, very very big number, bigger than any number you'd ordinarily think of as a number, but anyway it's at the end of (e.g.) the list of natural numbers. One or two of the less coherent [if that's not an antoximoron] of the cranks just mumble that "actual/completed infinity" is nonsense - which is true, I suppose; but this is not a variety of "infinity" that mathematicians talks about. If you can force yourself to talk only of (e.g.) bitstrings with two ends, or lists of numbers that are unending, you may well clear up your own confusion. Hope this helps. Brian Chandler http://imaginatorium.org
From: David Marcus on 23 Jan 2007 13:47
Andy Smith wrote: > stephen(a)nomail.com writes > >Cantor's argument simply says that given a list of real numbers, finite > >or infinite, there exists a real not on the list. That is all. > >An immediate consequence is that there does not exist a list > >that contains all the reals. > > > >> OK, well I do see the argument better now, but if that was an argument > >> that I had suggested for the first time, you would be on me like wolves > >> ... > > > >You still seem to be missing it. > > Maybe. The issue wasn't with a finite list, it was whether you could > have an infinite list when all the indices of the rows i.e. all natural > numbers, must be finite ... resolved by considering the list as "an > infinite set" just as the set of "all natural numbers" can be considered > as "an infinite set", even though no member of the natural numbers are > infinite; "an infinite set" is an abstract mental concept. Let's ignore sets and start at the beginning. The natural numbers are the following numbers: 0 1 2 3 .... The dots mean we keep going "forever". All of these numbers are finite. Probably we should just take that as the definition of the word "finite". So, saying a "finite natural number" is redundant. Now, I ask you: how many natural numbers are there? One? Two? Five? Ten? One hundred? Which natural number could be the answer to this question? Any of them? If your answer is the natural number n, this can't be right because all of 0,1,...,n are natural numbers and we see that there are n+1 of them. So, the answer to the question "How many natural numbers are there?" has to be "not a finite number" or "infinite". Note that we haven't said anything about sets. > If you see this as straightforward it is because your mindset has been > conditioned by your education to see this as normal. I can safely say > that if your concepts of infinite sets was placed in front of the > population at large 99 % would think that this is barking mad > doublethink ... Maybe. Maybe not. You might ask some of your friends. You might be surprised. I think that what I wrote above about natural numbers would be intuitively clear to a significant fraction of the population. And, I think they would also find what I write below believable. > Cantor provided a perspective to view infinity, and his insight > underpins, as I understand it, modern set theory. It may be consistent, > but I don't see that the philosophical rational is trivial - and, as I > understand it, in Cantor's day there were many eminent and far from > stupid mathematicians who couldn't get a handle on it. From posts on > this site I can see that their descendants are still here and active ... Well, the number of cranks is actually quite small. As for Cantor, his great idea was that it made sense to compare sets by whether you could biject or inject them. Before him, people thought that this idea didn't work. However, I think we can understand the fact that (in some sense) there are more reals than naturals without mentioning sets. A list is a bunch of rows where each row is labeled with a natural number, e.g., 0. adfna 1. afdkj 2. dfajhadf 3. adfj;df .... The question is can we construct a list such that every real number in [0,1] appears on the list? We argue as follows. Let's agree to write real numbers in [0,1] as infinite decimals. Some infinite decimals have two representations, one ending in 0's, the other ending in 9's. Let's agree to use the one ending in 0's, except let's use 0.999... for 1. Suppose we have a list of reals: 0. 0.1234... 1. 0.4893... 2. 0.3839... .... For convenience, let's number the digits (after the decimal point) starting with zero. We construct a real x as follows. Let the n-th digit of x be 4 if the n-th digit of the n-th number on the list is 5. Otherwise, let the n-th digit of x be 5. Is x on our list? Well, it can't be the 0th number because it has a different 0th digit. It can't be the 1th number because it has a different 1th digit. In fact, it can't be any of the numbers. So, it isn't on our list. (Note that x can't be equal to a number on our list without the digits matching because the only way it could do that would be if it ended in 9's.) Therefore, given any list of real numbers in [0,1], we see there is a real number in [0,1] that is not on the list. Hence, no such list can be complete. Notice that I haven't used the word "set" or said anything about the "sizes" (or "cardinalities") of "infinite sets". > If I was asked to sum it up, at present I would say that my > understanding is that you can't have an actually infinite integer, What in the world would an "actually infinite integer" be and where did you get the idea there could be such a thing? It seems so counterintuitive. > but > reals can be defined as having an actually infinite binary > representation .. (with apologies for the adjective "actually"). Isn't it intuitive that the decimal representation for 1/3 "never ends"? > So no surprise that the reals are "uncountable". On the contrary, I think the fact that the reals are "uncountable" is very surprising. The algebraic numbers are countable. -- David Marcus |