Cantor Confusion
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From: stephen on 23 Jan 2007 11:11 Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: > So Cantor's argument doesn't rely on locating an actually infinite > member of his list as some limit process n->oo, he just posits that an > actually infinite list can exist and is complete, and then shows it > can't be (or possibly alternatively that an actually infinite list can't > exist?) 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. Stephen
From: Andy Smith on 23 Jan 2007 11:19 G. Frege <nomail(a)invalid.?.invalid> writes >> >> - you can consider an actually infinite set but no >> actually infinite member of it. A key distinction. >> >Sorry, can't follow you here. > >But it seems to me that you are going astray, again. :-) > All that I meant was that one consider e.g. the set of all natural numbers ( as a mental construct, I can say/imagine/visualise that). But, there is no infinite natural number. So you can have an infinite set N but no infinite member of it. Aristotle would not have liked the statement "the set of all natural numbers" because that is a completed infinity. Actually Zeno helps me here because when I see the series 1/2 + 1/4 + 1/8 ... I can see that it has an actually infinite number of terms but yet sums to 1, so that is my "infinity in a bag". A psychological prop. >> >I was right. :-) > I never doubted that ! -- Andy Smith
From: Andy Smith on 23 Jan 2007 12:15 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. 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". :) -- Andy Smith
From: MoeBlee on 23 Jan 2007 12:17 Andy Smith wrote: > There is no greatest natural number. 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. > > Cantor's hypothetical numbered list of the reals is also finite ? Given many usual formulations of the argument, usually the list is not finite, but rather it is denumerable. We can suppose the list is not finite since it is trivial that the set of real numbers is not finite, so to prove that the set of real numbers is uncountable it suffices to show that the set of real numbers is not denumerable. Thus, we might as well consider an arbitrary denumerable list of real numbers and show that there is at least one real number not on the list. > 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? If the list is a finite list of denumerable sequences, then the anti-diagonal would be a finite sequence (you can't "stretch" past what is given, so there's not actually a "true" diagonal nor anti-diagonal (as long as 'diagonal' and 'anti-diagonal' are supposed to be infinite). > As I had previously understood it Cantor's argument relied on a > hypothetically complete set of reals, No, we do NOT have to suppose the list is of all real numbers. We can suppose it is a countable list (finite or denumerable) or, we can just cut the finite case since it is trivial, and suppose the list is denumerable. THEN we PROVE that the list is not of all real numbers. > with an actually infinite number > of rows, Yes, a denumerable number of rows. (But don't forget we don't need to talk about things like "rows". All we need is a denumerable list of denumerable sequences.) > and then showing that there was a real not included in the > actually infinite list. Right. > But since you cannot have an actually infinite > natural number, you cannot have an actually infinite list, No, we can have an infinite list of natural numbers. But, unlike a real number, a natural number is ITSELF not representable as a denumerable sequence in the way that a reals is representable in such a system of representations. Isn't it clear that the naturals are represented by FINITE sequences of digits while the reals are represented by DENUMERABLE sequences of digits? > so the > argument is invalid? (I am sure that it isn't, but that is what I am > trying to understand). The argument breaks down for natural numbers since the "rows" are not infinite in length, and even if we arrange so that there is a diagonal cutting down infinitely, the anti-diagonal is a real number (a denumerable binary sequence) and not a natural number. So the argument applied to natural numbers would just show that for any list of natural numbers we can construct an anti-diagonal that is a real number, which does not prove that the list misses any NATURAL numbers. MoeBlee
From: G. Frege on 23 Jan 2007 12:21
On Tue, 23 Jan 2007 16:19:27 GMT, Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: > > All that I meant was that one consider e.g. the set of all natural > numbers (as a mental construct, I can say/imagine/visualise that). > But, there is no infinite natural number. > Ok. > > So you can have an infinite set N but no infinite member of it. > Right. I have pondered about that question some time (motivated by the inability of many cranks to comprehend that "state of affairs"). Many cranks (M�ckenheim for example) claim that (from a logical point of view) it's not possible for a set of natural numbers to be infinite if only containing finite numbers. Well, let's see... Assume (for the sake of the argument) that in the set of _all_ natural numbers, denotes by IN*, there were infinite natural numbers (though I don't have the slightest idea how such numbers would look like). Now let's construct a set IN the following way: IN = {n e IN* | n is a finite number}. Then IN is the set of all finite natural numbers. It MUST be, since it is constructed that way. You might think of it the following way too: IN = IN* - {n e IN* | n is infinite} IN is the set of _all_ natural numbers /minus/ (set theoretic minus) _all_ the infinite natural numbers that are in IN*. Hence IN is the reminder: the set of all natural numbers that are not infinite, hence finite. Now the big question: Is IN finite then? C a n it be finite? If the cranks were right, it _should_ be! After all it consist of exactly all the natural numbers that are finite. You might try to find the answer to this question yourself. > > Aristotle would not have liked the statement "the set of all natural > numbers" because that is a completed infinity. > Well, actually we don't really know, since Aristotle didn't know /set theory/ (as we know it). Think about it: Concerning _physical_ objects (our physical reality) Aristotle might have a point (who knows?). On the other hand, the existence of set theory SHOWs that it is possible to think in coherent ways about "the infinite" (regardless of the question if Aristotle would like it or not). > > Actually Zeno helps me here because when I see the series 1/2 + 1/4 + > 1/8 ... I can see that it has an actually infinite number of terms ... > Though modern math (->Real Analysis) successfully eliminated exactly those type of reasoning/argumentation. :-) Usually in math we DON'T consider an infinite number of terms. F. -- E-mail: info<at>simple-line<dot>de |