Cantor Confusion
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From: Andy Smith on 23 Jan 2007 15:22 imaginatorium(a)despammed.com writes >> >> 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. > >If you want to know what it feels like to be labelled a "crank", just >start going on and on about mindsets being "conditioned". (If there is >such a thing as "feeling" after your head has been bitten off.) Yeah, >yeah, we have all memorised this stuff, which we mumble to ourselves at >breakfast to make sure we don't forget... > Actually you miss the point. I wasn't saying that this was wrong, just that you (i.e. mathematicians in general) had achieved a mental perspective not commonly shared, and that this perspective was not an intuitive one ... > > >> 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". > >And if you want to make progress, have respect for technical terms. >"Uncountable" means something very specific. Does 1/7 have an "actually >infinite binary representation" in your terms? Decimal fractions for >"most" rational numbers go on without ending - but the rationals are >not uncountable. > Sure. But (almost) all real numbers have a random infinite bit sequence, and any one instance of those requires an infinite number of bits to specify it (is what I meant). -- Andy Smith
From: Virgil on 23 Jan 2007 15:29 In article <1169550000.063305.137070(a)s48g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > > > Your tree-union of two trees is _defined_ to be the tree having the > > greatest of both trees depths. This is not a set-theoretical union. > > It is, because in the union every element appearing twice is > eliminated. If tree1 is (m1, e1) = { m1, {m1,e1)) and tree2 is (m2, e2) = { m2, {m2,e2)), where mi is the ith set of nodes and ei is the ith set of edges, then tree1 U tree2 = {m1, m2, {m1,e1}, {m2,e1}}, which is not even an ordered pair, much less a tree. While the amalgamation of several trees into a single tree that WM is suggesting is possible, it is not possible by forming a union of the separate trees. > > Again: I defined it so that everybody can constuct the finite union of > finite trees. To call it a union, when it cannot be a union, is misleading. .. > > And even if this were never heard of in set theory. Why shouldn't we > consider it as a new topic? Give this construction a name that is not already in use in a conflicting way.
From: Andy Smith on 23 Jan 2007 15:32 In message <1169577812.281281.78450(a)l53g2000cwa.googlegroups.com>, imaginatorium(a)despammed.com writes >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). Too much Aristotle. Yes, I can conceive of the "set of all natural numbers" as an entity, and see that it has no greatest member. An infinite set is different from it's constituents, as an abstract concept. And I can now see that (1/2,3/4,7/8, ...) forms an infinite sequence with no greatest member. I am thinking of changing my first name to "Winston" by deed poll. > >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. > Thanks. Got sorted out by various other denizens as well... Cheers -- Andy Smith
From: David Marcus on 23 Jan 2007 15:38 Andy Smith wrote: > imaginatorium(a)despammed.com writes > >> > >> 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. > > > >If you want to know what it feels like to be labelled a "crank", just > >start going on and on about mindsets being "conditioned". (If there is > >such a thing as "feeling" after your head has been bitten off.) Yeah, > >yeah, we have all memorised this stuff, which we mumble to ourselves at > >breakfast to make sure we don't forget... > > > Actually you miss the point. I wasn't saying that this was wrong, just > that you (i.e. mathematicians in general) had achieved a mental > perspective not commonly shared, and that this perspective was not an > intuitive one ... But we disagree. We think it is very intuitive. I think if you can free yourself from your misconceptions and go back to what you thought when you were younger, you would find it intuitive, too. > >> 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". > > > >And if you want to make progress, have respect for technical terms. > >"Uncountable" means something very specific. Does 1/7 have an "actually > >infinite binary representation" in your terms? Decimal fractions for > >"most" rational numbers go on without ending - but the rationals are > >not uncountable. > > > Sure. But (almost) all real numbers have a random infinite bit sequence, > and any one instance of those requires an infinite number of bits to > specify it (is what I meant). Why should that observation have anything to do with whether the reals are countable or uncountable? -- David Marcus
From: Virgil on 23 Jan 2007 15:41
In article <1169550664.526893.323320(a)l53g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1169472355.486545.309380(a)q2g2000cwa.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > Cantor, using his original diagonal argument, and anyone else who wants > > > > to emulate him, can show that any /list/ of paths in a complete infinite > > > > binary tree is incomplete. > > > > > > So you do not believe in complete trees? > > > > Non sequitur. What I believe is that complete infinite trees always have > > a set of paths which is actual but unlistable. > > > We talked about Cantor's diagonal method. You said one could apply it > in the complete tree. Actaully, Cantor himself applied it to a model of the complete tree even before it was applied to the reals. The set of paths of the complete tree corresponds in an obvious way to the set of all infinite binary strings, with left branchings represented by 0's and right branchings by 1's. Call this set of all such binary strings S. Given any listing of infinite binary strings (function from the set of naturals to the set of all such strings), say f:N -> S, then for every n in N, f(n) is the nth string. Define the binary string b so that for every digit position n, the nth digit of b is (1 minus the nth digit of f(n)). Then b =/= f(n) for every n in N. So that every listing is incomplete, and the whole set is unlistable, or, equivalently, uncountable. Until WM can provide a complete listing of the set of all endless binary sequences, the above proof but Cantor himself remains valid. |