Cantor Confusion
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From: G. Frege on 24 Jan 2007 02:40 On Wed, 24 Jan 2007 00:27:32 GMT, Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: > > To be more explicit, to represent (=address) [any] real [number] in say > [0,1] you would need as many bits for your integers as the reals occupy > But that would require integers with an [...] infinite bit length > e.g. say, the reflection of the reals about the decimal point to give > "numbers" like ...1101 > No. That's nonsense. Consider the representation of the rational number 0.123: 0.123 . Here we do NOT need the integers 123 or 321 to represent it. What we actually use here are the _numerals_ '0', '1', '2', '3' for the digits of 0.123 (and a '.' symbol). Likewise for a the representation of a real number with countable infinitely many digits: 3.1415927... Here each and any digit is one of '0', '1', ..., '9'. F. -- E-mail: info<at>simple-line<dot>de
From: G. Frege on 24 Jan 2007 02:43 On Tue, 23 Jan 2007 21:55:14 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > If that's what he means, then I'll agree he could be close. It isn't a > proof, but as a heuristic it is OK. > Though by using (almost) the same heuristic he might conclude that rational numbers aren't countable too. Well... F. -- E-mail: info<at>simple-line<dot>de
From: mueckenh on 24 Jan 2007 03:16 Franziska Neugebauer schrieb: > You have to choose whether you want to see a tree as > > a) the standard tree, T = (M, E) > b) the "Virgil"-tree. T = (set of paths having certain properties). > > Which definition do you choose? I defined the trees which I use by nodes and edges. 0. / \ 0 1 / \ / \ 0 1 0 1 | | | | 0 1 0 1 | | | | 0 1 0 1 . . . . . . . . . . . . This is a tree of type weeping willow, T(1), with edges 0 and 1. It is easy to see how the general definition of T(n) looks like but it is tedious to write it down here. However, who is unable to understand it, will be unable to follow my thoughts and should refrain from trying so. The corresponding cut tree T(1) is 0. / \ 0 1 We see that the union of two trees is the larger one. We see further that the union is the set theoretic union. And by enumerating the nodes 0 1 2 3 4 5 6 .... we prove that the union of all nodes and, threfore, of all levels and of all finite trees does exist in the original set theoretic meaning. > If so then the set of paths of G is cardinally greater than the > std-union of sets of paths of the trees in the set of all finite trees > V*. The set of paths of a tree is solely given by its nodes and edges. Same nodes and edges result in same paths. Everything else is nonsense. > > What is a segment of paths? A finite initial segment of a path is a finite part of a path from level 0 to level n. > > Since the union of V* is the std-union we know that that U V* > contains only "finite" paths. Correct. Hence 0.[10] is _not_ in UV*. Right words but wrong conclusion. You should say: If 0.010101... exists, then it is in a tree which contains all nodes and, therefore, all alternations 0 to 1 and 1 to 0. Your observation shows that 0.010101... does not exist in such a tree (the union of all finite trees), so it does not exist at all. I include some other posts of you because their contents are very similar to the former one, and therefore, fit together > > You have misunderstood the induction principle. It is not made for > "counting over to the infinite". It is valid for all existing natural numbers. Counting over to the infinite is nonsense. Counting occurs in the finite. > > If tree-union was the std-union of ZFC then by the axiom of union we can > state that U V* is defined. But that requires a commitment to Virgil's > definition of trees, which is not the standard-meaning of trees. Virgil dreams of different sets of paths in one and the same tree. I would not believe in his ideas. > > > If you try to construct the tree with n levels, do you fail at some > > number of levels? No. Therefore the union is defined for every n. > > But not for V*. Recall: V* = { T(n) | n e omega } is the infinite set of > all finite binary trees. It is a set which can be formed if the set of all n exists. > > > > > The union of all finite trees. > > So you implicitly commit to Virgil's trees? I.e. a tree is a set of > paths. The union of all finite trees is the complete infinite tree. It contains the set of all paths which are the unions of initial segments (finite paths of finite trees) because the paths are sets of nodes and proper subsets of the trees. This set is the same set of paths which is in the complete infinite tree T. > > Only under Virgil's definition this union is identical to the > standard-set-theoretical notion of union. If so UV* is not identical to > the full set of paths of the infinite binary tree G given by me in a > elsewhere. Virgil's UV* only comprises finite paths. Every UV* comprises only finite paths and comprises all infinite paths existing. > > > Further, both Virgil and William understand that the union defined by > > me is identical to the complete tree T as far as nodes and edges and > > levels are concerned. > > Nodes, edges and levels. Shaken or stirred? The levels are identical. The nodes are idebtical. The edges are identical. Correct or not? > > > They merely doubt the identity of path due to some inexplicable > > religious belief in a death religion. > > I must admit that great part of the discussion is to subtle for you. You are right. I will never understand how one can be so blind to see more paths in the tree T than in the tree T. > >> A U B = { x | x e {M_A, E_A} v x e {M_B, E_B} } > >> = { M_A, E_A, M_B, E_B } > >> > >> This union is not a tree. > > > > unless M_A, E_A and M_B, E_B have same elements at all levels they > > have in common. > > Of course. This in this special case (a = a U a) you are right. > If A /= B then the A U B is not a tree. Wrong. If A c B or B c A, then A U B is a tree, namely B or A, repsectively. > >> > The nodes can be enumerated in various ways. Here is a very simple > >> > method: > >> > 1 > >> > 11, 12 > >> > 21, 22, 23, 24, > >> > >> A fomal version of this "numbering" I have given in > >> 45b483f8$0$97267$892e7fe2(a)authen.yellow.readfreenews.net Fine. Then you should know it. > >> > >> > ... > > Please take a decision: I did already. > > > I would prefer David Marcus' definition. We can than examine the sets of > paths separately from the trees. I do not read this kind of writers. > > The union of two natural numbers is defined to be the larger one. This > > is a set theoretic union. > > 1. You are in error about what is defined first in ZFC. I did not say what was first. (There is no time in mathematics.) I said what is fact. It is possible to define the union of two finite numbers by the maximum of both. This can be extended to the union of all numbers. So it is with finite trees too. > The union of two > sets a and b is defined > > a U b = { x | x e a v x e b } > > first. Thereafter for two distinct ordinals a and b "larger" is simply > defined as > > a < b := a c b Because the number n is by definition is n = {0, 1, 2, ..., n-1}. This leads unavoidably to n U m = {0, 1, 2, ..., m-1} for m > n. And it leads to Un = {0, 1, 2, ...} = N. Exactly the same situation we have in the tree for the union of two or all distinct trees. > >> The only thing it has in common is the name "union" (equivocation). > >> > >> The extension of your tree-union to V* = { T(n) | n e N } fails as I > >> have pointed out already three times (or even more) because max (N) > >> does not exist. > > > > You can repeat it for 300 times without getting correct. > > If you assume that a max(N) exists you leave contemporary set theory. > Any "contradiction" is then at the expense of yours. I do not assume that max(N) exists. I only assume that N exists.,i.e., all n exist. >From this follows that the union of all levels L(n) exists. >From this it follows that the union of all finite trees exists. > > > > > What about the union of levels? > > What about the union of initial segments of one path? > > > >> Again. U V* is not defined. > > > > Again: I defined it so that everybody can construct the finite union of > > finite trees. > > You have not defined UV* (the union of actually _all_ finite trees). > > > This construction does not come to an end. > > In ZFC there is no time and no processes. You must be writing on > something entirely different. Have you scrapped your plan to show a > contradiction in contemporary set theory? "Infinite" means "not ending", "unending" or "not coming to an end". "Construction" is a notion that belongs to current mathematics. "A construction which does not come to an end" is simply the verbal description of "an infinite construction". Regards, WM
From: G. Frege on 24 Jan 2007 03:15 On Tue, 23 Jan 2007 23:48:47 GMT, Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: >> >> How do you go from "have ...1111" to concluding that "...1111" denotes a >> natural number? We said above what a natural number is. Why do you think >> the string "...11111" identifies or names a natural number? >> > I just meant, that on a simplistic lay view of infinity, that there is > no reason that you shouldn't have an infinite number of digits in a > [natural] number. OK, that isn't correct, but on a simple view of infinity > it seems plausible. Doesn't work with natural numbers. > Funny thing is: There ARE numbers that have such a form! :-) (Though not the naturals of course.) Those numbers are called "p-adic numbers". See: http://en.wikipedia.org/wiki/P-adic_number >>> >>> ... or that the Universe has to have had a beginning (everything >>> has to succeed to something previous, so there must be an origin, so you >>> can't have infinite negative time - but can conceive of an infinite >>> future time, Aristotle again. >>> Aristotle or not. This argument is _faulty_. (Actually, that can be shown in full rigor with help of the means of formal logic.) The integers deliver a model which "falsifies" Aristotle's argument. They just deliver a "counterexample". The claim: "...everything has to succeed to something previous, so there must be an origin..." Now consider the integers (on the number line): obviously every (each and any) integer is the successor of a previous integer. But still there is no _origin_ (of integers) there! >> >> Huh? >> > People cannot conceive of an infinite past ... > Well, that's a _psychological_ question. Neither a logical nor a mathematical one. You shouldn't mix up these things! (Moreover *I* c a n.) > > but are content with an infinite future. > Right. Though why, oh why, shouldn't this ability be "symmetrical" concerning say, the birth of Christ? ... -2 -1 (0) 1 2 3 ... (*) (Actually, there's no year 0.) > > There is something in common here with the natural numbers and the > Peano axioms ... > Well, looking at my picture (*) I'd say, there's something in common here with the integers. (Even better: with the _real numbers_.) > > ... there are no negative numbers ... > Huh? :-o What the heck is -1 ? :-o > > all numbers are the successor of some successor to 0. > Oh, you are talking about the natural numbers. :-) > > Perhaps you would object to the idea of an infinite past on the grounds > that this would imply the existence of infinite integers ... > No, it doesn't imply that. For the same reason why the (countable) infinite sequence of natural numbers 1, 2, 3, ... doesn't imply the existence of an infinite natural number. Consider the two number lines: 0 1 2 3 |---|---|---|--- ... 0 1 2 3 |---|---|---|--- ... (Note that there are no infinite numbers on this lines.) Now change the names (=/= '0') of the first line: X |-> -X 0 -1 -2 -3 |---|---|---|--- ... 0 1 2 3 |---|---|---|--- ... Now turn around the first line: -3 -2 -1 0 ...|---|---|---| 0 1 2 3 |---|---|---|--- ... Finally glue them together: -3 -2 -1 0 1 2 3 ...|---|---|---|---|---|---|--- You see, there's no infinite negative integer on the line of all integers. F. P.S. Of course, if you take it as an axiom that there is no "actual infinity". Then the counterexample from above doesn't work any more. Clearly there isn't such a thing as a "potential infinite" past (since the past is, or was, actual). On the other hand there's a potential infinite future (one might think). -- E-mail: info<at>simple-line<dot>de
From: G. Frege on 24 Jan 2007 03:19
On Tue, 23 Jan 2007 19:45:39 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >> >> People cannot conceive of an infinite past, [...] >> > Why not? I believe that was the usual assumption before the Big Bang > was discovered. > Not really... Remember? "In the beginning God created the heavens and the earth. Now the earth was formless and empty, darkness was over the surface of the deep, and the Spirit of God was hovering over the waters." This happened about 6000 years before Christ's birth, or so. F. -- E-mail: info<at>simple-line<dot>de |