An uncountable countable set
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From: Tony Orlow on 19 Aug 2006 18:12 mueckenh(a)rz.fh-augsburg.de wrote: > imaginatorium(a)despammed.com schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: >>> Franziska Neugebauer schrieb: >>> >>> >>>>>>> Therefore there are not infinitely many difference[s] of 1 >>>>>>> between natural numbers. >>>>>> Your consequent is proven false (see below). Therefore your >>>>>> implication is false, too. >>>>> You are in error. >>>> Where precisely is the error? >>> The assertion that infinitely many differences of 1 can be provided by >>> finite natural numbers. > > Why? Here it is: The staircase is not an independent individual, but it > is nothing than the set of stairs. Without a stair surpassing the > height H the staircase cannot surpass height H. Without a stair > surpassing every finite height the staircase cannot surpass every > finite height. (surpassing every finite = infinite). But the length > surpasses every finite length. >> You've been repeating this moronic nonsense for so long it's obviously >> hopeless, but just answer me a simple question. In your view of these >> "finite natural numbers", I presume 1 is included, and 2, and 3, and so >> on. If you were to simply count them, 1, 2, 3, 4, and so on, never >> stopping unless you came to an end, do you in fact think you _would_ >> reach an end? Any suggestions as to what this end would look like, >> given that it would entail a natural number such that adding 1 somehow >> failed to happen. > > No. This moronic nonsense would only be necessary if the complete set > of natural numbers would actually exist and have a cardinal number > larger than any natural number. > > Regards, WM > The best we can do in assuming any infinite number is to assign a value to the number of points in the unit interval, and try to use that for proper measure, and see if it works. It does. :) TO
From: Virgil on 19 Aug 2006 18:43 In article <ec7oro$qmd$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > In article <ec7gek$frg$1(a)ruby.cit.cornell.edu>, > > Tony Orlow <aeo6(a)cornell.edu> wrote: > > > > > >>> You said that a set exists if all its elements do exist. If so, then > >>> all natural numbers do exist and make up an infinite number without one > >>> of them being infinite. > >> Inductively provable: Any set of consecutive naturals starting at 1 > >> always has its largest element equal to size of the set. > > > > Equally provable: Any finitely bounded set of consecutive ordinals > > starting with the first always has a largest element less than its > > "size". > > And, your position is that, because the ordinals start with 0, rather > than 1, the addition of that single element makes the set larger than > any finite element contained therein? It merely points out the illogic of TO's argument. If one is to argue from the finite to the infinite, as TO so often does, one must consider all finite possibilities, not merely the ones which support your own particular conclusion about infinity. Not that TO has also ignored all these finite cases in which the largest natural in a set is greater that the number of element in that set. And there re a lot more of these that sets where the largest equals the number of members. So it would seem that there is not enough of a correlation between set size and maximum members to draw any conclusions.
From: Virgil on 19 Aug 2006 18:47 In article <ec7p2k$quv$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > > In any case, there are lots of injections, f, from N to P(N). > > > > Consider f(n) = {n}, for example. > > In which case one easily sees that > > M(f) = {}. > > > > Which exists quite nicely, thank you very much! > > But Monsieur, what about the injection from P(N) into N I am not familiar with that particular pipe dream of TO's. When TO can show an injection from P(S) to S for any finite set S, only then will he have any grounds for suggesting such an injection for infinite S's.
From: Virgil on 19 Aug 2006 18:57 In article <ec7pmg$rj2$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > The sequence of edges leading up to any edge is finite. So a finite > > sequence of 0's and 1's, 0 for left child, 1 for right child, uniquely > > identifies each edge, but it takes an endless sequence of 0's and 1's to > > identify any endless path. > > > > Thus in infinite binary trees, the set of edges is countable but the set > > of endless paths is not. > > Ah yes, we've been over this. Since one extra path is created for every > two edges added to the tree, there are clearly half as many paths as > edges (floored, for a balanced tree, of course). That only holds for finite trees and for terminal edges in such trees. But in infinite trees there are no terminal edges, so TO's analysis again fails. > It is unconscionably > stupid to regard the number of paths as being "uncountable" and > therefore greater than the "countable" number of edges, when clearly > there are fewer paths than edges once the first three nodes are added to > the root node, or binary point. Does To then choose to ignore that there is an explicit bijection between the set of paths and an uncountable set and an explicit bijection between the set of edges and a countable set? >The proof you have offered in the past > is dishonest in the sense that it bijects the edges with the naturals > using one interpretation of the tree, and then bijects the paths with > the reals using another incompatible interpretation. Until TO can show that either "interpretation" says anything false about infinite binary trees or the bijections described, his complaint is mere sour grapes. The issue is not the "interpretations" but the bijections. If the bijections are as advertized, then the proof holds despite TO's displeasure at its validity.
From: Virgil on 19 Aug 2006 19:01
In article <ec7prp$rj2$2(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > In article <1155998568.859179.65830(a)m79g2000cwm.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > >> Franziska Neugebauer schrieb: > >> > >>> You have agreed to that this limit does not exist ("There is no L in > >>> N"). So the sum is at least _not_ _finite_ (not in omega). > >>> > >> There is no L, nowhere. It is wrong to say that L is in |N and it is > >> wrong to say that L is larger than any n e |N. Actual infinity does not > >> exist! I told you that already several times. > > > > But "Mueckenh" has not been able to give us any compelling reason to > > believe him, and we have compelling reasons not to. > > That's true! While he correctly illustrates that the infinite set of > naturals cannot exist without infinite naturals That is precisely the part that WE have no compelling reason to believe and compelling reasons to disbelieve. That TO chooses to fly without a net, axiomatically speaking, does not mean that these who are more serious about set theory choose to operate in that haphazard way. |