An uncountable countable set
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From: Virgil on 21 Jul 2006 13:32 In article <1153478388.042474.295770(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1153301881.393720.57330(a)75g2000cwc.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > > > > > If the *+ sum of the list is a unary representation of some 1's, > > > > > then, > > > > > by the construction of the unary numbers of the list, this sum must > > > > > be > > > > > a unary numer of the list. > > > > > > > > As long as your list is finite. > > > > > > As long as it contains *numbers* which can be distinguised from each > > > other and which obey trichotomy. > > > > No, as long as your list is finite. > > The numbers count the lines. As long as there are only finite numbers, > the list is finite. In what axiom system does this allegedly hold? It does not hold in any axiom system in common use, such as ZF or NBG. And absent some axiom system being assumed, nothing can be proved. > But as you have not the understanding, consider the related case: > A finite number covers what it indexes. This does not depend on how > much numbers are in the list. > >From this discovery you may obtain the result for what you call an > infinity list. We do not call anything an infinity list, though we call some lists infinite lists when they have no last member. > > > > > Suppose the list is infinite, and the > > > > *+ sum of all members of the list is a member of the list. That would > > > > mean that the *+ sum of the list is the last element of the list, > > > > contradicting that the list is infinite. > > > > > > If we assume that non-terminating fractions have aleph_0 digits, then > > > this case is realized by > > > > > > 0.1 > > > 0.11 > > > 0.111 > > > ... > > > 0.111... > > > > > > Here the last number of an infinite list is in this list. > > > > I thought we were talking about natural numbers. I have not yet seen a > > definition that calls the last one a natural number. > > I never said so. > > > You should *not* > > switch between representations during the process. > > It is important for the discovery. Any "discovery" based on such MISrepresentation is fake.
From: Virgil on 21 Jul 2006 13:48 In article <1153478871.303029.5360(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > Dear Wolfgang, > > > > I would appreciate if you not cut away that parts of my previous posts > > you are trying to argue against. It may also be helpful if you read > > before cutting. > > Dear Franziska, I cut what I regard as unnecessary and not helpful in > order to make reading more comfortable. Your counting starting from > zero is superfluous and does not support your arguing. > > > > > 2 The *-sum of the sequence does not necessarily "inherit" the > > properties from their individual sequence members. If this > > "inherentance" is kind of a "general principle" you should refer to a > > recognized source. > > > > But there is even a finite counterexample: > > > > 101 > > 011 > > 110 > > ---- *-sum > > 111 > > > > The property "does not contain at least one 0" is not inherited by > > the *-sum. So the "inheritance principle" is not so obvious. > > But it is obvious for *all* *finite* *unary* number. And only finite > unary numbers are involved in the second list below: > > The *- sum of > > 0.1 > 0.11 > 0.111 > ... > 0.111... > > is > > 0.111... > > i.e. the result is a number of the list. > > > The *- sum of > > 0.1 > 0.11 > 0.111 > ... > > is > > 0.111... > > The result is allegedly not in the list. If I define "*+" on every "list" of ordinals, L, as the smallest ordinal not greater than all members of L, , more briefly as SUP(L), then I get precisely "mueckenh"'s "dual" result with no contradictions at all. Note that, for ordinals, if a set of ordinals, L, is bounded above in either ZF or NBG, then there is a unique ordinal, SUP(L). > > This is a contradiction, because by tertium non datur either the first > case or the second must be true (if aleph_0 is a number with respect to > trichotomy with natural numbers). As "meuckenh" is dealing with ordinal properties of naturals, not merely cardinal properties, he must abide by the ordinal properties of those ordinals. And for ordinals there is no conflict. Different sets of ordinals can have the same SUP without conflict.
From: Virgil on 21 Jul 2006 13:58 In article <1153479160.430937.310460(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > What is false unter the abovementioned presumption omega = { 0, 1, > > 2, ... }? > > > I am not willing to discuss this strange counting. > 0.1 means 1 and the first (1) digit after the point is a 1. > 0.11 means 2 and the first (1) and the second (2) digit after the point > are 1's. > > > > When it is not important "what I mean", why do you discuss with me? This > > is rather insulting. > > This remark concerned only the rather strange set-theoretic counting. > Other opinions of yours are often welcome, as you know. In other words, "mueckenh" refuses to consider anything which shows him wrong, however right that thing may be.
From: Virgil on 21 Jul 2006 14:08 In article <1153479321.840071.258130(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter wrote: > You again refrain from answering questions. The definition of 0.999... > is > as follows: > 0.999... = lim{n -> oo} sum{k = 1 .. n} 10^(-k) > You said that definition is silly. What is silly about it? > _______________ > WM: > Sorry, could you help me with "silly". My find function shows with > "silly" only the sentence: "It is only in modern set theory that such > silly things as the rejection of the binary tree occur." > > _______________ > Dik T. Winter wrote: > what indices in the limit above are undefined? > ________________________- > WM: > Those which cannot be enumerated by natural numbers. Short answer: none. > ________________ > Dik T. Winter wrote: > > Because, and I state it again, there is a difference between sequential > > processes and simultaneous processes. > ________________ > WM: > I know that in the infinite binary tree *every* split is realized by > the presence of two edges and would be impossible without them. Non-responsive. When "mueckenh" gets in over his head, he changes the subject. > > | > o > /\ > > Nothing can be more simultaneous than this knowledge which is present > without considering any sequential process and which concerns the whole > tree. Those, like "mueckenh", who chose to ignore concrete proofs that they are wrong, are wrong. > > The tree "in its length" is nothing else than a Cantor-List. The only > difference is that the tree guarantees the presence of every "diagonal > number". > _______________ > Dik T. Winter wrote: > Four your count of edges and > paths you need limits, but those limits do not exist. > _________________ > WM: > Oh, the limit of 1 + 1/2 + 1/4 + ... = 2 is not existing? Strange. > > There is no question that every path can be interpreted as the > representation of a real number. And now "those limits do not exist"? > Very strange. And you are then saying, by that remark, that the reals can be put into bijective correspondence with the naturals, and that there are sets which can be put into bijection with their power sets. > _____________ > Dik T. Winter wrote: > Again, you need something like a limit here, because adding nodes is a > sequential process. > __________________ > WM: > Adding lines in Cantor's list is not a sequential process? AS any list of reals must be 'completed' before being presented to Cantor for analysis, its method of construction is irrelevant. Cantor merely shows that any such list, ONCE CONSTRUCTED, is incomplete.
From: Virgil on 21 Jul 2006 14:12
In article <1153482158.663876.226910(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > What your > > example shows is that an infinite sequence of transpositions can destroy > > well-orderedness. What is the difference? The only conclusion is that > > Cantor's statement is incorrect or that the reading of Cantor's statement > > is incorrect. Take your pick. > > Your hinting at any doubt about the clear meaning of Cantor's statement > is outrageous. No more so than many of "mueckenh"'s outrageous claims. > > >I have found already a few instances where > > Cantor's statements are in conflict with modern set theory. > > There are many. See for instance for a very simple one: "On Cantor's > proof of continuity-preserving manifolds" on > http://www.fh-augsburg.de/~mueckenh/ > > > And I am not > > surprised. He found the building blocks but was not entirely sure about > > the way to go. His articles were research in progress, as it happens. > > And now it happes that all his way turns out as a deadlock. > > > > And now for a debunking of a myth. Nowhere (at least I could not find > > any place) has Cantor used the diagonal argument to show that the reals > > are not countable. His proof about the reals shows that a complete, > > densely ordered set is not countable. That one does not use the diagonal > > argument at all. > > So far we agreed recently. > > > His diagonal proof shows that the set of infinite > > sequences of two symbols is not countable, and as an extension, that the > > powerset of a set has cardinality strictly larger than the cardinality > > of the original set. > > > > So all your arguments about 0.999... = 1.000... are *not* directed against > > Cantor. > > Wrong. My arguing is directed against the complete existence of the set > of all naturals, the set of all digits of a real number, the actual > infinity, the first transfinite number. That all is purest Cantor. It is now a lot of other people too. Cantor was only the first of many. Nowadays, you have also to fight von Neumann, in which fight you will always lose. > > Regards, WM |