An uncountable countable set
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From: Virgil on 10 Jul 2006 15:15 In article <1152543944.017463.146190(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Christian Clason schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > There is no axiom that uncountable sets exist. In the contrary, it can > > > be shown that, given ZFC is free of contradictions, it must have a > > > countable model. This is a result of Skolem, who, therefore, argued > > > that all the evidence that had been given for the existence of > > > uncountable sets was inconclusive. > > > > And yet, within V, it is easily provable that uncountable sets exist. > > These "proofs" are wrong. At least can be sown that the opposite > results can also be obtained: uncountable sets can also be proved > countable. Not within the the model in which they wewre created. One has to go outside it, and then one has incorporated other sets, which were not in the model which are uncountable in that extension of the model. > > > The > > _logical_ conclusion is that the notion of countable and uncountable has > > different meaning "inside and outside V" (since the "meaning" of the > > element relation is also part of the model). > > Yes, a model where the bijection cannot be defined. Nice. A bijection > can be formulated with less than 100 letters. But those letters are not > available in all those models? Countable infinity is not enough to > supply so many letters? Ridiculous! That "mueckenh", who claims such things as a simultaneous well ordering and size ordering of the rationals,should call things which are so far beyond his understanding ridiculous, is ridiculousness squared. > > >This is known as the Skolem > > paradox, which is discussed in detail in most books on logic or elementary > > model theory. > > Why, do you think, was Skolem an intuitionist? > Why was it his opinion that uncountability was nonsense (he said > "inconclusive" - at those times the habits were not yet spoiled by the > internet). See, among other places, http://www.ltn.lv/~podnieks/gta.html
From: Virgil on 10 Jul 2006 15:22 In article <1152544026.187966.227320(a)s13g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > > >> mueckenh(a)rz.fh-augsburg.de wrote: > > >> > > >> > The set ordered by size and by natural indices is the limit of my > > >> > transpositions. > > >> > > >> 1. What is a limit of transpositions? > > >> 2. How do you prove its existence and uniqueness? > > > > > > How does Cantor prove the existence and uniqueness of the well-ordered > > > set of rationals or > > > > Once again: What is a limit of transpositions? How do *you* (not Cantor) > > prove its existence and uniqueness? > > Me, not Cantor? If you agree that Cantor's ideas are wrong, I need not > prove anything else, because I prove just this. > > If you believe that Cantor's ideas are correct, then I can utilize them > too. You could if you understood them, which you clearly do not. > These ideas imply that a countable set is exhausted, if the > element to be mapped on n e |N is determined for every n. Cantor's rule implies that when something applies equally and independently to every member of a set, it need not also be verified individually, member by member, for each member. >This is the > case for the set of my transpositions, which is countable and has order > type omega. No fixed sequence of transpositions will impose order on an arbitrarily ordered sequence of objects.
From: Virgil on 10 Jul 2006 15:38 In article <1152544194.679686.187540(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > Are you writing about an infinite sequence of transpositions? > > As you were not present, when I explained it first, I repeat it: > > Start with a well-ordered set of all positive rationals. The initial > indices remain with the rationals, but I use current indices m, n in > the following definition. > > A transposition (m,n) means: If the elements with current number m and > n are not in order by size, exchange these elements, if they are > already ordered by size, do nothing. That is not what "transposition" means in standard usage. > > Now the following set of transpositions is applied: > > (1, 2) > (2, 3) > (1, 2) > (3, 4) > (2, 3) > (1, 2) > ... > > The set of transpositions is countable and has order type omega. It is > a sequence. And every term of this sequence is well defined, i.e., if > one wants to calculate how many transpositions it will take to have the > first j elements of he well-ordered set ordered by size too, she can do > that. Diese Methode leidet keinen Stillstand. One never gets ready, but > that is due to the fact that the well-order has no last element. All > elements which do exist in the well-order will be in the order by > magnitude. Except that after each (1,2), there will only be a finite initial subsequence in numerical order and an infinite terminal sequence not yet ordered. Using 1 origin indexing, the nth occurrence of (1,2) occurs at the (n^2 + n)/2 th position in the list of "transpostions". At which (n^2 + n)/2 th operation is the entire list ordered?
From: Virgil on 10 Jul 2006 15:40 In article <1152544294.936976.98720(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > So the elements of omega do exist "statically"? > > Neither nor. But if one claims their existence, I reserve the right to > let them exist at my convenience. > > Regards, WM And if "mueckenh" claims existence of any mathematical or logical skills, I reserve equal right to deny they exist at my convenience.
From: Virgil on 10 Jul 2006 15:44
In article <1152544884.364304.157620(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > > >> > Either K is in the list (which would contradict analysis in the > > >> > same way as my original example), then there is an n with 10^(-n) = > > >> > 0. Or K is not in the list. > > >> > > >> K is not in the list. > > >> > > >> > Then there must be a position which cannot be enumerated by natural > > >> > numbers > > >> > > >> non sequitur. All positions are indexed by definition of decimal > > >> representation. > > > > > > All positions are indexed <==> K is in the list. It is a logical > > > equivalence for linear sets like my list. > > > > What kind of "logic" is employed here? > > That one which deserves its name, and which you unfortunately seem to > be not familiar with. The logic of a linear sequence like this sequence > of list numbers > > 0.1 > 0.11 > 0.111 > ... "Mueckenh" insists that that sequence has a double meaning, both as unary representations of naturals and as binary, or other based, rationals, and requires that 0.111... satisfy both interpretations simultaneously. Ergo, it is garbage. |