An uncountable countable set
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From: Dik T. Winter on 2 Sep 2006 23:23 In article <1157227790.457614.271810(a)74g2000cwt.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1156842898.805375.169610(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > > > > You gave an example how a number of the form 0,111...1 with > > > > > n digits indexes the n-th digit but does not cover all digits > > > > > with m =<n ??? > > > > > > > > No, because that does not exist. > > > > > > How then can you believe and assert that the number 0.111... which has > > > *only finitely indexable positions* could be completely indexed but not > > > covered. > > > > Because there are infinitely many finitely indexable positions. Do you > > see the way in which the first number you give "0.111...1" differs from > > the second number you give "0.111..."? > > For indexing we have exactly the same question. Does not make sense. Again, do you see the way in which those two numbers differ? Why do you never give a straight answer to such plain questions? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 2 Sep 2006 23:20 In article <1157227721.868719.283000(a)p79g2000cwp.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > You are trying to make it difficult, using UTF-8. But this one takes the > > cake. > > What is UTF-8? From your article: Content-Type: text/plain; charset="utf-8" > > Again translated (why do you post so much German in an English speaking > > newsgroup while you should know that most readers are not able to read > > German?): > > Because I have the German text available and because I do not want to > be blamed of mistranslating. So you can blame other persons of mistranslating? > > Cantor: So while a changing quantity x that successively takes the > > various values of finite numbers 1, 2, 3, ..., v, ... , is a > > potential infinite, on the other hand, a through the axioms completely > > determined set (N) of all integral finite number is an example of an > > actually finite quantity. > > Not through the axioms, but through "a law" (ein Gesetz). What is the difference? He uses his (I think, but that is from memory) second completion law. In current terminology, that is an axiom. > > Nice that you found the quote I have alluded to, and that you did deny > > of existing, but that I could not find back. > > > > What Cantor is stating here (and I did already indicate that in an > > earlier response), is, translated to current set theory: > > The set N is potentially infinite, > > No. The changing quantity is here a variable. > > >the size of N is actually infinite. > > (In current terminology a set is not a quantity.) > > Cantor's changing quantity is a variable. A set is never potentially > infinite according to Cantor. You missed my "in current set theory"? To be honest. If you did not go any further than Cantor, you may find quite a few contradictions. This does not say anything about set theory as developed beyond Cantor. Stating that set theory is inconsistent because Cantor was inconsistent is simply false. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: David R Tribble on 3 Sep 2006 11:52 mueckenh wrote: >> As long as the set of natural numbers includes only finite numbers, its >> cardinal number is also finite. > David R Tribble schrieb: >> What is that cardinal number? Do you have a name for it? > mueckenh wrote: > The set is potentially infinite. That contradicts what you stated above, "... its cardinal number is also finite". > It has no cardinal number in the sense of set theory. Then it is not a set, since cardinality is a property of all sets. > All we can attach to it is the number of elements known > or existing. Disregarding physical constraints ... I was not aware that abstract mathematical concepts (e.g., sets) had any physical constraints. > ... we can assume that the > elements of the set of natural numbers are counted by the largest > natural number temporarily known. With respect to physical constraints > the number of elements is less than 10^100. So how many numbers are in this sequence?: 1, 2, 3, ..., 10^100, 10^100+1 What exactly is it about your set theory that physically constrains your sets to being less than some arbitrarily chosen maximum size? David R Tribble schrieb: >> Tony calls it "alpha", the cardinality of the finite but "unbounded" >> set of all finite naturals, and also the largest natural. It appears >> to have the curious property of being a finite natural but having >> no finite successor. He's never provided an estimate of how >> large it is, though. Perhaps you have a better idea of this? > mueckenh wrote: > Sorry, there is no largest natural and no natural is infinite. Therefore there can be no finite cardinal for the set of naturals.
From: mueckenh on 4 Sep 2006 06:44 Virgil schrieb: > > It is a potentially infinite set, as described by Peano and by all the > > mathematicians before Cantor. > > Does removing 1 from {1, 2, 3, ..., w} change it from an actually > infinite to a potentially infinite set? No. omega has by far another quality (and quantity) than 1. > > > Look at my last post concerning the final proof that 0.111... with > > actually infinitely many digits does not exist. > > It does in the world of mathematics, which is the only one that counts. > Pun intended. If you cannot read and understand proofs you may assert this. Regards, WM
From: mueckenh on 4 Sep 2006 06:47
Virgil schrieb: > Cantor's "diagonal" in its original form applied only to the set of all > binary sequences, i.e ., infinite sequences whose terms were from a set > of two objects, and not to real numbers of decimal digits. This is > essentially the same as the set of all functions from N to {0,1}, where > N is understood to be an endless set of natural numbers. > His theorem says that the set of all such functions could not be listed > sequentially without leaving some out. And my tree shows that the complete set can be listed in form of a tree. > > What is the difference between my tree and Cantor's list. > > Lists are sequences which do not branch. Trees branch. What is the difference concerning completeness of number representations? > > The problem is the following: You assert that the digit positions of > > 1/3 are countable as well as the levels of my tree (all of them), but > > you deny that the edges at these levels are not all countable. That is > > wrong because for every n-th level you count, the number 2^n of edges > > can also be counted. There is no limit. > > It is only the set of individualy infinite paths in that tree that are > uncountable, The set of all edges and the set of all nodes are both > countable. And the set of paths is provably not larger than the set of edges, because any branching off require another edge. Look at Dik's explanation. He has at least understood this fact and therefore, in order to save set theory, he denies the existence of infinite paths (in my tree - not in Cantor's list. That is his inconsistency. In principle he is right, because there is no infinite path anywhere, neither in my tree nor in Cantor's list.) Regards, WM |