Cantor Confusion
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From: Eckard Blumschein on 8 Dec 2006 13:25 On 12/7/2006 8:24 AM, David Marcus wrote: > Eckard Blumschein wrote: >> Why do you not admit the possibility that countability of >> a set requires countable numbers. Doesn't it make sense? > > Of course, it doesn't make sense. The reasons are: > > 1. You haven't defined/explained what a "countable number" is. Countability is self explaining. I argue: The current deviating use in set theory is not necessary if set theory turns out to be a nice fancy. > > 2. There is already a standard definition of "countability of a set" and > it seems unlikely that you can define "countable number" in such a way > to make your statement true. I know it and consider it pretty narrow minded. >> I didn't find a single counter-example. > > That doesn't prove anything. In mathematics, we prove things. See my reply to Virgil. >
From: Virgil on 8 Dec 2006 13:47 In article <db3a4$45792f8b$82a1e228$30957(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > In article <68588$45791ff3$82a1e228$8581(a)news2.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>It's quite simple. Set Theory can not be the foundation for mathematics, > >>because NOT EVERYTHING IS A SET. E.g. a calculation is mathematics > > > > A calculation is an application of mathematics, but may actually be > > physics, or chemistry, or merely commerce. > > Geez! Can mathematics be separated from its "applications" in this way? > > Han de Bruijn When a sales clerk makes change on a purchase, is that sales clerk "doing" mathematics or merely applying it? When a child does his arithmetic homework on a calculator is he or she really "doing" mathematics?
From: Eckard Blumschein on 8 Dec 2006 13:52 On 12/7/2006 8:18 AM, David Marcus wrote: > Eckard Blumschein wrote: >> On 12/5/2006 2:13 PM, Bob Kolker wrote: >> > For the latest time. Uncountability is a property of sets, not >> > individual numbers. >> >> I know this widespread view. > > So you claim. However, last time I asked you to give the standard > definitions, you failed. Care to try again? Define "countable" and > "uncountable". Do you believe someone who is urged to say the words of pater noster ... will immediately become a believer? >> > There is no such thing as an uncountable real >> > number. >> >> Real numbers according to DA2 are uncountable altogether. People like >> you will not grasp that. > > And, people like you don't listen when we point out that "DA2" does not > define/construct/characterize the real numbers. O.K. I agree. Cantor had first to chose an appropriate definition and then to make sure that it was fulfilled in his DA2. > Of course, if you > actually bothered to read a modern book, you could learn this for > yourself. At the moment I am reading Fraekel et al. Foundations of Set Theory. 1958. It does not reveal any solid fundamental, just endless attempts to repair an illusion. > >> > Countability >> > /Uncountability are properties of -sets-, not individuals. >> >> Do not reiterate what I know but deny. > > How come you get to deny things, but we don't? Doesn't seem fair. Well, I am unbiasedly looking for strong arguments. Presently I tend to summarize what I found out as follows: Dedekind and Cantor made a big mistake when they on one side categorically equated rational and real numbers but simultaneously proved that real numbers are uncountable while rationals are countable. Fabrication of the notion cardinality and interpretation of uncountable as more than countable led to horrific speculations (e.g. CH and AC) and ample bizarre mutilation of mathematical terminology. Being made as a crutch for saving the formerly naive belief in nonsensical set theory, Hilbert's extention of Euclid's axiomatic method nonetheless seems to be reasonable, in principle. Perhaps, Ebbinghaus was correct when he quoted Lessing: An obvious error eventually led to something valuable. This is not the overdue revolution but rather a cautious first step towards reformation.
From: Eckard Blumschein on 8 Dec 2006 13:53 On 12/7/2006 8:14 AM, David Marcus wrote: > Eckard Blumschein wrote: >> Nevertheless, all these arguments of mine are most likely flawless. > > Wow! That has got to be your most absurd statement yet! Weak statements invite for refutation. Good luck
From: cbrown on 8 Dec 2006 13:55
Virgil wrote: > In article <1165564379.213843.105070(a)j72g2000cwa.googlegroups.com>, > cbrown(a)cbrownsystems.com wrote: > > > Virgil wrote: > > > In article <1165552164.399063.145330(a)f1g2000cwa.googlegroups.com>, > > > cbrown(a)cbrownsystems.com wrote: > > > > > > > Virgil wrote: > > > > > In article <MPG.1fe27d2a5a4ae60a989a01(a)news.rcn.com>, > > > > > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > > > > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > I do nothing. The tree cares that even in the limit the number of > > > > > > > paths > > > > > > > cannot become uncountable. 2^n remains the cardinal number of a > > > > > > > countale set, even in the limit n --> oo. That's why I devised the > > > > > > > tree! > > > > > > > > > > > > The number of paths in a tree of height n is 2^n. Are you saying that > > > > > > the number of paths in the infinite tree is lim_{n -> oo} 2^n? > > > > > > > > > > Actually, if one interprets 2^n as the number of functions from > > > > > {0,1,2,...,n-1} to {0,1} in NBG, or. equivalently. as the number of > > > > > functions from {1,2,3,...,n} to {1,2}, then 2^n -> 2^N as n increases > > > > > without limit and 2^N, interpreted that same way, is an uncountable > > > > > set. > > > > > > > > What do you mean by "the number" of functions? > > > > > > I mean the cardinality of the set of all such functions, of course. > > > > > > > > > > In what sense does the > > > > sequence of natural numbers (1,2,4,8, ..., 2^n, ...), approach "the > > > > number" of the set of functions N->{0,1}? > > > > > > For each finite n in NBG, n = {0,1,2,...,n-1}, so the set 2^n consists > > > of all functions from n to {0,1} = 2, and is of cardinality equal to the > > > number of such functions. > > > > > > So as n --> N, 2^n --> 2^N. > > > > Ah! So, because (1,2,4, ..., 2^n, ...) is a subsequence of (1,2,3, ..., > > n, ...), then as n->N, n->2^N. > > That is neither what I said nor what I meant. > > In set notation (rather than number notation) give non-empty sets A and > B, the shorthand notation for the set of all functions from B to A is > A^B. A slightly less compact notation, used in the category of sets and > functions, would be Hom(A,B). > > Thus Card(A^B) = Card(A)^Card(B) for all finite sets A and B. > > And Card({0,1})^{0,1,...,n-1}) = Card({0,1})^Card({0,1,...,n-1}) = 2^n > for all natural numbers, Card({0,1,...,n-1}) = n in N. > Who could disagree with the above? And by these definitions, we don't need or use limits at all to evaluate Card(2^N.). > Then as {0,1,...,n-1} --> N, {0,1}^{0,1,...,n-1} --> {0,1}^N > and 2^n --> 2^N Why not simply say, "by definition, n = N implies 2^n = 2^N"? It's that troublesome "-->" that I am questioning. In what sense does the sequence of sets ((0), (0,1), ..., (0..n-1), ...) "approach" the set N? In what sense does the sequence of cardinalities (1, 2, 4, ..., 2^n, ...) "approach" Card(2^N)? Cheers - Chas |