Cantor Confusion
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From: Virgil on 6 Dec 2006 23:18 In article <4576FC0A.7000004(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > According to the axiom of extensionality, a set has been determined by > its elements. Why do you not admit the possibility that countability of > a set requires countable numbers. The countability of a set requires no more than it have countably MANY members, but is totally independent of the nature of those members. For example: The set of square roots of prime naturals is as countable as the set of prime naturals itself by an obvious bijection, even though according to EB each of those square roots is "uncountable" in some weird anti-mathematical sense.
From: Virgil on 6 Dec 2006 23:24 In article <4576FD8D.60408(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > It started with the question how to deal with the nil in case of > splitting IR into IR+ and IR-. I got as many different and definitve > answers as there are possibilities. I expected that there is only one > correct answer, and I found a reasoning that compellingly yields just > one answer in case of rationals and a different one in case of reals. It might depend on the purpose of performing the split. I can think of several ways to deal with it, at least one of which should be satisfactory for any given purpose. If, for example, one is trying to create Dedekind cuts, about the only requirement is that if one puts 0 in one of IR+ or IR-, one goes the same way with the boundary rational at every rational cut.
From: imaginatorium on 6 Dec 2006 23:25 David R Tribble wrote: > stephen wrote: > >> The idea that 25 is ever going to be anything but 25 is absolutely ridiculous. > >> The idea that a set ever changes is equally ridiculous. > > > > Brian Chandler wrote: > > Obviously not a FORTRAN programmer... > > > > SUBROUTINE CHANGE(A, B) > > IF(A .EQ. 25) A = B > > RETURN > > END > > > > Now try: > > CALL CHANGE(25, 17) > > :-) > > But I still don't see how calling a subroutine changes the value of 25. > Unless you have a memory overwrite bug. At a certain time, in a certain place, with a certain compiler, it was true that: all subroutine arguments are passed by reference (including constants!), and therefore after the above call, the value of the expression 25 .LE. 20 is TRUE, unless of course you have changed 20 to something under 17. > A better programming metaphor is that numbers and sets are constants, > not variables. Metaphor schmetaphor. In the present environment, of attempting to reason with cranks, metaphors _always_ lead to disaster. Brian Chandler http://imaginatorium.org
From: Virgil on 6 Dec 2006 23:26 In article <4576FE21.8040606(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/6/2006 2:05 AM, Virgil wrote: > > In article <1165345832.735910.255620(a)79g2000cws.googlegroups.com>, > > "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > > > >> Bob Kolker wrote: > >> > There is no such thing as a countable integer, countable rational or > >> > countable real. > >> > >> In the sense you're trying to get across to the other poster, I > >> understand your point. But, just for the record, in a technical sense > >> in set theory, as integers, rational numbers, and real numbers are > >> themselves sets, it does make sense to say whether one of them is > >> countable or not. For example, where integers are defined as > >> equivalence classes of natural numbers, each integer is itself a > >> denumerable set. I am not necessarily endorsing anything the other > >> poster has said; I'm just adding the technical note that in a strict > >> set theoretic sense, even numbers are sets and thus it is meaningful to > >> talk about the cardinality of a number. > >> > >> MoeBlee > > > > In that particular sense, each real number as a Dedekind cut, being a > > partition of the rationals into two sets, always has cardinality 2, > > while each member of a Dedekind cut is countably infinite. > > > > But each real number as a set of equivalent Cauchy sequencesis > > uncountable. > > Conclusion: Dedekind cut's are self-delusive. Why is any set with two clearly defined members self-delusive?
From: Lester Zick on 7 Dec 2006 00:01
On 6 Dec 2006 11:40:21 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > >mueckenh(a)rz.fh-augsburg.de wrote: >> stephen(a)nomail.com schrieb: >> >> > If you think sets grow, then you do not understand set theory. >> >> I know that in modern set theory sets do not grow. > >It's good that you finally came around to that conclusion. > >> But I heavily doubt the relevance of modern set theory. > >It's relevant to me as an axiomatization of ordinary mathematics as >well as of meta-mathematics and also interesting onto itself as an >abstract study. Have we decided yet what ordinary mathematics is exactly? Is set "theory" all of ordinary mathematics? Set mathematikers have this rather nasty habit of saying one thing and claiming another. ~v~~ |