Cantor Confusion
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From: stephen on 7 Dec 2006 12:09 David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > stephen(a)nomail.com wrote: >> Nobody but you has talked about "growing" sets. Sets, like numbers, do not >> grow. You, like many other people who do not understand set theory, >> think of sets as mutable objects, that change as we perform operations >> on them. This is akin to thinking that numbers change when we perform >> addition. If I add 3 to 7, neither 3 or 7 changes. > It is such an odd belief. Why use a set for something that a function is > naturally for? I don't really understand why cranks insist on using sets > for everything, while at the same time insisting that sets are useless > or illogical or whatever. I do not think it is that odd. In everyday usage, the word "set" is used to denote something that changes. But then again, so is the word "number". The number of people in a room may change, but that does not imply that a specific number, such as 5, changes. Most people seem to have an abstract enough concept of number that the common usage does not confuse them. However they do not apply this abstraction to sets, so if someone says the set of people in the room changes, they think a specific set changes. Stephen
From: Lester Zick on 7 Dec 2006 13:06 On Thu, 7 Dec 2006 02:04:34 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Bob Kolker wrote: >> Eckard Blumschein wrote: >> >> > Roughly speaking, it just claims that a set is unambiguously determined >> > by its elements. If i recall correctly A=B<-->(A in B and B in A) >> > >> > Perhaps the Delphi oracle provided less possibilities of tweaked >> > interpretation betwixed and between potential and actual infinity. >> >> What is "potential" infinity. Can you define it rigorously? > >Even a non-rigorous defintion would be a start. Well since according to David a definition is "only an abbreviation" how about "X"? ~v~~
From: Lester Zick on 7 Dec 2006 13:07 On Thu, 7 Dec 2006 02:18:36 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> 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". Since according to you "definitions are only abbreviations" how about def(countable)="Y" and def(uncountable)="Z"? ~v~~
From: Lester Zick on 7 Dec 2006 13:10 On Thu, 7 Dec 2006 02:46:06 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >mueckenh(a)rz.fh-augsburg.de wrote: >> Franziska Neugebauer schrieb: >> > mueckenh(a)rz.fh-augsburg.de wrote: >> > >> > > And the power set of this set is a finite set too. And so on, in >> > > infinity ... (potential infinity , of course) >> > >> > You still have not yet understood the concept of inifinite sets. >> >> I have understood the concept and its failure. > >If you've really understood it, please demonstrate this. State a theorem >and give its proof in standard set theory. Why standard set "theory". I mean what exactly makes standard set "theory" definitive and exhaustive? Is it true? And if not why should anyone analyze sets in such terms to the exclusion of other terms? ~v~~
From: Lester Zick on 7 Dec 2006 13:17
On Thu, 7 Dec 2006 03:00:21 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Bob Kolker wrote: >> MoeBlee wrote: >> > 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. >> >> an element of a ring or a semi-group is a set? > >If you use ZFC (or something similar) as your foundation for >mathematics, then everything is a set. Of course, while solid >foundations are good to have, if you are living on an upper floor, you >may prefer to ignore what is going on in the basement. So you're saying that set "theory" is all of mathematics? Of course since what you say isn't necessarily true that's not exactly a ringing endorsement of set "theory". ~v~~ |