Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Lester Zick on 8 Dec 2006 17:02 On Fri, 08 Dec 2006 09:19:01 +0100, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >Lester Zick wrote: > >> On Thu, 7 Dec 2006 03:00:21 -0500, David Marcus >> <DavidMarcus(a)alumdotmit.edu> wrote: >> >>>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". > >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, but >it's not a set. Set theory may be of limited use, but it's supremacy is >complete nonsense, and will be overruled in time. Well you know there is a sense in which every "thing" may in some sense represent a set of predicates. However it is pretentious in the extreme to suggest that those predicates are in any sense points. I can see set analysis as a valuable tool just not to the exclusion of general analytical techniques. ~v~~
From: mueckenh on 8 Dec 2006 17:03 William Hughes schrieb: > Recall this post from Dec 1 > > We extend this to potentially infinite sets: > > A function from the potentially infinite set A to the > potentially infinite set B is a potentially infinite set of > ordered pairs (a,b) such that a is an element of A and b is > an element of B. A function, according to modern mathematics, is a set, actually fixed and complete. The expression "variable" is merely a relict from ancient times when people knew that the objects of mathematics do not exist in some nirvana but have to be present in a mind where not everything can be present simultaneously. > > We can now define bijections on potentially infinite sets Only if we consider them being actually infinite. But that would exclude them from being potentially infinite. Regards, WM
From: Lester Zick on 8 Dec 2006 17:07 On Fri, 8 Dec 2006 14:29:26 +0000 (UTC), stephen(a)nomail.com wrote: >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 > >Sure it can, just as computer science can. Do you think the fact that computer >programs exist to model chemical reactions makes computer science part of chemistry, >or vice versa? Well the problem here, Stephen, is that we don't quite know what computer "science" is. Is it merely the science of computers in the sense of hardware? Or does it include software? And if the latter what makes one kind of software more scientific than another? It looks to me like all computer "science" studies are programming techniques and calls the results computer "science". ~v~~
From: mueckenh on 8 Dec 2006 17:07 Dik T. Winter schrieb: > In article <1165403362.548786.220370(a)16g2000cwy.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > > Everybody knows what the number of ther EC states is. > > > > > > That is *not* what I did ask you. You state that it is simply a matter > > > of definition how one interprets "to grow" and "number", > > > > sure. > > > > > and I asked you > > > to provide definitions. Moreover, the number of the EC states is not > > > fixed, so you can only state what the number of the EC states is at a > > > particular point in time. > > > > The number of EC states is "the number of EC states". It is simply a > > notion which can be equal to a natural number. > > Again you have provided neither a definition of "number", nor of "grow". > Are you unable to do so? In common parlance, but that is not mathematics. > In mathematics functions can grow in relation to their argument, but not > the entities they denote. Functions cannot grow, according to modern mathematics. The expression "variable" is merely a relict from ancient times when people knew that the objects of mathematics do not exist in some nirvana but have to be present in a mind where not everything can be present simultaneously. Regards, WM
From: Virgil on 8 Dec 2006 17:09
In article <4579BF5B.5010406(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/8/2006 8:04 PM, Virgil wrote: > > In article <45793B7C.4050105(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > with higher abstraction. > > >> While the imaginary numbers are obviously different from the ordinary > >> numbers, the corresponding distinction between rationals and reals is > >> more subtle. > > > > Rationals are ratios of integers, non-rational reals are not ratios of > > integers. That should not be too subtle even for EB. > > Do not try cheating. Rationals are ratios of integers, irrrational reals are not ratios of integers. That is the definition of rationals and irrationals. How is invoking definitions cheating? Or is is just that EB does not like being shown to be wrong? > > > > It is most unhelpful to misuse words whose common meanings tend to > > mislead one about their technical meanings. > > In this case the common meaning is quite appropriate. Or can you suggest > better words? Rational and irrational work for me. > > > >> The difference between rationals and reals corresponds to the difference > >> between potentially infinite and perfectly infinite. > > > > Since in such set theories as ZF or NBG or NF there do not exist any > > such things as potentially infinite sets but there do exist infinite > > sets, the distinction is irrelevant in those set theories. > > This is perhaps the most important achievement of these systems of > axioms, maybe it has even positive aspects too. Except that it is only within such set theories that real can b found at all. > > > And in those > > theories each real is a set just as each rational is a set and each > > natural is a set. > > How to get this information out from the axioms? By constructing models of the reals. > > > > If EB wishes to produce an axiomatic system which distinguishes between > > potential and actual, let's see him do it. > > Admittedly, I would not be in position for doing so, I do not intend it, > and I do not even criticise the effective neglect of the difference in > mathematical pratice. Nonetheless I suggest to clarify that > 1) There is a categorial difference between rational and real numbers Perhaps, but not in the way EB claims. > 2) Corresponding reasoning of mine is not wrong It is mathematically and logic ally wrong however right it may seem to an engineer. > 3) It is overdue to delete nonsensical basics and terminology from books > and lectures (in particular the distinction between ordinals and > cardinals, the notion cardinality, the German word ueberabzaehlbar, > transfinte numbers, all alephs, etc.). We English speaking mathematicians are quite willing to delete all purely German mathematics, so long as we can keep whatever bits have adopted in English. This will include all of Cantor and Dedkind, but none of EB. |