Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Lester Zick on 8 Dec 2006 13:57 On Fri, 08 Dec 2006 01:53:46 -0700, Virgil <virgil(a)comcast.net> wrote: >In article <68588$45791ff3$82a1e228$8581(a)news2.tudelft.nl>, > 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 > >A calculation is an application of mathematics, but may actually be >physics, or chemistry, or merely commerce. > >Accounting involves calculations, and while its cacluations involved may >be in a sense mathematical, they are not Mathematics. So what is mathematics ~v~~
From: Eckard Blumschein on 8 Dec 2006 13:59 On 12/7/2006 8:03 AM, David Marcus wrote: > Eckard Blumschein wrote: >> On 12/4/2006 8:47 PM, Virgil wrote: >> > In article <1165238765.397374.303270(a)79g2000cws.googlegroups.com>, >> > mueckenh(a)rz.fh-augsburg.de wrote: >> >> >> Most "mathematicians" even don't know what potentially infinite is. >> > >> > As it is a useless idea, such ignorance is bliss. And WM's sinful >> > attempts to destroy that innocence is reprehensible. >> >> Cantor still understood that the Aristorelian potentially infinite point >> of view is quite different from actual infinity. > > Don't you think it a bit much for you to be telling us what Cantor > understood? Simply read it. > >> The formerly Archimedean axiom of infinity describes the potential >> infinity. >> >> Blissful ignorance of mathematicians does not utter complains if the >> axiom of (possibly infinite) extensionality claims the existence of a >> set which has to include all of its elements. >> >> According to my reasoning this does neither clearly include nor clearly >> exclude the actual infinity, i.e. all elements together. >> Nobody complains. Obviously, the fiction of actual infinity is merely >> required from theoretical point of view. Nobody really needs it in >> practice. This preserved ambiguity lead to the theoretical imperfections >> I reported. > > Quite right: neither "potential infinity" nor "actual infinity" occur in > modern mathematics. Time to leave the Antiquarian Bookshop and join the > 21st century. The basic errors mainly date back to 1872 and German megalomania. >
From: Eckard Blumschein on 8 Dec 2006 14:00 On 12/7/2006 7:54 AM, David Marcus wrote: > Eckard Blumschein wrote: >> On 12/1/2006 3:03 PM, mueckenh(a)rz.fh-augsburg.de wrote: >> >> > If you use the term "set" (like for instance for your set A) as defined >> > in set theory, then all the elements are "there" (where ever that may >> > be). Therefore you cannot describe potential infinity by means of ZF or >> > NBG set theory, unless you use completely different definitions of >> > "set" etc. >> >> The trick with these axioms is: They do not really define the notion >> set. > > Why not? Although, the word "characterize" would be better than > "define". > >> Notice: Cantor's untennable definition > > Which definition are you referring to? See an other reply of mine today. > >> has not been substituted by a new and correct one but the oracle-like axioms. >
From: Virgil on 8 Dec 2006 14:04 In article <45793B7C.4050105(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/8/2006 1:54 AM, Dik T. Winter wrote: > > In article <45781DC1.1000804(a)et.uni-magdeburg.de> Eckard Blumschein > > <blumschein(a)et.uni-magdeburg.de> writes: > > > On 12/7/2006 1:54 AM, Dik T. Winter wrote: > > ... > > > > Oh, well. In Bourbaki's mathematics R+ and R- both contain 0. So you > > > > are a follower of Bourbaki after all? But of course the 0's are the > > > > same. If they were different you would have quite a few problems with > > > > limits and continuity. > > > > > > According to my reasoning, any really real number is not unique but must > > > rather be void because even the tiniest interval is thought to contain > > > indefinitely not just many rational numbers but indefinitely much of > > > real numbers. > > > > But what you state here for real numbers is also valid for rational > > numbers. > > So you make not clear why in one case they are "void" (whatever that may > > mean) and in the other case they are not void. > > I beg your pardon. Though the matter is clear to me, I fear it is > difficult to explain because mathematicians are not used to think in > terms of notions with higher abstraction. On the contrary, mathematicians are used to thinking on much higher levels of abstraction than most non-mathematicians. It is the nature of mathematics to abstract. > So meanwhile every dull mathematician tells and is > even teaching that all numbers are fictitious. Wrong. mathematicians never call numbers fictitious. It is only dull non-mathematicians who insist on trying to teach their grandmothers to suck eggs. > > 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. > Perhaps it is most helpful to declare the reals just > fictions, while the rationals, including naturals and integers, are > genuine numbers. It is most unhelpful to misuse words whose common meanings tend to mislead one about their technical meanings. > > 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. And in those theories each real is a set just as each rational is a set and each natural is a set. If EB wishes to produce an axiomatic system which distinguishes between potential and actual, let's see him do it. But absent such a system, there is nowhere that such a distinction exists. > > Infinity is in some sense the opposite of being infinite. Sanity is clearly the opposite of being EB.
From: Lester Zick on 8 Dec 2006 14:06
On Fri, 08 Dec 2006 13:00:50 +0100, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >On 12/7/2006 10:51 PM, Lester Zick wrote: >> On Thu, 07 Dec 2006 10:43:31 -0800, Mark Nudelman >> <markn(a)greenwoodsoftware.com> wrote: >> >>>On 12/7/2006 3:15 AM, Eckard Blumschein wrote: >>>> Just try and refute: >>>> >>>> Reals according to DA2 are fictitious >>>> Reals according to DA2 are fictitious >>>> With fictitious I meant: They must not have a directly approachable >>>> numerical address. This was the basis for the 2nd DA by Cantor afer an >>>> idea by Emil du Bois-Raymond. >>>> >>>> Good luck >>>> >>> >>>Can you define what you mean by a "directly approachable numerical address"? >> >> The definition of "directly approachable numerical address" is "Q" >> since according to David Marcus definitions are "only abbreviations". >> >> ~v~~ > >Excellent. > >Can you tell me what ~v~~ abbreviates? Sure. It's my law of contradiction in symbolic form: "not or not not". Since "not not" is self contradictory "not" must be universally true of everything. I adopted this as my logo and signature about a year ago because some logic techies were giving me a hard time about not having a symbolic reduction for my verbal expressions of the law. Then I realized some of the modern logic symbolism I was seeing around would fit the bill quite nicely. The rest is history. ~v~~ |