Cantor Confusion
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From: Virgil on 5 Dec 2006 15:08 In article <1165323140.395876.228500(a)79g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > stephen(a)nomail.com schrieb: > > > > 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, > > Do you really think that there are people who do not understand set > theory (if they try)? No one but WM can say for certain whether WM has really tried to understand axiomatic set theory, but it is evident that he has either failed or is being deliberately crankish. > Do you need this conviction for your self-respect? Nothing WM can say or do will affect any mathematician's self-respect, whether amateur or professional, but it can, and does, effect the nature of their opinion of WM.
From: Virgil on 5 Dec 2006 15:20 In article <45756F17.1010409(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/5/2006 1:20 AM, Virgil wrote: > > In article <45746B98.5040606(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 12/1/2006 8:20 PM, Virgil wrote: > >> > In article <1164967792.130794.251330(a)j72g2000cwa.googlegroups.com>, > >> > mueckenh(a)rz.fh-augsburg.de wrote: > >> > >> >> May be if you apply your personal definition of potentially infinity, > >> >> but not if you apply the generally accepted definition. > >> > > >> > What "generally accepted" meaning is that? Most mathematicians do not > >> > accept that a set can be "potentially" infinite without being actually > >> > so. > >> > >> I see it quite differently: Potentially and actually infinite points of > >> view mutually exclude each other as do countable and uncountable, > >> rational and irrational. > > > > Except that countable and uncountable coexist within the same set theory > > and rational and irrational coexist within the same real umber field. > > > Cantor's DA2 illustrates that there is no such field/list of real numbers. EB conflates "list" with "set". Nothing in any axiomatic set theory I am aware of requires sets to be lists, or even listable. > Isn't this "coexistence" on the same low level of abstraction a basic > though hard to unveil intentional mistake by Dedekind? What "coexistence"? > Dedekind argued: As naturals can be extended to the integers in order to > allow subtraction and include negative numbers, and integers can be > extended to rationals in order to allow division and include fractions, > so rationals can perhaps be extended to reals in order to allow > non-linear operations and include irrationals. As Dedekind (and others) demonstrated precisely how to construct each of these extensions, his arguments conclude with Q.E.F. > Being mislead by the idea of a dotted line of numbers, he overlooked two > aspects. First of all, the irrationals cannot be located numerically. The "cuts" can be defined precisely. Locating the numbers exactly on a number line may be a a problem for engineers, but is not one for mathematicians. > Secondly, the irrationals are not an addendum to the reals but the other > way round, the reals vanish completely within the sauce of irreals. That "sauce of irreals" is unknown to mathematics. What engineers cook up is not our problem. > The irrationals are at best fictitious numbers because they do not have > an exact numerical representation available. All numbers are equally fictitious, mere creations of the mind.
From: Virgil on 5 Dec 2006 15:23 In article <45757068.8060203(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/5/2006 1:17 AM, Virgil wrote: > > In article <45746ACD.1020008(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 12/1/2006 8:27 PM, Virgil wrote: > >> > In article <45700481.7010300(a)et.uni-magdeburg.de>, > >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> >> Why should mathematics be esoteric? > >> > > >> > Not all of it is. Various bits of it come at various levels of > >> > abstraction, and even children understand the least esoteric bits. > >> > >> For my feeling, Dedekind and Cantor were lacking power of abstraction. > > > > From present evidence, they had a great deal more "power of abstraction" > > than EB has. > > Do not confuse Cantor's virtue of belief in god given sets with my power > of abstraction. Cantor's religious beliefs are as irrelevant as EB's beliefs in his own infallibility. > Cantor said: Je le vois, mais je ne le crois pas. Obviously he didn't > infinity. What is "he didn't infinity" supposed to mean?
From: Virgil on 5 Dec 2006 15:43 In article <1165324497.159718.12520(a)80g2000cwy.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > cbrown(a)cbrownsystems.com schrieb: > > > > > I think, nobody would oppose to dividing the edges merely in two halves > > > each. If the series 1 + 1/2 + 1/4 + ... yields 2, then we can extend > > > this knowledge to bijections too. > > > > If I say that the sets {a} and {p,q} have the same cardinality, > > I do not follow this "path". Only for 1/2 edge together with another > 1/2 edge together I assert to have 1 edge with cardinal number 1. And > that is correct. For each path, WM requires an infinite series of edges (or nodes), but it is well known that the set of infinite series, i.e., all functions from N to any set of more that one element, is uncountable, so that WM's alleged bijection between the set of edges ant paths is actually a bijection between the set of infinite sequences of edges and paths. WM's attempts to impose this swindle as a valid bijection is pathetic.
From: Virgil on 5 Dec 2006 15:46
In article <1165324818.172734.85540(a)j44g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1165237393.288598.129130(a)j44g2000cwa.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > William Hughes schrieb: > > > > > > > > > > > > > > > Informally we have that a potentially infinite set is a set > > > > which is always finite, but to which we can add an element > > > > whenever we want. We say that x is an element of > > > > the potentially infinite set if we can add enough elements > > > > to get to x. > > > > > > Yes. In particular this method of adding elements guarantees that such > > > a set can never be uncountable. > > > > But having "added an element to it" produces a different set according > > to the axiom of extensionality. > > Please learn: Common set theory is not possible with potentially > infinite sets. But it is quite possible with inductively closed sets. |