Cantor Confusion
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From: Eckard Blumschein on 7 Dec 2006 07:39 On 12/7/2006 4:18 AM, Virgil wrote: > In article <4576D8A1.1030807(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/5/2006 10:54 PM, Virgil wrote: >> > In article <4575A562.60602(a)et.uni-magdeburg.de>, >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> > >> > >> >> 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. >> > >> > EB should read what axioms of extensionality actually says before >> > pontificating. >> >> 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) > > Then it does not state the truism that a set must contain all its > elements as EB originally implied. Truism?
From: mueckenh on 7 Dec 2006 07:39 Bob Kolker schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > Those who pretend to be able of knowing every integer and, therefore, > > to imagine the whole actually infinite set. > > If there were an integer I could not imagine, there would have to be a > least such integer. Well, it isn't 0 or 1. So the least integer I could > not image has a predecessor. But I can imagine it. I could also imagine > adding one to it. > > Contradiction. > > Bob Kolker You cannot imagine the integer [pi*10^10^100]. Contracontradiction. Regards, WM
From: Eckard Blumschein on 7 Dec 2006 07:39 On 12/7/2006 4:10 AM, Virgil wrote: > In article <4576D4C7.4070207(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/5/2006 10:41 PM, Virgil wrote: >> > In article <457580FA.1030700(a)et.uni-magdeburg.de>, >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> >> >> >> That there are more rationals than reals? >> > >> > WE do not make any such assumption. >> >> I perhaps meant more reals than rationals. I apologize. >> >> Dedekind as well as Cantor started from this idea. Cantor even >> fabricated the notion cardinality in order to quantify the putative >> difference in size. Considering the rationals a subset of the reals, >> dull people conclude that there must be more reals than rationals. > > There certainly cannot be fewer, or even just as many, so what other > option is open/ the 4th one
From: mueckenh on 7 Dec 2006 07:41 Dik T. Winter schrieb: > In article <1165492756.322548.255040(a)j72g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1165421463.339178.48680(a)j44g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > ... > > > The set in the quantifiers you are using is not finite either. The > > > quantifier are not over a single line, but over the set of natural > > > numbers. > > > > For finite natural numbers, we have finite lines only. It is not the > > question of a single line. Every line is finite. Therefore there is no > > line where quantifier reversal could not be applied. > > But the quantifier reversal is *not* applied to individual lines, it > is applied to the set of natural numbers. No. Regards, WM
From: Han de Bruijn on 7 Dec 2006 07:46
mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: >>Can you provide a quote from Hrbacek and Jech where they state that >>everything is a set? > > [ ... ] So the only objects with which we are concerned from now on > are sets. > > But this point of view is also entertained in many other modern books: > In ZFC everything is a set. Exactly! Set Theory rules the world of mathematics, inasmuch as Marx's ungeheure Waren-sammlung rules our capitalistic economies. I prefer to call this temporary phenomenon the Storehouse or Supermarket Paradigm: http://hdebruijn.soo.dto.tudelft.nl/jaar2006/Supermarkt.jpg Mainstream mathematics just licks the hand by which it's fed. Wouldn't dare to act otherwise. This paradigm was introduced by Cantor, the son of a merchant, the son of a storehouse owner. What a coincidence, huh .. Han de Bruijn |