Galileo's Paradox
in [Math]
|
From: Eckard Blumschein on 27 Nov 2006 04:10 On 11/24/2006 11:50 PM, Richard Tobin wrote: > In article <45675E35.9090008(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >>According to Spinoza, infinity is something that cannot be enlarged >>(and also not exhausted). It is a quality, not a quantity. > > He may have found that definition useful, but mathematicians don't.> > -- Richard Dedekind uttered and explained his illusory intention. So I understand why most mathematicians do not like logical consequences concerning the basics of their discipline. Regards, Eckard
From: Eckard Blumschein on 27 Nov 2006 04:21 On 11/25/2006 1:18 AM, Virgil wrote: >> According to Spinoza, infinity is something that cannot be enlarged >> (and also not exhausted). It is a quality, not a quantity. > > And how did Spinoza become an authority on the mathematical meanings of > words? > > In English, at least, we do not stick to a strict "one word, one > meaning" regimen, but allow the choice from several meanings to be > determined by context. The distinction between two mutually excluding qualities like countable and uncountable (by DA2) is not a play with words and meanings. > > And in mathematical contexts, non-mathematical meaning are irrelevant. While set theory is only something inside mathematics, proponents of it claim to speak on behalf of mathematics. While mathematics is only something inside science, Cantor got very popular when promising: The essence of mathematics is just its freedom. No. Time has proven him wrong. While still celebrated, his transfinite integers and his aleph_2 did never find any useful application.
From: Virgil on 27 Nov 2006 04:23 In article <456AAA3F.9010802(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/24/2006 11:25 PM, Virgil wrote: > > In article <45675C18.5090705(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 11/24/2006 12:35 PM, Six wrote: > >> > GALILEO'S PARADOX > >> > >> > >> I already wrote elsewhere that Galilei eventually found out: > >> The relations smaller, equally large, and larger do not hold for > >> infinite quantities. > > > > Depends on how one defines them. Cantor's definition works quite well > > for infinite sets, at least in ZFC and NBG. > > I wonder if the relations smaller, equally large, and larger require any > comment or definition. I also got not aware of a redefinition by Cantor. Cantor defined them , in terms of cardinalities, quite adequately. > > > >> Dedekind and Cantor intended to make possible the impossible and just > >> enchant irrationals into numbers of the same kind as rational numbers. > >> If this would be possible, then there were indeed more real than > >> rational numbers. > > > > It was and there are. > > Amen. Thus EB agrees that it was and there are!
From: Virgil on 27 Nov 2006 04:30 In article <456AAB6C.5010703(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/24/2006 11:50 PM, Richard Tobin wrote: > > In article <45675E35.9090008(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >>According to Spinoza, infinity is something that cannot be enlarged > >>(and also not exhausted). It is a quality, not a quantity. > > > > He may have found that definition useful, but mathematicians don't.> > > -- Richard > > Dedekind uttered and explained his illusory intention. So I understand > why most mathematicians do not like logical consequences concerning the > basics of their discipline. > > Regards, Eckard On the contrary, mathematicians deal with the logical consequences of their assumptions with great care, a form of behavior EB would benefit from emulating.
From: Eckard Blumschein on 27 Nov 2006 04:51
On 11/25/2006 1:20 AM, Virgil wrote: > In article <45675EBD.5000706(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 11/24/2006 2:03 PM, Bob Kolker wrote: >> > Six Letters wrote: >> >> > Same size (i.e. same cardinality) means there is a one to one >> > correspondence between the sets. That is the definition of "same size". >> >> Do you mean somebody here is not familiar with all of your wise utterances? > > EB seems to remain ignorant of them, however often and by whomever they > are repeated. Cantorians equate "same quality to never end" with "same size". Fraenkel wrote of generalized size. Should we understand infinity like something of huge size? Children may think so. Intelligent people may express the quality of oo by agreeing on oo + a = oo. Cantor called his putative size oo omega. Then he tried to show that there are numbers in excess of omega and started to count aleph_0, aleph_1, ... Maybe you will consider this exciting, hard to retrace, and correct. To me it is ridiculous, transparent, and pointless. |