Galileo's Paradox
in [Math]
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From: Virgil on 24 Nov 2006 17:25 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. > > 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.
From: Richard Tobin on 24 Nov 2006 17:50 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 -- "Consideration shall be given to the need for as many as 32 characters in some alphabets" - X3.4, 1963.
From: Virgil on 24 Nov 2006 19:18 In article <45675E35.9090008(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/24/2006 1:20 PM, Richard Tobin wrote: > > > >>* If you have two sets of infinite size, is the union of these sets > >>than larger than infinity? What would larger than infinity mean? > > > > That this is not a stumbling block should be clear of you replace > > "infinite" with "big". The union of two big sets can be bigger than > > either of the original big sets. Like "big", "infinity" is not a > > number; it's a description of certain numbers. > > 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. And in mathematical contexts, non-mathematical meaning are irrelevant.
From: Virgil on 24 Nov 2006 19:20 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.
From: Bob Kolker on 24 Nov 2006 19:42
Eckard Blumschein 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. Cardinal numbers are comparable. Bob Kolker |