Galileo's Paradox
in [Math]
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From: Eckard Blumschein on 12 Dec 2006 08:07 I will perhaps no longer reply to nonsensical replies. On 12/11/2006 9:29 PM, Virgil wrote: > In article <457D53C3.3060308(a)et.uni-magdeburg.de>, >> > "uncountable" mean? >> >> Uncountable is >> definitely not a property of numbers. Numbers are always countable. >> Nonetheless a single real "number" is uncountable. > > Typical self-contradiction. "Uncountable" of an object means that it is > a set whose members cannot be injected into the set of naturals. Uncountable is the opposite of countable. Therefore it has also to include non-sets. > Which > Dedekind cuts does EB claim are uncountable by this definition? E.g. sqrt(2).
From: Eckard Blumschein on 12 Dec 2006 08:20 On 12/11/2006 9:40 PM, Virgil wrote: > How does EB think he can deduce that N = oo? I did not claim deducing N = oo. > Then let us eliminate all "genuine" reals, which according to EB are > individually uncountable objects and restrict ourselves to what EB calls > putative reals, the Dedekind cuts, each of which is, as a set, countable > with cardinality 2. You did not understand anything.
From: Eckard Blumschein on 12 Dec 2006 08:23 On 12/11/2006 9:44 PM, Virgil wrote: > In article <457D59C5.90007(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/6/2006 12:18 PM, Bob Kolker wrote: >> > Eckard Blumschein wrote: >> >> >> >> >> >> A subset inside the reals is comparable to a piece of sugar within tea. >> >> >> > >> > No. The elements of a piece of sugar are not tea. >> > >> > A is a subset of B if and only if every element in A is an element in B. >> > >> > Why do you make a simple concept more difficult with inept and inapt >> > analogies, when a straightforward definition is at hand? >> > >> > Bob Kolker >> >> You seems to like straigtforward definitions instead of thinking. >> Definitions are not always reasonable. Do you know what 1/2 and 1/4 >> definition caused some people to remove a signboard Sarah Heydrich and >> replace it by S. Heydrich? >> >> My metaphor sugar in tea refers to the really reals, those which were >> assumed for DA2 and those which resulted like the power set of the >> naturals. > > If we do away entirely with what EB calls DA2, We still have the same > set of Dedekind-cut reals as before, with each such real being a > finite, therefore countable, set, but the set of all of them being, by > Cantor's first proof, uncountable. Cantor's first diagonal argument (stolen from Cauchy) does not show uncountability of the reals but merely countability of the rationals.
From: Eckard Blumschein on 12 Dec 2006 08:31 On 12/11/2006 9:59 PM, Virgil wrote: >> > that your own understanding of mathematics is superior to that of >> > thousands of others who have spent much more time and effort and talent >> > in gaining their understanding than you have. >> >> Concerning time you may be correct. However, do not underestimate the >> talents by Galilei, Spinoza, Gauss, Kronecker, Poincar�, Wittgenstein >> and many others. > > Just as Hilbert could improve on Euclid's geometry, those who follow can > improve on what those who have gone before have wrought, but not the > reverse. Kronecker was the teacher of Cantor. Normally, a good student has to be in the end better than his teacher. Kronecker called Cantor in public someone who spoils the younger generation, and he was correct with this judgement.
From: Eckard Blumschein on 12 Dec 2006 08:38
On 12/11/2006 7:10 PM, Tony Orlow wrote: > Eckard Blumschein wrote: >> More is just inappropriate as to describe something infinite. There are >> not more naturals than rationals. There are not equaly many of them, >> there are not less naturals than rationals. >> >> > So, you wouldn't agree that every natural is a rational, Every natural is a rational, yes. > and that there are "more" rationals besides those? No. In this even the dullest Cantorians agree with me. They say: There is a bijection between both. I add: Galilei: incomparable |