Galileo's Paradox
in [Math]
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From: Virgil on 12 Dec 2006 15:28 In article <457EA849.6030606(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > If Galilei was correct If even Homer nodded, and Euclid , too, what makes Galileo Galilei impervious to error? > > Axioms are arbitrary. Accepting axioms is bound to reasonable basics. > This basis is lacking in case of ZFC. It is at least equally lacking in the assumptions EB would force on everyone. > Therefore your question has a > distracting aspect. The axiom of power set was constructed in order to > make sure that there are subsets of the real numbers. Actually, the entire construction of the Dedekind model can be done in ZFC or NBG without invoking the power set axiom once. > > The reals according to DA2 are categorically different from the reals > according to mandatory definitions. Then roundfile DA2. We don't need it as long as we have DA1. > > V> How is that relevant to your "calculating binomial coefficients" claim? > > The whole ZFC issue tries to maintain what I consider the D&C illusion > that real numbers are quite normal numbers (in the language of C. > numbers with full civil rights within the kingdom ...). It is obvious > that one can only calculate all variations if one has actually all > elements. > > > Avoiding a question by asking an unrelated one is a copout. > > I did not avoid questions on condition they are meant honestly and do > not just waste our time. EB considers having to answer questions for which he does not have ready answers a waste of time. But it is the questions he does not answer that show the holes in his philososphy. > >> This is Bolzano's religious thinking. > > > > It is far preferrable to EB's religious thinking. > > Bolzano was a priest, and his mathematics was closely related to his > belief in god. It is on a par with EB's belief in his own version of nonsense.
From: Virgil on 12 Dec 2006 15:31 In article <457EA97A.3030909(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > 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). I count that as one irrational. So, having been counted, it is hardly uncountable. Note that the word "uncountable" has a meaning in mathematical discussions that overrides any of EB's puerile attempts to redefine it.
From: Virgil on 12 Dec 2006 15:33 In article <457EACA0.60506(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > 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. That I do not kowtow to nonsense does not limit my understanding of things that make good sense.
From: Virgil on 12 Dec 2006 15:38 In article <457EAD62.1050108(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/11/2006 9:44 PM, Virgil wrote: > > 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. I am not talking of any sort of "diagonal arguments" but of Cantor's first proof, wherever he got it from, that the reals are not countable. As far as I can recall, the rationals were totally irrelevant in that proof.
From: Virgil on 12 Dec 2006 15:40
In article <457EAF3C.9090701(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/11/2006 9:59 PM, Virgil wrote: > > 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. It is often the case that when the teacher is surpassed by his student he becomes bitter and spiteful, as Kronecker did. |