An uncountable countable set
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From: David R Tribble on 26 Sep 2006 19:11 Tony Orlow wrote: > For the sake of this argument, we can talk about infinite reals, of > which infinite whole numbers are a subset. What are these "infinite reals" and "infinite whole numbers" that you speak of so much? If you've got a set containing the finite naturals and the "infinite naturals", how do you define it? N is the set containing 0 and all of its successors, so what is your set?
From: Ross A. Finlayson on 26 Sep 2006 19:14 Virgil wrote: > In article <1158591295.350485.163410(a)m73g2000cwd.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > Nothing has changed. There is no complete set of natural numbers. Any > > set that can be established is a finite set. Hence, the probability to > > select a number divisible by 3 is 1/3 or very very close to 1/3. > > That presumes that the allegedly finite set of naturals that can be > constructed is nearly uniform with respect to divisibility by 3 at > least, and probably by other numbers as well. What is the justification > for this assumption? Yeah! You might feel comfortable to think that, but there are obvious counterexamples in the reals with the limiting analysis. So, when you say Robinson and others you mean Robinson's NSA and IST, Internal Set Theory, when I say Robinson and others I mean Robinson and everybody else, in talking about infinitesimals. Obviously you know I say to look at Schmieden and Laugwitz, because I have my own real numbers. Also don't forget Conway and the surreal numbers and of course your potential and actual infinity and so on and so forth. Nice thing about NSA you get infinite integers, the hyperintegers, which are no different than the integers. It helps if you know everything about cardinal numbers. Ross
From: Dik T. Winter on 26 Sep 2006 19:53 In article <5d871$4518de76$82a1e228$8051(a)news1.tudelft.nl> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > Dik T. Winter wrote: > > > In article <1159211074.494116.142040(a)e3g2000cwe.googlegroups.com> > > Han.deBruijn(a)DTO.TUDelft.NL writes: > > ... > > > I'm still flabbergasted why those difficult proofs as for Fermat's Last > > > Theorem or the Poincare Conjecture are not proved then with the full > > > power of modern computers. > > > > Perhaps because computers can only be used to prove finitely many cases > > and that FLT and Poincare are not tangible to be reduced to finitely > > many cases (like 4CT)? > > Has Pythagoras ever been proved automatically? Not that I know of, and (to me) it appears unlikely. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: cbrown on 26 Sep 2006 20:10 Tony Orlow wrote: > Han de Bruijn wrote: > > Tony Orlow wrote: > > > >> Virgil wrote: > >>> > >>> Mathematicians know better. > >> > >> Define "better". Those that work in various areas of science share a > >> notion which defines science. Theories which have no means of > >> verification are not science, but philosophy. In mathematics, > >> verification really consists of corroboration by other means, > >> agreement between different approaches. In science, where you find a > >> contradiction with your theory, it needs revision. So, the scientific > >> approach to mathematics requires some criterion for universal > >> consistency, as measured by the predictions of the various theories > >> that comprise it. Where two theories collide, one or both is in error. > >> I think that's better. > > > > Precisely ! In mathematics, there are contradictory approaches, such as > > constructivism (Brouwer) against axiomatism (Hilbert). Its practicioners > > are asked to be "nice" to each other and to "reconciliate" the different > > points of view, which turns out to be a hopeless task. Such a situation > > would be unthinkable if mathematics aimed to be a science. > > > > Han de Bruijn > > > > Well, Han, I'm not sure I agree with the statement that reconciliation > is hopeless. Is it hopeless to reconcile the wave nature of elementary > entities with their particle nature? No, the same dual natural exists > for every object. It's just that, at that scale, the wave nature is as > prominent as the particle nature. If the two contradict each other, the > universe isn't consistent. But it is. At least science acknowledges that. > > There is confusion about my "definition" of infinitesimals, because I > can see the validity both in nilpotent infinitesimals and in those that > are further infinitely divisible. Actually, there is confusion about your "definition" of infinitesimals, because your "definition" isn't a mathematical definition - it's "something that's infinitely small", whatever that is supposed to imply. > It's a matter of application, the > former lending themselves to finite measure and the latter to infinite > recursion. It's a matter of scale, where the relatively infinitesimal > has no measure. > Since you haven't defined what you mean by "finite measure" or "infinite recursion" in mathematical terms, I have no idea what the distinction is here. > Constructivism and Axiomatism are two sides of a coin. Do you think that constructivists don't use axioms? I think instead of "Axiomatism", HdB meant "Formalism". Cheers - Chas
From: imaginatorium on 26 Sep 2006 23:33
David R Tribble wrote: > Tony Orlow wrote: > > For the sake of this argument, we can talk about infinite reals, of > > which infinite whole numbers are a subset. > > What are these "infinite reals" and "infinite whole numbers" that you > speak of so much? > > If you've got a set containing the finite naturals and the "infinite > naturals", how do you define it? N is the set containing 0 and all > of its successors, so what is your set? Oh, I should think it's the same, only if you keep an open mind, and try harder. (How'm I doing, Tony?) Brian Chandler http://imaginatorium.org |