An uncountable countable set
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From: cbrown on 21 Sep 2006 01:08 stephen(a)nomail.com wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > > > <snip> > > >> I did give another curve with the same "Tlimit" as the staircase in the > >> limit, which produced an interesting result, giving weight to the notion > >> that an infinitesimal is something distinct from 0, whose square is not > >> distinct from 0. > > > Suppose we let B represent Big'un; then B*1/B = 1, where 1/B is an > > infinitesimal. Then what you are saying means > > > 1/B = 1/B > > 1*1/B = 1/B > > (B*1/B)*1/B = 1/B > > B*(1/B*1/B) = 1/B > > B*(1/B^2) = 1/B > > > Since 1/B is infinitesimal, its square is not distinct from 0; so... > > > B*(0) = 1/B > > 0 = 1/B > > > So 1/B is identical to 0. Where is my error? > > Assuming that Tony's definition of infinitesimal has anything > to do with any standard definition of infinitesimal. :) If TO claims that every T-infinitesimal, when T-squared, is not T-distinct fom T-0, I'm T-willing to T-take him at his T-word :). Cheers - Chas
From: Virgil on 21 Sep 2006 01:20 In article <45119c60$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >>> It isn't. What does "successor to N" even mean? > >> Ask von Neumann. It is the set of all naturals, > > > > What are you talking about? The successor of w is not w. The successor > > of w is wu{w}. > > The size of the set x is the value What "value" > and it contains every natural less > than x. Cardinalities, as cardianlities, do not contain anything. > Is this not the generalizable scheme? No! > > > > >> and any ordinal being > >> the set of all preceding naturals, > > > > What are you talking about? An ordinal is the set of all preceding > > ordinals, not necessarily the set of all preceding naturals. > > Aren't finite ordinals and cardinals and naturals the same thing? Perty > much. > > Really no > need to pretend there's any smallest infinity, is there? And where does TO propose to find one smaller than Aleph_0.
From: Virgil on 21 Sep 2006 01:26 In article <45119eae(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> Omega is successor to N > > > > No, it is not not. Why do you keep repeating your error? > > In the limit ordinal sense, it's what comes after the set of finite > naturals. The "limit ordinal sense" says that a limit ordinal is not "the successor" of any ordinals. They are, in fact, the only ordinals, other than {}, which are not successors of other ordinals. So That TO has it backwards. AS usual.
From: Han de Bruijn on 21 Sep 2006 02:57 Tony Orlow wrote: > Virgil wrote: > >> The axioms system of Euclidean geometry is inconsistent with that of >> various non-Euclidean geometries. and there are a lot of other places >> where one system contradicts another. > > What happened is that the parallel postulate was downgraded to a > postulate, rather than a law. Whether two parallel lines meets at 0, 1 > or two places determines the type of space we are discussing. To say > that the parallel postulate applied and also DIDN'T apply to a GIVEN > space would contradictory. I would like to add that these non-Euclidian geometries have a _model_ inside Euclidian geometry. (If I remember well, the hyperbolic case is covered by Euclidian geometry inside a circle). It's a bit off-topic but I have the following question. The mathematical discipline called 'analytical geometry' - as teached in our schools - is the algebraic equivalent of Euclidian geometry. OK ? Parallellism of two straight lines can be detected algebraically. And moreover, two straight lines CAN be parallel. How then can 'analytical geometry' translate into an algebraic equivalent for non-Eulcidian geometries? Han de Bruijn
From: Han de Bruijn on 21 Sep 2006 03:05
imaginatorium(a)despammed.com wrote: >>>>Han de Bruijn wrote: >>> >>>>>Precisely! Mathematicians get confused by the idea of a "bijection", >>>>>which is an Equivalence Relation, which in turn is a "generalization" >>>>>of "common equality" (yes: the one in a = b). But the funny thing is >>>>>that EQUALITY HAS NEVER BEEN DEFINED. > > Idiot. Do you know the definition of an equivalence relation? Do you > claim that bijection is not an equivalence relation? I'm not an idiot. I *know* the definition of an equivalence relation. And I know that a bijection is an equivalence relation. Mind the ""s. > (I think that was a different crank - here's Tony...) And wash your mouth. Han de Bruijn |