Uncountability of the Rationals?
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From: David R Tribble on 5 Jun 2010 12:37 Tony Orlow wrote: > Perhaps simple bijection as a proof of equinumerosity is superficial. > That's also a possibility. :) If you have two sets, and every single member of both are paired together, in what logical sense can you say that they are not really (only "superficially") equinumerous?
From: David R Tribble on 5 Jun 2010 12:44 Virgil wrote: >> Note that the reals with their standard order satisfy Tony's definition >> of "sequence" though there is not even any explicit well-ordering of them. > Tony Orlow wrote: > There are always the H-Riffics. Remember "Well Ordering the Reals"? Yeah. Remember how several of us demonstrated that the H-riffics is only a countably infinite set, and omits vast subsets of the reals (e.g., all the multiples of powers of integers k, where k is not 2)?
From: David R Tribble on 5 Jun 2010 12:48 Virgil wrote: >> Both the rationals and the reals, with their usual orders, satisfy YOUR >> definition of sequences, and while the rationals, with a suitable but >> different ordering may be a sequence, there is no ordering on the reals >> which is known to make them into a sequence, at least for any generally >> accepted definition of "sequence". > Tony Orlow wrote: > Surely you remember the T-Riffics? Yeah. Surely you remember how you could never come up with a self-consistent notation for them? Or a self-consistent definition for incrementing from one T-riffic to the next? Or several other missing critical pieces of your theory?
From: Virgil on 5 Jun 2010 12:48 In article <ed1b8d08-87d2-427b-bb10-9fcc46cd823c(a)d8g2000yqf.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Perhaps simple bijection as a proof of equinumerosity is superficial. > That's also a possibility. :) It is certainly adequate as such a proof of equal cardinality.
From: Virgil on 5 Jun 2010 12:53
In article <9083dd54-a2f1-46af-ab41-421b3c253ea9(a)k39g2000yqb.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 4, 4:24�pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > On Jun 4, 3:20�pm, David R Tribble <da...(a)tribble.com> wrote: > > > > > Tony Orlow wrote: > > > > One might think there were something like aleph_0^2 rationals, but > > > > that's not standard theory. > > > > > Actually, there are Aleph_0^2 rationals. And Aleph_0^2 = Aleph_0. > > > > Orlow can't be bothered to learn such basics. > > > > MoeBlee > > Piffle to you both. I already stated that very fact very early in this > thread. Don't start crying "quantifier dyslexia". You know better. > > Tony To declare, as TO does above, that the cardinality of the rationals being equal to aleph_0^2 is NOT part of the standard theory, is just plain wrong! |