Uncountability of the Rationals?
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From: David R Tribble on 5 Jun 2010 14:13 Tony Orlow wrote: >> Perhaps simple bijection as a proof of equinumerosity is superficial. >> That's also a possibility. :) > David R Tribble wrote: >> 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? > Tony Orlow wrote: > I can say so in the sense that one may be a proper subset of the > other, or is less dense in the natural quantitative order, ... Yes. But how does that make them not equinumerous? > .. or that > bijection is only part of a problem which is only properly solved by > taking into account the mapping function between the two sets. Take > your choice. The problem being, what, exactly? What we mean by "size of an infinite set"?
From: David R Tribble on 5 Jun 2010 14:22 Tony Orlow wrote: >> There are always the H-Riffics. Remember "Well Ordering the Reals"? > David R Tribble wrote: >> 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)? > Tony Orlow wrote: > Sure when using 2 as a base, the numbers you mention are uncountably > distant from the beginning of the uncountable sequence. But then, I am > using "uncountable sequence" in a rather nonstandard way. It's more basic than that. Your H-riffic set completely omits most of the reals, such as any multiple of any integral power of 3 (e.g., 3, 1/3, 27, etc., ad infinitum). Besides, your set is only countable (which is obvious from its very definition), so it can't possibly contain all the reals. It doesn't even contain all the rationals.
From: Virgil on 5 Jun 2010 23:17 In article <ae9e776b-2cad-4b09-b6d9-908b2f23978d(a)y12g2000vbr.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 5, 12:44�pm, David R Tribble <da...(a)tribble.com> wrote: > > 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)? > > Sure when using 2 as a base, the numbers you mention are uncountably > distant from the beginning of the uncountable sequence. But then, I am > using "uncountable sequence" in a rather nonstandard way. So non-standard that not even Tony Orlow has any idea what he is talking about.
From: Virgil on 5 Jun 2010 23:18 In article <8c60ceed-ef2c-4f49-b3e0-fa669e0c1e33(a)s41g2000vba.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 5, 12:48�pm, David R Tribble <da...(a)tribble.com> wrote: > > 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? > > That doesn't ring a bell. > > :) TOny Does FOR me!
From: Virgil on 5 Jun 2010 23:22
In article <60e3188a-0cc8-4a43-abf6-0cd75570bdad(a)o15g2000vbb.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 5, 12:53�pm, Virgil <Vir...(a)home.esc> wrote: > > In article > > <9083dd54-a2f1-46af-ab41-421b3c253...(a)k39g2000yqb.googlegroups.com>, > > �Tony Orlow <t...(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. > > > > 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! > > I didn't say that was its cardinality, and if I had, it wouldn't > matter because aleph_0^2=aleph_0 in standard theory. Perhaps Tony should not have said "One might think there were something like aleph_0^2 rationals, but that's not standard theory" when it is PRECISELY standard theory that there are PRECISELY aleph_0^2 rationals. |