Uncountability of the Rationals?
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From: Tim Little on 8 Jun 2010 02:00 On 2010-06-08, Tony Orlow <tony(a)lightlink.com> wrote: > I don't even know the basics? Correct. > Excuse me, but there are several unanswered questions, even in your > provincial neighborhood. There is the continuum hypothesis What in particular do you think is unanswered about the continuum hypothesis? > there is a non-existent but "provably extant" well ordering of the > reals What is nonexistent about a well ordering of the reals? > Remember my big challenge? What is the width of a countably infinite > complete list of digital numbers? In what salient way does your "big challenge" differ from "what is the width of a snarflek"? > This is the same question as CH. Only in your own little logically inconsistent corner of mathematical theory. > That's all solved in my quadrant of the cosmos. Well of course it's solved in your quadrant - every inconsistent theory has proofs for all propositions. - Tim
From: Tim Little on 8 Jun 2010 02:12 On 2010-06-08, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 7, 7:54 pm, David R Tribble <da...(a)tribble.com> wrote: >> Indeed. Here are a few ways to write the set of naturals, >> all of them being exactly the same set: >> >> { 0, 1, 2, 3, 4, 5, 6, ... } >> { 0, 2, 1, 4, 3, 6, 5, ... } >> { 0, 2, 4, 6, ..., 1, 3, 5, 7, ... } >> { 0, 1, 3, 4, 6, 7, 9, 10, ..., 2, 5, 8, 11, ... } >> { ..., 5, 4, 3, 2, 1, 0 } >> { 0, ..., 4, 3, 2, 1 } > > Why are all of these infinite sets you cite some kind of sequence? What do you mean by "all of these infinite sets"? There's only 1 infinite set mentioned there, just different ways of writing it. The ways of writing it listed there are "some kinds of sequence" because that's how ASCII text is represented, and has nothing to do with any properties of the set itself. - Tim
From: Tim Little on 8 Jun 2010 02:15 On 2010-06-08, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 7, 5:29 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: >> On Jun 5, 1:06 pm, Tony Orlow <t...(a)lightlink.com> wrote: >> > or is less dense in the natural quantitative order >> There is no general definition that we have (let alone that you have >> provided) of "the natural quantitative order". > > It's simply a matter of the axioms of order. Examine them alone for a > bit. "The axioms of order" provide nothing more than a few restrictions on otherwise arbitrary relations. Any relation satisfying the axioms is an ordering. Which of the many possible relations for any given set is "the natural quantitative order"? - Tim
From: Tim Little on 8 Jun 2010 04:44 On 2010-06-08, Tony Orlow <tony(a)lightlink.com> wrote: > A quantitatively ordered set is one wherein each member is either > greater than, less than, or equal to every other member. Yes/no? I'd say no. That just means a "total order", which is quite arbitrary in general. The natural numbers can be totally ordered alphabetically by their English names, for example. > If it is not clear to you, then please ask a clarifying > question. > > Under the criterion of difference in the infinite case as a basis > for a formulaic inequality, the difference that either remains or > grows (more usually) stands as a basis for application of the '<' > symbol. My request for clarification: "What on Earth - or in your quadrant of the cosmos, as applicable - is that supposed to mean?" > Yes, there is no explicitly defined infinite set which is not also a > sequence, or some recursive structure with even more than one > dimension (child per level). As with many of your posts, the usual meanings attached to the mathematical terms you use make your sentence trivially false or nonsensical. So what are you actually trying to say? > In any case, you might want to give me a reason why you think infinite- > case induction does not work for espressions of inequality on > formulas of x based on differences that do not have a limit of 0 as x > increases without bound. Usually they increase without bound. For > instance, for x>2: > > x+2<x*2<x^2<2^x<x^x You say that x^2 < 2^x. Try x = 3 or 4. > You know IFR makes vastly more distinctions than cardinality. You > know infinite-case induction is currently workable, I know no such things. - Tim
From: Tim Little on 8 Jun 2010 04:47
On 2010-06-08, David R Tribble <david(a)tribble.com> wrote: > You're reading more into "set" than is there. Again, you're > making yourself look foolish, acting as though you really > don't understand what a "set" is. Given many of TO's past posts, I think the latter really is the case. He has numerous times in the past confused sets with specific ordered representations of those sets, as he has again here. - Tim |