An uncountable countable set
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From: Lester Zick on 19 Oct 2006 17:30 On 19 Oct 2006 11:28:49 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Han de Bruijn wrote: >> The confusion stems from the fact that I cannot and shall not understand >> the _infinite_ counterparts of the finite cardinals and ordinals. > >How can you understand if you won't read a book that explains it? (By >the way, Halmos is a good book, but it's just an overview; it doesn't >give you the full explanations that you need.) Or you could just give us the short version. >So you seem to think it is better to spout nonsense on the Internet >about a subject you cannot possibly understand since you insist that >you won't. And you seem to think it is better to spout nonsense on the internet about a subject you cannot possibly understand, Moe. You like set theory in its modern math reification. Others don't. So what? The group is called sci.math not sci.modernmath.settheory. ~v~~
From: MoeBlee on 19 Oct 2006 17:35 Lester Zick wrote: > And if set theorists ever get around to formulating a mathematical > theory do let us know. There are a whole bunch of recursively axiomatized mathematical theories that are set theories. Z set theory (and its variants) has been in good stead as an axiomatic theory since Skolem mentioned how to handle the previously too vague notion of a definite property. MoeBlee
From: Lester Zick on 19 Oct 2006 17:35 On 19 Oct 2006 11:20:24 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Han de Bruijn wrote: >> MoeBlee wrote: >> >> > Han de Bruijn wrote: >> > >> >>But why are the finite ordinals not equivalent with the naturals (I mean >> >>in mainstream mathematics)? >> > >> > The set of finite ordinals IS the set of natural numbers. >> > >> > x is a natural number <-> x is a finite ordinal. >> > >> > Why don't you just read a textbook? >> >> It's all very confusing. Because there also "exist" infinite ordinals, >> they say. > >You won't read a textbook in set theory because it's confusing? Or because it confuses definitions. > So, >you'd rather remain confused and completely ignorant about set theory, >while spouting nonsense about it every day on the Internet? Personally I'd rather not be confused by a theory espousing confusing definitions for no better reason than they're confusing. And I'd rather spout nonsense about theories which spout nonsense about mathematics than kowtow to their confusion. >Anyway, what is confusing about the theorem that there exist finite >ordinals and that there exist infinite ordinals? Well very little is confusing except the idea that finite cardinals are the same as finite ordinals. There we seem to have some confusion. ~v~~
From: MoeBlee on 19 Oct 2006 17:40 Lester Zick wrote: > On 19 Oct 2006 11:28:49 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > > You like set > theory in its modern math reification. Others don't. So what? The > group is called sci.math not sci.modernmath.settheory. I don't insist that conversations must be about set theory. However, when various incorrect things are said about set theory, then I may see fit to comment, as well as I may see fit to mention set theoretic approaches to certain mathematical subjects. In the present case, a poster mentioned certain things about set theory, then I responded, then he asked me questions, then I gave answers, then you commented on my answers. That hardly presents me as insisting that the scope of this newsgroup be confined to set theory. MoeBlee
From: David R Tribble on 19 Oct 2006 19:25
Tony Orlow wrote: >> That doesn't seem "real", and the axiom of choice aside, I don't see >> there being any well ordering of the reals. The closest one can come is >> the H-riffic numbers. :) > David R Tribble wrote: >> Hardly. The H-riffics are a simple countable subset of the reals. >> Anyone mathematically inclined can come up with such a set. > Tony Orlow wrote: >> You never paid enough attention to understand them. They cover the reals. > David R Tribble wrote: >> They omit an uncountable number of reals. Any power of 3, for example, >> which you never showed as being a member of them. Show us how 3 fits >> into the set, then we'll talk about "covering the reals". > Tony Orlow wrote: >> 3 is an unending string, just like 1/3 is in base-10. Rusin confirmed >> that about two years ago. But, you're right, I need to construct a >> formal proof of the equivalence between the H-riffics and the reals. > David R Tribble wrote: >> Your definition of your H-riffic numbers excludes unending strings. > Tony Orlow wrote: > Since when? Do the digital reals exclude unending strings? You misunderstand. Your H-riffics are simply finite-length paths (a.k.a. the nodes) of a binary tree. Your definition precludes infinite-length paths as H-riffic numbers. David R Tribble wrote: >> So 3 can't be a valid H-riffic, and neither can any of its successors. > Tony Orlow wrote: > Nice fantasy, but that's all it is. I suppose 1/3 doesn't exist in > decimal either. As I said, you misundertsand. Please demonstrate how 3 (or any multiple or power of 3, for that matter) meets your defintion of an H-riffic number. You claim it (they) do, and I'm asking you for proof. David R Tribble wrote: >> I know you don't get this, but go back and read your own definition. >> Every H-riffic corresponds to a node in an infinite, but countable, >> binary tree. > Tony Orlow wrote: > No, like the reals, it corresponds to a path in the tree. No, read your own definition again. Each H-riffic is a finite node along a path in a binary tree. David R Tribble wrote: >> The H-riffics is only a countable subset of the reals, and omits an >> uncountable number of reals. > Tony Orlow wrote: > Just like all finite-length reals. That is only a countable set. Exactly. The H-riffics exclude an uncountable number of reals, and thus do not cover all the reals. Tony Orlow wrote: > So, the digital reals are not the reals? Tell it to Cantor the Diagonal. Irrelevant. All of the reals can be written in digital form. But the reals with non-terminating non-repeating fractions form the uncountable set of irrationals that comprise most of the reals. |