Uncountability of the Rationals?
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From: Jesse F. Hughes on 11 Jun 2010 12:57 Tony Orlow <tony(a)lightlink.com> writes: > On Jun 10, 11:35 pm, David R Tribble <da...(a)tribble.com> wrote: >> MoeBlee wrote: >> >> MoeBlee: So you can't compare the airplane, which >> >> accomplishes a certain purpose, with your ball, which does not >> >> accomplish that purpose. >> >> ToeKnee: You just don't get it. Sigh. >> >> ChanceFor PrinceOfPull: There goes MoeBlee again silencing alternative >> >> theories. >> >> Transfer Principle (L Walker) wrote: >> >> > [...] In this analogy, >> > MoeBlee is forcing everyone to fly only on American >> > Airlines and not Virgin or any other airline. >> > I like to believe that a wonderful theory can come out >> > of allowing proper subsets to have distinct sizes, but >> > MoeBlee won't even let us _build_ the plane, much less >> > fly it anywhere. >> >> Bad, bad, naughty MoeBlee, preventing you from considering >> other theories like that. >> >> Sorry, but how exactly is he forcing you to use only ZFC? >> Perhaps you have some secret knowledge about MoeBlee's hidden >> powers over the Internet, as but far as the rest of us are concerned, >> this is a free forum for suggesting new ideas. >> >> It's Tony who keeps claiming that his bigulosity, T-riffics, IFC, and >> all the rest fit well within (or as simple extensions of) standard >> theory, >> in spite of all continuing evidence to the contrary. > > I never claimed any such thing. My ideas are clearly at odds with > transfinite set theory. I make no apologies for that, nor should I be > expected to. Please don't misrepresent my position. Thanks you for > your time and attention. Well, I guess I'm confused, too. So, you're *not* working in ZFC? What set theory are you using? Do you have that settled yet? -- Jesse F. Hughes "The Hammer has arrived." -- James S. Harris, Feb. 14 2006
From: MoeBlee on 11 Jun 2010 13:07 On Jun 11, 7:15 am, Tony Orlow <t...(a)lightlink.com> wrote: > On Jun 10, 2:42 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jun 10, 1:35 pm, Tony Orlow <t...(a)lightlink.com> wrote: > > > > and I > > > have to retreat > > > You retreat every time you fail to define your terminology in a non- > > circular way, every time you fail to answer the substantive questions > > but instead blame the actually informed people here for what you claim > > to be their obtuseness or dishonesty, every time you fail to answer > > the substantive questions but instead resort to things like corny "yo' > > mamma" talk. > No, I tend to retreat when the criticism gets too personal and harsh > and starts to seriously erode my motivation and self esteem. Fair enough, but you ALSO retreat into the forms I mentioned even when pressed in a personally neutral way. Also, you should be self-aware enough (I doubt that you are) to notice that often YOU engender the negatively personal aspects when you respond to people with your customary dismissiveness, charges that others (including well informed and talented mathematicians) are too obtuse or dishonest to understand you, juvenile sarcasm, and ridiculous bumptiousness. > BTW, you > mention a goal of set theory being to have a "size" for every set. I > would like you consider a slightly modified goal: To exactly order > sets according to size, and to distinguish between those that have an > absolute size and those that do not. Whatever "absolute size" might mean, then fine, no one's stopping your from pursuing such a goal. On the other hand, when you present gobbeldygook, then you can expect that mathematically informed people will call you on it. > For me, no countably infinite set > has an absolute size, A countably infinite set is, by definition, 1-1 with w. If you wish to have a system that defines as well "absolute size" then step right up and tell it to us...primitives, axioms, definitions. > but only one relative to other countably > infinite sets, especially the standard N, starting at 0. Uncountably > infinite sets can have an absolute size in terms of zillions. Both > kinds of infinity can be multiply tiered using infinite case induction > on all the arithmetic functions used in bijections. As above... > Also, sometimes I retreat to rethink something, when an issue in my > thinking is pointed out, which has happened several times, and which I > don't regret. I welcome logical criticism. When it gets personal, I > can get personal back, but I'm trying to behave myself. Are you? I've many times attempted to engage you in a personally neutral way. It's never worked. Quickly the conversation becomes personal, often as I steer it that way, because the elephant in the room is that you just don't have the intellectual and personal maturity to participate in a coherent mathematical or even math/philosophical discussion. But I'm always willing to try again. If you like, tell me from (a new) beginning just what you propose. Please don't just SWAMP me with a ton of UNDEFINED verbiage. What I think you fail to appreciate is how arrogant you are when you throw a bunch of undefined verbiage at people. The effect of that is that ONLY YOU have a definitive call in what does or does not hold in your "mathematics". That defies the opposing intellectual standard that a mathematical system should allow anyone to prove theorems and devise additional definitions in it INDEPENDENT of the particular human being who first proposed the system. Because of the way you present matters, only you can say, from case to case, what inferences we are to draw, because your mathematics is for the most part merely a bunch of ideas swarming in your personal brain somewhat expressed or sketched or alluded to with undefined terminology. Thus your part in the conversation is lacking in the basic mathematical/intellectual consideration of presenting a system from which OTHER PEOPLE may themselves draw inferences without having to consult you as its oracle. And even if you are not at the stage of presenting a system, but rather sketching some philosophical math/ontological notions, then still what is lacking from your are (1) an understanding of the work that's already been done so that you can put your own notions in perspective that way, as well as that your critiques of already presented mathematics are ill-premised, and (2) a sincere effort to find language, even if only illustrative, that would allow other people to approach some grasp of what it is you want in some system- later-to-be-stated; as instead you mainly barrage us with undefined math sounding terminology. So to start anew, if you wish: (1) What logic at this time do you propose for carrying out your mathematics. Classical, multi-valued, para-consistent, some other logic that you've devised? (2) Do you intend for your mathematics to provide a mathematics sufficient for the sciences? (3) Do you contend that your mathematical notions are in some sense true of some ultimate (for lack of a better word) physical or abstract world - a reality - that other mathematics (such as ZFC, constructivist mathematics, et. al) are not true to said reality? MoeBlee
From: MoeBlee on 11 Jun 2010 13:23 On Jun 11, 8:10 am, Tony Orlow <t...(a)lightlink.com> wrote: > On Jun 10, 5:06 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jun 10, 2:19 pm, Tony Orlow <t...(a)lightlink.com> wrote: > > > > > You retreat every time you fail to define your terminology in a non- > > > > circular way, every time you fail to answer the substantive questions > > > > but instead blame the actually informed people here for what you claim > > > > to be their obtuseness or dishonesty, every time you fail to answer > > > > the substantive questions but instead resort to things like corny "yo' > > > > mamma" talk. > > > Please cite an instance recently where I've used any circular logic. > > > I said circular TERMINOLOGY. You have no primitives, so your > > terminology is never grounded. It either goes circularly (and I showed > > you instances of that a few years ago), or it just dangles at terms > > themselves not defined not primitive. > > Yes, go back a few years. But your terminology remains undefined to THIS DAY. > > Or, please, I'd love for you to list your exact primitives: constants, > > predicate symbols, function symbols (or words that signal such > > formalisms). > > > A few years ago you made a pretense of doing so, but it was an > > embarrassing mess. I even took a good amount of time to help you out. > > It ended up with you getting impatient and childishly defensive as you > > ordinarily do, so that nothing really was accomplished. > Moe - You seem to think that coming up with a complete alternative to > transfinite set theory from the ground up is a simple matter to be > easily spewed out. Then when I try to tell you where it's going, you > deride me for not explaining every detail, including how logic works. > So, you'll just have to wait for the book. No, I realize the extent of such a project. I've even mentioned that even if you don't have a formal system worked out, you could at least provide some coherent overview. But you don't. MoeBlee
From: MoeBlee on 11 Jun 2010 13:43 On Jun 11, 8:20 am, Tony Orlow <t...(a)lightlink.com> wrote: > The inverse function rule. If a set of reals is bijected with a > segment of the naturals I.e., for a countable subset of R ('R' being the set of real numbers - the real numbers as we ordinarily understand them?) > using an invertible formula, What is an "invertible formula"? And the terms you use to answer that question, do they have defintions too? Does ANYTHING you say end with axioms and primitives or AT LEAST some language that is informal (thus not necessarily in sequence back to primitives) but COMMON enough that other people may grasp your notions without resort to special PERSONAL terminology? > then that inverse > formula may be used to calculate the number of elements in the set > within any given value range. "may be used to calculate" (do you mean there is a recursive function, or is your notion of 'calculate' something different from Church- Turing?) "number of elements" "value range" > Where a<b, I take it '<' is the standard ordering on R. > and where S is a set mapped > from the naturals using f(n)=x, I take it S is a subset of R. f is the bijection you mentioned earlier? > and where there exists an inverse > function g(x)=n such that f(g(x))=g(f(x))=x, I.e., where g is the inverse of f. > the number of elements in > the set S within the range [a,b] is given by floor(g(b))-ceiling(g(a)) > +1. Try it on any finite set. Now imagine applying it to the range > [0,omega]. As far as I can tell, this is not well defined, since it is RELATIVE to f (g being the inverse of f). So what we would have is "number of elements of [a b] PER f" is [...]. But, even worse (to the extent I understand what you're trying to say) there IS NO bijection between [a b] and any segment of w, perforce no bijection between S and a segment of w. Would you please restate this whole thing in standard terminology, or if mixed with your own terminology, please define to either standard terminology or to primitives. MoeBlee
From: MoeBlee on 11 Jun 2010 13:58
On Jun 11, 8:33 am, Tony Orlow <t...(a)lightlink.com> wrote: > so please don't shove that in my mouth. You complain about that the conversations get pesonal. But here is an example (and a minor one to choose among more important ones) of your rhetoric turning quickly bellicose. "shove that in my mouth", as if the poster making whatever arguments is a "shoving in your mouth". > You just don't like my perspective on the > matter. Again, as usual, Tony precludes that the objections of other posters are well meant or fairly thought out, but merely that the other poster is dogmatically opposed. > > Specifically, given an arithmetic formula f(x) on all real x, > > on what basis can you assert that it automatically applies > > to any non-real x? > > On the basis that a difference without a limit of zero in the infinite > case remains in the infinite case, and stands as the basis for > quantitatively ordering two infinite values expressed in terms of such > functions on the variable. That's infinite-case induction. I'm stupid and dishonest and mean spirited, because I don't know what the above is supposed to mean. Weird, though, that I can understand graduate texts in mathematics and follow along with the meanings of the author even though I'm not even a math student. It must be that Orlow is just too smart even for someone who, without even a formal education in mathematics, can grasp a fair amount of graduate level material. > > Explaining how this is the case for a simple example would > > help immensely. For example, take a simple real function: > > f(x) = x + 1 > > Okay. > > > Now explain how this function retains its arithmetic meaning > > when extended to non-real (infinite) x. Or even how it can > > be extended in this way at all. > > Okay. You would use this function, say, to map the naturals starting > at 0 with the naturals starting at 1. For every element x of the first > set exists an element y in the second such that y=x+1, right? No, he said f is a function on R. So your f is a different function, okay. > Now, the > inverse function is x=y-1, correct? It's f(x)=x-1. But the domains are different, so it's only an inverse on w\{0}. > In the infinite case (barring > limit ordinals, which do not figure into my theory) this difference > does not decrease to 0, but stays at a constant value of 1. WHAT is "the infinite case"? > Using IFR, > then, we can say that set y has 1 fewer elements than set x, which is > what we would intuitively desire, since it is missing element '0'. For > me, x+1>x for all x, not just for the finite. That's what you DESIRE to have from some theory you say you're writing. By the way, where '<' is the ordering on ordinals, we do have x+1>x for all x. MoeBlee |