Uncountability of the Rationals?
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From: Tony Orlow on 13 Jun 2010 11:10 On Jun 11, 6:56 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 11, 5:11 pm, Tony Orlow <t...(a)lightlink.com> wrote: > > > > > > > On Jun 11, 12:45 pm, David R Tribble <da...(a)tribble.com> wrote: > > > > Tony Orlow wrote: > > > > 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. For me, no countably infinite set > > > > has an absolute size, but only one relative to other countably > > > > infinite sets, especially the standard N, starting at 0. > > > > Wouldn't it be easier to say that some chosen countably infinite > > > set has a standard "absolute" size, and then that all other such > > > sets have a size that is some fraction of that size? Thus every > > > set then has an "absolute" size, based on the standard set size. > > > That might be easier, and would lead, as suggested, to natural > > density. > > With the word "fraction" in scare quotes, and allowing also over the > size of the "absolute" it's along the lines of ZFC cardinality. Actually a finite fraction or multiple of omega is "equicardinal" with omega. > > > However, that doesn't address the relations between infinite > > sets that have other ratios or offsets besides the finite. > > Okay, but we take w as the "basic infinite size" and find that there > are other sets that have larger infinite size. Yes, that is what you do. You have an acceptable proof that there exists some higher infinity than that of the naturals, basically based on power set, or more generally, N=S^L, however, as long as S and L are countable, so is N, the way I see it. I know that's wrong according to your theory. I make no apologies. > > > It is much > > more fruitful > > Fruitful in WHAT sense? Toward WHAT end? In that it makes more distinctions between infinite sets, observes where it cannot absolutely measure size and compares parametrically, and fails to offend real intuitions regarding the realtion between sets and proper subsets. > > > to look for a unit infinity based on uncountable > > infinity, which can be much more easily combined with measure and > > topology. > > How so? By making the count of the "uncountable" set completely consistent with spatial measure, down to the "point", or really, "fluxion". > > (I'm skeptical that you even know what a topology is.) I understand your skepticism. > > Anyway, in ZFC we already do have a least uncountably infinite size. > Take that as the "basic infinite size" if you like. Then w is one > infinite size less. CH means there is a question about c=aleph_1. Bigulosity does not leave such questions unanswered. > > > > For convenience, you'd want to choose the "largest" countably > > > infinite set for the standard "absolute" set size. This would > > > probably be N, since it contains all of the countable naturals. > > > (Another logical choice would be Z.) Given this, you could then > > > say that every countably infinite set does indeed have an > > > "absolute" set size.\ > > I'd take the least (where 'least' means is a subset of all others). I > don't know how w would be reckoned as the largest. > > > But, for me, they do not. If N starts at the first location with 1, > > Then N is w\{0}. w can start anywhere according to your theory and it doesn't matter. N starts with 1 in my theory. Sorry for my previous error of 1, by suggesting it may start at 0. It's the most common mistake in computation. Oopsles! > > > and then the second with 2, etc, then the nth natural is n, and unless > > a set has n elements, there is no nth one. If there are omega > > naturals, then there is an omegath, > > You simply POSIT that. It's not the case in ordinary mathematics, so > if you want it, then you need axioms to prove it from. Consider it a model of the naturals. n is the nth element of N. If the initial sequence of size n exists, there exists the nth one. Therefore, if the size is omega, then omega is a member of the set. However, omega cannot be in the set, because the set contains all, and only, the finite naturals, and no natural can possibly be the size, because there is always a natural after it, since the set has no end. Potential, but not actuality. > > > which by the order=value > > definition of that set must equal omega. However omega cannot exist > > within the set. Consider this. Are the following two statements > > logically equivalent? > > > AneN n<omega > > > ~EneN n>=omega > > No, they are not LOGICALLY equivalent. They are equivalent in certain > THEORIES, such as ZFC. Oh. I thought you kinda liked ZFC... (sheesh) > > > The first says there is no ntaural that can be the size of the entire > > set of naturals. That's true. > > It entails what you just said; I wouldn't say that is what it says in > itself. Instead? > > > The second says that there is no n in N that is greater than or equal > > to this "size". > > Okay, I'll play along with that. > > > Does either imply, actually, that such a size exists? No. > > We have an AXIOM that entails that there is a set that has as members > all and only the positive natural numbers (what you call 'N'). You do not have an axiom definining the operator on x: |x|. If you do, present it as a starting point, so I can get across my extnesion in parlance you can understand. > > > Both assert > > that no natural number will suffice. However, this is not to say that > > there exists ANY number which does. > > The formulas you mentioned don't entail that there is such an ordinal > number, true. But we do prove from our AXIOMS that there exists such > an ordinal number. Uh huh. How so? > > > Logically, you must concede, > > neither statement proves that omega exists as a number. > > There's really nothing to concede. It's never been at issue. Eyep, s'kinda central, ackchooley. > > > I am not > > obligated to imagine such a number, or give it any credibility. > > Of course you're not. You're not obligated to imagine anything at all. Nope > > > When > > it comes to the countably infinite, we can only look at this as some > > kind of limit, > > It is a "limit" in a certain special sense; it is a limit ordinal. So > what? Ordinal schmordinal. > > > and for most discriminating results, consider them as > > Big-O measures or something similar. > > Too bad I don't have my dictionary of Orlowism to look up 'Big-o > measure'. Fuckin' google it. Big O measure. Study computation for bananaface's sake. http://www.itl.nist.gov/div897/sqg/dads/HTML/bigOnotation.html > > Anyway, in all your verbiage I still don't see a substantive point > other than, pretty much, this: > > The existence of a set of all the positive natural numbers doesn't > follow from the formulas you mentioned. > > Yes, we agree. > > MoeBlee- Hide quoted text - > Wow cool. Oy
From: Virgil on 13 Jun 2010 14:35 In article <ada300e3-f8d1-4ace-b40e-d15c968cb0dd(a)z10g2000yqb.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 11, 12:45�pm, David R Tribble <da...(a)tribble.com> wrote: > > Tony Orlow wrote: > > > 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. For me, no countably infinite set > > > has an absolute size, but only one relative to other countably > > > infinite sets, especially the standard N, starting at 0. > > > > Wouldn't it be easier to say that some chosen countably infinite > > set has a standard "absolute" size, and then that all other such > > sets have a size that is some fraction of that size? Thus every > > set then has an "absolute" size, based on the standard set size. > > No, it is easier to establish some absolute unit infinity for the > number of points in the unit line segment, and wed the uncountable to > measure, while treating the uncountably infinite as more of an > undetermined size, parametrically compared over iterations. Nonsense is better than sense only in children's books. > > > > > For convenience, you'd want to choose the "largest" countably > > infinite set for the standard "absolute" set size. This would > > probably be N, since it contains all of the countable naturals. > > (Another logical choice would be Z.) Given this, you could then > > say that every countably infinite set does indeed have an > > "absolute" set size. > > Actually, in Bigulosity, there are much larger countably infinite > sets. The square roots of the naturals have a higher "density" than > the naturals. In terms of natural density, they probably have a > measure of oo? Not sure. For me, there are omega^2 number of square > roots of naturals within the range of N. However N, starting from 1 > (not 0 as I mistakenly said before) is the standard countable > infinity, compared to which all others are measured, according to IFR. However, so far there have been no theorems or other work in Bigulosity theory of any value to anyone, other than merely for bragging purposes. > > > > > Of course I've glossed over all the important details, but that's > > really your job to provide. > > I think I've responded appropriately. Not with anything that shows "Bigulosity" to have any mathematical value whatsoever.
From: Virgil on 13 Jun 2010 14:41 In article <b64f1e03-f862-4255-82ec-14a31e8c10a6(a)i28g2000yqa.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 11, 2:47�pm, David R Tribble <da...(a)tribble.com> wrote: > > Well, the list of all finite-length digit strings contains > > only finite-length strings, and is a countably infinite list. > > The list of all infinite-length digit strings contains > > only infinite-length strings, and is an uncountably infinite > > list. > > I'm sorry, Tribbs, but are you implying that these strings are longer > than countably infinite, or are you sying the lists are equally wide, > OR, are you saying there are different "kinds" of countably infinite > widths? Please expliculationalize. David's exposition is already clear enough to anyone with any mathematical acumen. And he is clearly NOT saying any of the things you suggest, as anyone who can read can tell.
From: Virgil on 13 Jun 2010 14:45 In article <b3a0465e-25c4-4dff-84d3-aa541d678f4b(a)h13g2000yqm.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 11, 3:33�pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > On Jun 11, 2:07�pm, Tony Orlow <t...(a)lightlink.com> wrote: > > > > > > > > > > > > > On Jun 11, 8:47�am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > > > > > Tony Orlow <t...(a)lightlink.com> writes: > > > > >> > (1) It can be classically inductively proven that f(x)>g(x) for > > > > >> > every > > > > >> > real value greater than some finite x (the base case), and > > > > > > >> Does that imply that these are real functions? > > > > > > > Yes, they can be any invertible arithmetic functions. Does that pose > > > > > a > > > > > problem for you? Probably, since I am applying them to infinite > > > > > values. However, that's what infinite case induction is for. > > > > > > No, your description of infinite case induction only gave certain > > > > sufficient conditions for concluding that f(omega) > g(omega). �It does > > > > *not* give any reason to believe that a function f defined on natural > > > > (or real) numbers can be extended to infinite sets in some unique way. > > > > You are presuming that without stating it explicitly. > > > > > Please reread what I wrote above. I am saying that for IFR the two > > > mapping functions are mutual inverses, which means they form a > > > bijection. I was saying that your problem with my idea occurs when I > > > apply these functions to infinite values, because you don't consider > > > them to be defined for nonstandard values. I am telling you that this > > > application to the infinite for the purposes of IFR is acheived > > > through the application of ICI. Do you understand what I wrote now? > > > > What's "ICI"? > > (sigh) Infinite Case Induction. > > > > > > Okay, do you disagree with this statement? > > > > > AneN n<omega > > > > What does '<' stand for? > > Same thing it always does. Normal order. Again: > > a<b ^ b<c -> a<c > ~(a<b ^b<a) > ~(a<b v b<a) = (a=b) There are all sorts of "normal orders" on large sets, so WHICH normal order do you refer to? > I don't have > lice, or nits. We do not take your unproved assurances for other things, so why should we take that one?
From: Virgil on 13 Jun 2010 14:50
In article <0f7e5b06-128e-4bf2-a0b2-810f0cb81f32(a)r27g2000yqb.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 11, 3:49�pm, Virgil <Vir...(a)home.esc> wrote: > > In article > > <e84e2ffc-fc2a-4008-8ff6-e9d097aed...(a)g19g2000yqc.googlegroups.com>, > > �Tony Orlow <t...(a)lightlink.com> wrote: > > > > > > > > > > > > > On Jun 10, 3:40 pm, Virgil <Vir...(a)home.esc> wrote: > > > No, they are exactly invertible functions, as are used in any > > > bijection. IFR is simply an extension of bijection and an extension of > > > natural density which derives more distinctions than either of those. > > > > > > > (1) It can be classically inductively proven that f(x)>g(x) for every > > > > > real value greater than some finite x (the base case), and > > > > > > Does that imply that these are real functions? > > > > > Yes, they can be any invertible arithmetic functions. Does that pose a > > > problem for you? Probably, since I am applying them to infinite > > > values. However, that's what infinite case induction is for. > > > > Real functions do allow infinite quantities either as arguments or > > values, so that you are making false statements. > > > > Don't correct me incorrectly. Standard real functions take real value > parameters and return a real result. Where they have return an > infinite value, they are declare "undefined" or to have some infinite > "limit" at that point, and where any parameters are of infinite > nature, the modern mathematical procedure is to refer to the "limit" > of the function as one or more variables approach oo or -oo. To lie by > accusing someone is lying is...lame. > > > > > > > > (2) The difference between the two terms which establishes the > > > > > inequality does not have a limit of zero as x approaches infinity, > > > > > then > > > > > > What is the "arithmetic" of the set in which those differences occur? > > > > > The arithmetic of the set? The difference between f(x) and g(x) is > > > f(x)-g(x) for any given natural x. If the limit of f(x)-g(x) tends to > > > zero as x tends to oo, then infinite case induction fails. Otherwise > > > it holds. > > > > What if that "difference" were something like (-1)^x (which is quite > > possible for the difference between two monotonically increasing > > functions)? > > > > Hmmm, an example might be nice. Let's see. We have N mapped to > {2,1,4,3,6,5...}. Well, by the axiom of extensionality, that is N. > They are same in the infinite case, or any even case. It does not have > a limit of 0, so it may be discounted in ICI, but the axiom of > extensionality pretty much resolves the issue. > > > > > > > > > > (3) If omega is greater than any finite counting number, or c or > > > > > aleph_1 (whether they are the same or not), then the same rule also > > > > > applies to such large numbers. > > > > > > > Please let me know which step seems vague. > > > > All of them. > > > > > I hope I clarified some of that for you. > > > > Still All of them. > > Whatever that means. All of your steps were vague before, and all of them are vague STILL. |