Uncountability of the Rationals?
in [Math]
|
Prev: Collatz conjecture
Next: Beginner-ish question
From: David R Tribble on 11 Jun 2010 14:39 Tony Orlow wrote: >> E 0 >> E 1 >> 0<1 >> x<y -> E z: x<z<y >> >> This leads to some level of infinites between counting numbers. > David R Tribble wrote: >> That's hard to believe. The only thing that your definitions do >> is define two numbers, 0 and 1, and an order relation for them. >> Where are all of the rest of the reals? Or even the naturals? > Tony Orlow wrote: > How do you read the last statement? Between any two reals lies another > real. Lather, rinse, repeat. Does this not lead to an infinite number > of reals, or even rationals, between any two naturals? (sigh) Your rules don't say what z is. Are x, y, and z members of R, or N, or Z, or Q, or what? You don't say. You certainly don't say how you get from a pair of x and y to a particular z. Other than 0 and 1, you don't even say what x and y can be. However, assuming you mean that 0, 1, x, y, and z must be members of Q (rationals), then you end up with, in the least, the set of {z | 0 < z < 1}. Fractions of the form 0 < k/2^-n < 1 for positive integers k and n satisfy this property, i.e., every z is a positive integral multiple of some integral power of 1/2. Construction: Given x = k/2^n and y = j/2^m, then z = (mk + nj) / 2^(n+m-1). Example: x=3/4, y=5/8, z=11/16. We'll assume that x=0 and y=1 produces z=1/2 for simplicity. Of course, no z is greater than 1, which seems kind of limiting. If you really intend your system to include all of the reals, then say so.
From: David R Tribble on 11 Jun 2010 14:47 Tony Orlow wrote: >> I was wondering what thoughts you [Walker] had on the countably infinitely long >> complete list of digital strings, if anything. > David R Tribble wrote: >> That would be the countable list of all finite-length digital strings, >> wouldn't it? Or would it be the uncountable list of all infinite- >> length >> digital strings? > Tony Orlow wrote: > Um, excuse me, but is the list of all finite digital strings not > ultimately countably infinite in width? If it is finite, then please > state this finite width. If you mean what is the number of digits needed to list all possible finite-length digital strings, the answer is omega. So yes, the "width" of the list is countably infinite. > If it is countably infinite, then how does it > differ from your uncountably long list? 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.
From: Brian Chandler on 11 Jun 2010 14:57 Jesse F. Hughes wrote: > Tony Orlow <tony(a)lightlink.com> writes: > > > On Jun 11, 12:21 am, David R Tribble <da...(a)tribble.com> wrote: > >> I'm still waiting to hear what bigulosity/IFR says about the size > >> of the set of natural squares. Any results yet? > > > > I already did that. According to IFR it's sqrt(omega), which according > > to ICI is less than omega. > > So, it is a(n ordinal) number x such that x * x = omega, is that right? > > I'm not clever like you, but I'd wager that one can prove no such > ordinal exists, when * stands for ordinal multiplication. Does that > bother you? Come on! Of course it isn't an ordinal by any normal definition. Tony uses our words to help us along, but only with his meanings. I really think it is fairly clear how Tony really approaches all this: Think of numbers, small (like 3) and large (like 672), but also very very large (like 10^101). Now think that these numbers go on and on and on and on, and well, really speaking they never stop, but when they do, or at least when one gets in the general region of where they would stop (if they didn't go on without stopping, that is), well, these numbers are simply imponderably enormous. Let Bigun, or whatever today's name is, be the representatively simply most totally imponderably enormous of these numbers. Now the important thing is that while imp. enor., Bigun is still a respectable number that can be added divided, sqrted, and so on. Well, I've said this before, so I'll stop. Brian Chandler
From: Tony Orlow on 11 Jun 2010 15:07 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? Okay, do you disagree with this statement? AneN n<omega Do you disagree with this one? AneN 1<n -> n<n^2 Not so far, right? How's about we talk about the set of all set sizes, or "counts", and call it N+? Now, you undoubtedly disagree with this statement: AneN+ 1<n -> n<n^2 It runs completely counter to cardinality, but is a much cleaner extension of finite induction than transfinite induction IMHO. Just pointing out exactly where our differences lie. There's a concrete example. > > What you really want, but are not stating, is something like this: > > For every (invertible?) function f:R -> R, there is a unique extension > f':? -> ? (Ord -> Ord?) such that ... (what?). Exactly!!!! :D You hit the nail on the head. Or, with your head. For every countably infinite set S whose membership is defined by monotonically increasing function f:N -> R, which of necessity has real inverse function g:seS -> N, the count of elements in the range [a,b] is given by floor(g(b))-ceiling(g(a))+1. Do you care to dispute whether this works, in the finite case? Please do comment. Tony "I am one wave, in a vast ocean"
From: Tony Orlow on 11 Jun 2010 15:13
On Jun 11, 11:45 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Tony Orlow <t...(a)lightlink.com> writes: > > On Jun 10, 5:09 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> Tony Orlow <t...(a)lightlink.com> writes: > >> > On Jun 10, 3:05 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> >> Transfer Principle <lwal...(a)lausd.net> writes: > >> >> > To me, the difference between ZFC and non-ZFC theories > >> >> > is more analogous to the difference between a flight > >> >> > on American Airlines from JFK to LAX, and a flight on > >> >> > Virgin America from JFK to LAX. In this analogy, > >> >> > MoeBlee is forcing everyone to fly only on American > >> >> > Airlines and not Virgin or any other airline. > > >> >> No, he's not. Moe has *never* said that ZFC is the only acceptable > >> >> theory. > > >> >> > In another thread, he claims that he doesn't consider ZFC to be the > >> >> > best theory, but the fact that he compares ZFC to an airliner and > >> >> > other theories to rubber balls speaks for itself. > > >> >> The fact is that Tony, AP, etc., have *not* offered any coherent > >> >> mathematical theory at all *and you know it*. Thus, if Moe criticizes > >> >> their blatherings, then it is extraordinarily disingenuous to claim this > >> >> as evidence that he accepts only ZFC. > > >> >> -- > >> >> Jesse F. Hughes > > >> >> "Really, I'm not out to destroy Microsoft. That will just be a > >> >> completely unintentional side effect." -- Linus Torvalds > > >> > "Disingenuous" means "lying". I believe Transfer's comment falls into > >> > the category of a best-guess interpretation of Moe's motives. You, > >> > Moe, Virgie, The Tribble and others seem completely closed to the > >> > concept of any improvement on the standard obfuscation. For, "none > >> > shall drive us from the Garden which Cantor has created for us". If it > >> > doesn't produce fruit, it's time to plant a new bed, or at least > >> > fertilize. > > >> I have nothing against alternative definitions of set size, but you have > >> offered no clear definition at all. > > >> Nor do I have anything against alternative theories. Why should I? > >> Indeed, in a previous life, I did a bit of work in ZFA, which is an > >> anti-well-founded variant of ZFC. > > >> The criticisms that you receive are based primarily on the fact that you > >> have never offered a clear and complete definition of set size that > >> really competes with cardinality at all. Instead, you have a definition > >> that applies to *some* (not all) sets and --- according to your own > >> description --- gives different answers depending on one's perspective.. > > >> That's not a promising mathematical definition. > > > Then I guess natural density is not very promising either. Huh! > > Not as a general notion of set size, no. But it is useful in certain > contexts. > > Let E be the set of even numbers and S the set of squares. > > It's perfectly sensible to say that, while E, S and N have the same > cardinality, the natural density of E in N is 1/2, while the natural > density of S in N is 0. > > Who could criticize such a claim? Who can criticize Bigulosity for claiming to have different measures for, not only these three sets, but the sets of cubes, square roots and logs of natural numbers? Who could possibly reject a method that can accomplish that? I can't imagine.... > > Now, if I said instead that cardinality is foolish and we should use > natural density as a measure of set size, then of course I am being > silly. Natural density is no replacement for cardinality. It just > doesn't do the same job. It does a slightly better job, at least distinguishing infinities with a finite difference or ratio relative to omega. It misses greater distinctions which IFR detects. What's the complaint? You really have a JSH obsession, don't you? TOny |