Uncountability of the Rationals?
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From: Tony Orlow on 11 Jun 2010 09:11 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. > > -- > "It was over ten years ago that I was a lieutenant in the U.S. Army > and one day for some reason I thought to myself that I should be able > to figure out something brilliant. [...] Like, why can't I figure out > some math thing?" -- James S. Harris on inspirational moments.- Hide quoted text - > > - Show quoted text - Then I guess natural density is not very promising either. Huh! TOny
From: Tony Orlow on 11 Jun 2010 09:20 On Jun 10, 5:30 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > David R Tribble <da...(a)tribble.com> writes: > > > Tony Orlow wrote: > >> No, That's IFR in particular which is used on countable sets of reals > >> mapped from some segment of N. > > > So then what is the *general* case for IFR? > > Could someone remind me what IFR stands for? > > -- > "I've noticed [...] I routinely have been putting up flawed equations > with my surrogate factoring work. My take on it is that I have some > deep fear that the work is too dangerous and am sabotaging myself." > -- James S. Harris The inverse function rule. If a set of reals is bijected with a segment of the naturals using an invertible formula, then that inverse formula may be used to calculate the number of elements in the set within any given value range. Where a<b, and where S is a set mapped from the naturals using f(n)=x, and where there exists an inverse function g(x)=n such that f(g(x))=g(f(x))=x, 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]. I hope that is clear enough for you. TOny
From: Tony Orlow on 11 Jun 2010 09:22 On Jun 10, 10:25 pm, David R Tribble <da...(a)tribble.com> wrote: > Transfer Principle (LWalker) wrote: > >> In searching for a way to make the T-riffics more > >> rigorous, we can start by letting z, the symbol for > >> "zillion," be a primitive and go from there. > > Tony Orlow wrote: > > Yes. > > E 0 > > E 1 > > 0<1 > > x<y -> E z: x<z<y > > > This leads to some level of infinites between counting numbers. > > 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? 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) Tony
From: Tony Orlow on 11 Jun 2010 09:33 On Jun 10, 10:51 pm, David R Tribble <da...(a)tribble.com> wrote: > Tony Orlow wrote: > > Consider f(x) and g(x) to be any two given arithmetic formulae on x. If: > > > (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 > > > (2) The difference between the two terms which establishes the > > inequality does not have a limit of zero as x approaches infinity, > > then > > > (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. > > How do you get from any finite x to any non-finite x? Through an uncountable number of finite additions, multiplications, exponentiations or whatever, or through an uncountable addition, multiplication, exponentiation, etc. I don't believe in any smallest infinity, so please don't shove that in my mouth. > > Your assumption is, by fiat, that this is the case. However, > you have given no arithmetic justification for this. I have dozens of times. You just don't like my perspective on the matter. > > 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. > > 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? Now, the inverse function is x=y-1, correct? 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. 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. I'm not sure if that answered your question exactly, but hopefully it clarified it a bit. TOny
From: Tony Orlow on 11 Jun 2010 09:46
On Jun 10, 11:20 pm, FredJeffries <fredjeffr...(a)gmail.com> wrote: > On Jun 9, 6:51 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > > > > http://en.wikipedia.org/wiki/Schnirelmann_density > > > This looks a little heady to wade through right now but, "the > > Schnirelmann densities of the even numbers and the odd numbers, which > > one might expect to agree, are 0 and 1/2 respectively". That's nothing > > like Bigulosity. > > If I may please ask, according to Bigulosity, are there the same > number of even natural numbers as odd, which would seem to be > indicated by the figure: > > 0 <--> 1 > 2 <--> 3 > 4 <--> 5 > . > . > . > > OR is there one more even number than odd, which the following seems > to indicate: > > 0 > 2 <--> 1 > 4 <--> 3 > 6 <--> 5 > . > . > . > > ? It ultimately doesn't matter. If we start with 0 and map the naturals to the evens using y=2x, then x=y/2, and the evens are omega/2-0/2+1. If we start with 0, the odds map using y=2x+1, so x=(y-1)/2, or y/ 2-1/2. Since IFR uses the ceiling function at the top end, the extra difference of 1/2 disappears, and again we have that half of the naturals are odd, plus 1. In other words, they are equinumerous. What is the extra 1, you may ask? It really doesn't matter, since in any case the method allows us to order sets more precisely according to size. TOny |