Uncountability of the Rationals?
in [Math]
|
Prev: Collatz conjecture
Next: Beginner-ish question
From: Virgil on 11 Jun 2010 16:29 In article <7d090737-107b-483a-a648-0f22af3f05b9(a)y4g2000yqy.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > So, you can't even remember what it stands for or what it means, but > you are sure it's incosistent. Another new word!
From: Jesse F. Hughes on 11 Jun 2010 16:26 MoeBlee <jazzmobe(a)hotmail.com> writes: >> the count of elements in the range >> [a,b] > > Do you mean the count of [a b] (but there is no such countable > "count") or do you mean the count of > > {xeS | x e [a b]} > > ? > >> is given by floor(g(b))-ceiling(g(a))+1. > > But this is not well defined, because there are MANY functions f such > that f is an increasing function from N into R. And for each such f we > get a different g. I think that he wants f to be strictly increasing, in which case the set S determines the function f, no? f(0) = min(S) f(1) = min(S\{f(0)}) and so on. Far as I'm concerned, his first issue is that, unless a and b are in S, then g(a) and g(b) are undefined. Perhaps what he means is that |S n [a,b]| = g(max{x in S | x <= b}) - g(min{x in S | x >= a}) + 1. Unless I'm brainfarting, that is correct. In fact, it can be stated rather more simply: If S c R and g:S -> N is strictly increasing, then |S n [a,b]| = g(max{x in S | x <= b}) - g(min{x in S | x >= a}) + 1. Am I mistaken? -- Jesse F. Hughes "Mathematicians who read proofs of my results seem to basically lose some part of themselves, like it rips at their souls, and they are no longer quite right in the head." -- James S. Harris, Geek Cthulhu
From: Jesse F. Hughes on 11 Jun 2010 16:30 Tony Orlow <tony(a)lightlink.com> writes: > 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.... We criticize Bigulosity because we have not seen a clear, coherent definition of the term. We've seen ad hoc claims about its consequences, but not its most fundamental principles. >> 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? Can we start with the defining principles? Wait, never mind bigulosity. Can you tell me the axioms of your set theory? Let's start there. Unless you can tell me at least which set theory you use, there's not much to say about bigulosity. It's a half-thought. > You really have a JSH obsession, don't you? I could quit any time. I just don't want to. -- Jesse F. Hughes "Time and again, history has shown that people who think their beliefs trump reality lose, and lose badly. Luckily, I don't have to listen to you." -- James Harris on reality avoidance
From: Virgil on 11 Jun 2010 16:32 In article <654c57ae-d10b-4f99-8788-71fa261c1534(a)w31g2000yqb.googlegroups.com>, Tony Orlow <tony(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? But we don't believe it, and won't without better evidence than you have yet presented.
From: MoeBlee on 11 Jun 2010 16:36
On Jun 11, 3:26 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > MoeBlee <jazzm...(a)hotmail.com> writes: > I think that he wants f to be strictly increasing, in which case the set > S determines the function f, no? > > f(0) = min(S) > f(1) = min(S\{f(0)}) > > and so on. Right, that was my mistake. I corrected myself in a followup post. > Far as I'm concerned, his first issue is that, unless a and b are in S, > then g(a) and g(b) are undefined. > > Perhaps what he means is that > > |S n [a,b]| = g(max{x in S | x <= b}) - g(min{x in S | x >= a}) + 1. In my followup I formulated a bit differently, but on at least a quick glance mine looks equivalent to yours. MoeBlee |