Uncountability of the Rationals?
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From: Tony Orlow on 13 Jun 2010 09:02 On Jun 11, 4:26 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > MoeBlee <jazzm...(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. Yes, thank you. That looks pretty well boiled down, and I detect no flaw. Duly cut and pasted, and all credit due, Jesse. However, I see where this is leading, what with the min and max and all.... ;) > 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? No, I don't think so. The example with the inverses of the reals, which was sort of the second half of what was supposed to be one example, taught me something about that. I suggested a range of [6,50], but clearly that is not within the range of {1/n | neN}, and I ran into problems. It was a good question, which I praised highly. It's a compactified countably infinite set, not resident throughout the range of R, but contained within a finite range. So, indeed, it makes little sense to talk about the number of elements contained within a region that includes less than min(S) or greater than max(S). Furthermore, since it was monotonically decreasing as opposed to increasing, the lower end of the range should be taken as b and the higher as a, instead of the other way around with the formulation for a monotonically increasing sequence. I think we have established pretty well the Bigulosity foundation in the finite case. Do you agree? > > -- > 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- Hide quoted text - > Yeah, I agree with JXH. Every time you read one of my posts I steal a piece of your soul, and store all the pieces in a box in my closet, and one day I will sew together a mathematical Coat of Many Colors, and take my rightful place as ruler of all reality, as well as surreality. Yeah, that's what I'll do.... BTW, Jesse, if you use this as one of your random "sigs", you better indicate it was a joke or I'll sue you for libel. ;) Tony
From: Jesse F. Hughes on 13 Jun 2010 09:49 Tony Orlow <tony(a)lightlink.com> writes: >> 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? > > No, I don't think so. The example with the inverses of the reals, > which was sort of the second half of what was supposed to be one > example, taught me something about that. I suggested a range of > [6,50], but clearly that is not within the range of {1/n | neN}, and I > ran into problems. It was a good question, which I praised highly. > It's a compactified countably infinite set, not resident throughout > the range of R, but contained within a finite range. So, indeed, it > makes little sense to talk about the number of elements contained > within a region that includes less than min(S) or greater than max(S). > Furthermore, since it was monotonically decreasing as opposed to > increasing, the lower end of the range should be taken as b and the > higher as a, instead of the other way around with the formulation for > a monotonically increasing sequence. I think we have established > pretty well the Bigulosity foundation in the finite case. Do you > agree? I can't speak to the "Bigulosity foundation in the finite case", since I don't know where you're going from here, but at least the above theorem seems correct, as well as the dual, where "strictly increasing" is replaced by "strictly decreasing" and a and b interchanged in the right hand side of the equation. >> >> -- >> 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 >> > Yeah, I agree with JXH. Every time you read one of my posts I steal a > piece of your soul, and store all the pieces in a box in my closet, > and one day I will sew together a mathematical Coat of Many Colors, > and take my rightful place as ruler of all reality, as well as > surreality. Yeah, that's what I'll do.... > BTW, Jesse, if you use this as one of your random "sigs", you better > indicate it was a joke or I'll sue you for libel. ;) You needn't worry. Sorry, but you haven't risen to JSH's level of quote-worthiness yet. But don't let that stop you from trying. -- Jesse F. Hughes "I'm losing you and you are all I've got. Thanks a lot. Thanks a lot." --Johnny Cash
From: Jesse F. Hughes on 13 Jun 2010 09:54 I skipped most of this post, but would like to comment on one remarkable request you make below. Tony Orlow <tony(a)lightlink.com> writes: > On Jun 11, 4:17 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> But let's look at a case where g is indeed definable on R in a natural >> way. Let f:N -> R be defined by f(n) = n^2 and g:R -> N be the sqrt >> function. Then, for instance, S is the set of squares of natural >> numbers and >> >> S n [a,b] = floor(sqrt(b)) - ceiling(sqrt(a)) + 1. >> >> Yes, I think that looks correct, but for the assumption that g is >> somehow naturally extended to R. I think that what you have in mind >> really is a function f:R -> R with inverse g:R -> R. To see why your >> statement doesn't make much sense, let >> >> f:N -> R >> >> be defined by f(n) = nth prime. This is an increasing function, and >> hence there is an inverse g:Prime -> N, but it makes no sense to ask >> what g(b) is unless b is a prime. > > It certainly makes sense to ask how many prime numbers exist within > any given range, but until something new is discovered about primes, > the best that can be done with them is to talk about their aymptotic > relation to the full set of naturals as one approaches N. There is no > algrebraic mapping there. So, IFR doesn't apply. > > There are various ways to extend g to >> R. Here are three: >> >> g_1(b) = number of primes less than or equal to b. >> >> g_2(b) = 1 + number of primes less than b >> >> g_3(b) = g_1(b) if b <= 4 >> g_2(b) if b > 4 >> >> So, as we can see, you have not quite correctly stated the context of >> your claim. It will be another thing entirely to tell me what the >> claim means when we move to infinite arguments. > > Please state to me what the function is which maps N to the set of > primes, derive the inverse function (it's monotonically increasing, > for sure), and then you'll have your answer. Your question is mighty confused. The strictly increasing function which maps N to the set of primes is: f(n) = nth prime. That's a function. I don't have a simple algebraic formula that expresses it, but it is a function. It satisfies the only property a function requires, namely that each value of the domain is mapped to one value in the codomain. > It should come out to about 1/logx for the inverse function, at least > in the infinite case... > > TOny -- Jesse F. Hughes "I just define real numbers to be all those on the number line, as they were defined before Dedekind and Cauchy." -- Ross Finlayson's simple definition.
From: Tony Orlow on 13 Jun 2010 10:19 On Jun 11, 6:29 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 11, 4:58 pm, Tony Orlow <t...(a)lightlink.com> wrote: > > > > What's an invertible formula? > > > More specifically (sorry) a monotonically increasing real function, > > So we can throw out the confusing terminology 'invertible formula', > which seems to refer to formulas as opposed to functions. Yes, but I would like to include the monotonically decreasing and the finitely compactified countably infinite sets. > > > > > Try it on any finite set. Now imagine applying it to the range > > > > [0,omega]. > > So you mean > > f:w+ -> R > > and f is increasing ? > > Okay, all that means is that f\{<w f(w>} is a bounded inceasing > sequence in R, and f(w) is an upper bound of the range of f\{<w > f(w>}. > > Okay, what about it? > > MoeBlee Good question. Did you get the sarcastic tone?
From: Tony Orlow on 13 Jun 2010 10:22
On Jun 11, 6:38 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 11, 5:22 pm, Tony Orlow <t...(a)lightlink.com> wrote: > > > > > > > On Jun 11, 1:07 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > 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? > > > So far, I am using classical logic. If we could ever get into > > probabilistic logic I'd be amazed. > > > > (2) Do you intend for your mathematics to provide a mathematics > > > sufficient for the sciences? > > > Not necessarily. Perhaps in cosmological areas. > > Then if it turns out your theory does not provide for mathematics for > the sciences, what foundational theory do you propose for mathematics > for the sciences? > > And if it turns your theory doesn't provide for mathematics for the > sciences, then what reason do you give that anyone should be > interested in your theory AS AN ALTERNATIVE to ZFC? > > You understand that if we didn't look to a foundational theory (such > as ZFC) for a mathematics for the sciences then of course we'd be free > to dispense with a lot what people may think to be counterintuitive. > > Sure, we can invent all kinds of theories with all kinds of > interesting, satisfying, and intuition-friendly mathematical > ontologies. > > But merely having an interesting, satisfying, and intuition-friendly > mathematical ontology is not all we (editorial 'we') ordinarily expect > from a foundational theory. > > > > (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? > > > Mostly intuitive notions, but we'll see. ELiminating proper subset > > equinumerosity is worthwhile. > > You didn't answer that question. It is true that, "the part is smaller than the whole." > > MoeBlee- Hide quoted text - > > - Show quoted text - "Offense is a personal experience" - Kwang Chi |