Uncountability of the Rationals?
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From: Tony Orlow on 14 Jun 2010 10:21 On Jun 13, 6:59 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 13, 8:10 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > > > > > 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. > > We know what subset and proper subsets are. > > But for "fraction" one would think you have some notion of a division > OPERATION. > Please refer to "natural density", capable of detecting differences of a finite number of elements, or a finite ratio between infinite sizes. > > > > > > 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. > > We don't need to use N=S^L. Yes, and I don't need to eat onions, but they're good for me. > > > 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(f) 'N' stands for the set of natural numbers (or perhaps the set of > positive natural numbers) then of course N is countable. No, in "N=S^L", N is the size of the complete language including strings of length L constructed from a symbol set of size S. > > If 'N', 'S', and 'L' range over sets in general, then indeed it is NOT > the case that > > there is a bijection from a countable set onto the set of denumerable > sequences made from members of that countable set. For example there > is no bijection from w onto the set of denumerable sequences of > members of w. Well, that may yet need to be determined, but for now, suffice it that we agree the power set is larger than the base set, in every case. Whether one can construct an uncountable infinity from a countable one is a point of contention which is unlikely to be resolved here. > > So if you claim otherwise Then you contradict ZFC. That's fine. But at > least just be sensible enough not to claim that you don't and to (if > you would be so kind) say which axioms you reject. Why don't you tell me which axiom I've contradicted? > > > I make no apologies. > > Oh, you're such a brave maverick you are! Fear is not my forte. > > > > > It is much > > > > more fruitful > > > > Fruitful in WHAT sense? Toward WHAT end? > > You left off the line that 'it' refers to. I don't remember what 'it' > refers to. Bigulosity, I believe. Of course all the snippage and excessive verbiage doesn't help the flow of the conversation. > > > 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. > > And all that is fruitful toward what end? Towards a more intuitive understanding of infinite sets. > > > > > 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". > > We'll wait for your system one day. But you even admitted that you > don't know whether your theory will provide mathematics sufficient for > the sciences. Does the axiom of choice? No applications that I can see. That's the only axiom I see as relatively useless and rather misapplied. But then, that one's optional, no? > > ZFC provides for classical analysis and computability theory, which > provides a mathematics that is used to build such technology as used > for our computer communication right now. When Orlow theory does that, > please let us know. Hahaha. Our computational technology is based on George Boole's work, which preceded the flowering of set theory, and he wasn't even a mathematician. So, whatever. Transfinitology is a pipe dream from the opium era. > > > > 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. > > Bigulosity leaves unanswered what Bigulosity IS. Also, unanswered are > what is the set of axioms. What is "is", Mr. President? > > Meanwhile, ANY theory that can do as much arithmetic as Robinson > arithmetic will have "unanswered" questions. Since we don't know your > axioms, we don't know that your "theory" doesn't leave unanswered > questions even more basic than CH. Moreover, we can't even BEGIN to > discuss whether your theory is "consistent" unless you tell us just > what your theory is, what its axioms are. Dtsrt with IFR in the finite case. Get comfortable with it. > > Moreover, we can extend ZFC in certain ways that DO answer the > question of CH (though, of course, as with ANY sufficiently rich > theory, other unanswered questions would result). Yeah. That's a little dissatisfying, but I guess that's a matter of taste. > > > > > > 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. > > In the sense of isomorphism and in certain contexts, yes. > > But still, for purpose of definiteness in certain instances, we would > like to be specific. Standard omega is |N+| in Bigulosity (yes, this time I mean the positive naturals). > > > 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! > > Then, for you, N is the set of positive natural numbers, okay. > > I take it you agree w is the set of natural numbers, i.e., N u {0}/ > > And N+ is N u {N}, is that right? No, not at all. I previously meant *N, not N+, and assumed you took N+ to mean the positive naturals. *N, the hypernaturals, does not include omega as a specific member, though every finite function on omega falls within the range of *N. > > > > > > > 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. > > WHAT is a model of the naturals? As far as I understand models it's a specific application of the axioms, specifically the axiom of infinity, to create a set of elements isomorphic to N. > > Unless you say otherwise, I take a model (to be mathematically strict > per a certain definition) to be in the sense of mathematical logic. I think that's what I mean too. So? > > If you mean otherwise, then please define 'model'. > > > n is the nth element of N. If the > > initial sequence of size n exists, there exists the nth one. > > Fine. :) > > > Therefore, if the size is omega, then omega is a member of the set. > > You provide no LOGIC that goes from your correct premise to what > follows your 'therefore'. Is N not the initial sequence of N of size omega? If it exists, then by what you agreed to above, there exists an omegath member, which must be equal to omega, given the identity function between count and value which defines N+. > > You're just RE-positing your conclusion again. I've already heard it > from you a million times. You should get it by now. The logic is clear. You just agreed with the premise. The conclusion follows logically. > > If you want to prove your conclusion, then you need to add more > premises to your argument. It seems, at best, that you have some > unstated premises you're not including. State them, THEN we'll be on > our way to finding some axioms for you. > > Look again. It's a complete argument. Which premise do you disagree with? > > > 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. So, you don't disagree with this... Therefore, since omega must be in the set, but cannot exist in the set, omega is itself a contradiction, and cannot exist anywhere. > > > > > 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) > > Lose the 'sheesh' here when YOU are the ignoraums. No, logically those are the same two statements. Give me context outside of von Neumann where that doesn't hold. > > No one claims the the theorems of ZFC are logically true. Indeed, the > lhe logically true statements are the pure predicate calculus. To get > some mathematics, we need to add some statements that are (hopefully) > among themselves consistent, but NOT true in EVERY model. > > The statements you mentioned are NOT logically equivalent, rather they > are equivalent within certain theories. I.e., the equivalence is > proven from certain AXIOMS that are not themselves logically true but > rather true in all and only those models of the theory. > > Get an education, you boring ignoramus. Now you're being boorish. Let's do a little elementary logic review, Pal. Here are my two statements: AneN n<omega ~EneN n>=omega True or false, for statement S? AneN S <-> ~EneN ~S True or false for the normal order relation? n<omega <-> ~(n>=omega) Okay, now put those pieces together, and explain how the statements are not equivalent. Ignoramus. > > > > > 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? > > Instead what? What on Earth does it say to you? > > "there is no ntaural that can be the size of the entire set of > naturals" is entailed by (along with other obvious axioms) the > original statement you made but is not in itself equivalent with the > original statement you made. Of course it is! I'll let someone more reputable correct you. > > > > > 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. > > We have a DEFINITION of the '| |' operator. > > I've explained a THOUSAND times that our method of definition allows > reduction to the primitive language in all cases. > > PLEASE go inform yourself with a book on mathematical logic. It is > getting just exasperating talking with someone who won't sufficiently > inform himself to understand the basics of the subject. > So, you want me to answer the same questions over and over, and you won't answer one? You have no answer. Put up or shut up. > > > > > > 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- Hide quoted text - > There it is! Over there! Tony
From: Tony Orlow on 14 Jun 2010 10:28 On Jun 14, 12:47 am, Virgil <Vir...(a)home.esc> wrote: > In article > <b27319dd-1f53-49e5-a876-753b6301f...(a)g39g2000pri.googlegroups.com>, > Brian Chandler <imaginator...(a)despammed.com> wrote: > > > Seems to me something has gone wrong here... > > > David R Tribble wrote: > > > Virgil wrote: > > > >> Real functions do allow infinite quantities either as arguments or > > > >> values, so that you are making false statements. > > > Really? What does Virgil mean by "real functions"? One might guess > > (particular if one were as mathematically ignorant as Tony) that he > > means "real-valued functions". But to make V's statement true, I > > think, he would just have to mean "real functions" as opposed to > > "fake functions". Pretty confusing. > > In standard mathematical usage a "real function" is one whose domain and > codomain are both subsets of the set of real numbers. > And that standard usage meaning is what I mean by the phrase. Are there infinite real numbers? If not, then why did you say, "Real functions do allow infinite quantities either as arguments or values", and then accuse ME of making false statements???????? > > > > Tony Orlow wrote: > > > > Don't correct me incorrectly. Standard real functions take real value > > > > parameters and return a real result. Where they have return an > > > > infinite values... > > Real functions do not return "infinite values" because > "infinite values" are not real numbers. Then why did you say what you see above? I did not change it. That's what you said. Oy! Tony
From: Tony Orlow on 14 Jun 2010 10:31 On Jun 14, 1:21 am, Brian Chandler <imaginator...(a)despammed.com> wrote: > David R Tribble wrote: > > Tony Orlow wrote: > > > Actually, in Bigulosity, there are much larger countably infinite > > > sets. The square roots of the naturals have a higher "density" than > > > the naturals. > > > How is that? The two sets should be exactly the same size > > (your size(), not just cardinality). Every natural has a square root, > > and every natural square root has a corresponding natural. > > Can't you see? You're just using "cardinality" to point out a > bijection between naturals and their square roots. Bigulosity Theory, > or BT, simply rejects such arguments, in its search for the Answer. > > Consider any large number Q (perhaps a quillion): it is clear that in > the range 1-Q there are Q naturals, but there are (roughly, at least) > Q^2 real values x such that x^2 is a natural. Now appealing to ICI, > PCT, QD, and any other TLAs to hand, declare Tav* to be our unit > infinity, let Q tend to Tav, and lo! the answer we wanted. > > * Last letter of the Hebrew alphabet according to Wikipedia. > > It is true that BT, of which I am but a beginning student, leaves > nagging doubts. Suppose, for example we thought about the bigulosity > of the set NQ > > NQ = { {p,q} | p^2 = q e N} (set of pairs of naturals with their > roots) Done. You have an omega X omega^2 matix. You omega^3 elements. Next.... > > But I suspect that our Leader will determine that this is a Trick Set, > to which special rules apply. We may have to declare a new Unit > Infinity, or Recalibrate the naturals, or something. But you'll have > to wait for the answer till he comes back from lunch. > > Brian Chandler
From: Tony Orlow on 14 Jun 2010 10:36 On Jun 14, 1:32 am, David R Tribble <da...(a)tribble.com> wrote: > Brian Chandler wrote: > > Seems to me something has gone wrong here... > > Okay, before we all start talking at cross purposes here... > > Virgil wrote: > >> Real functions do allow infinite quantities either as arguments or > >> values, so that you are making false statements. > > Brian Chandler wrote: > > Really? What does Virgil mean by "real functions"? One might guess > > (particular if one were as mathematically ignorant as Tony) that he > > means "real-valued functions". But to make V's statement true, I > > think, he would just have to mean "real functions" as opposed to > > "fake functions". Pretty confusing. > > Yes, okay, Virgil probably meant "actual" functions, as opposed > to functions (or "formulae") in Tony's sense. And yes, functions > can certainly have non-real domains and values. > > Tony (and I) are talking about real-valued functions. > > > > > > Tony Orlow wrote: > >> 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. > > David R Tribble wrote: > >> A real-valued function, by definition, has a domain from which it > >> takes its arguments, and a range (or codomain) of values it "returns" > >> (i.e., maps those arguments to). The function maps each argument > >> value from the domain to a single result value in the range. > > >> If the function has no mapping for a particular argument, then the > >> function is not defined for that argument value, and that value is > >> not within the domain of the function. > > >> Likewise, real functions take real arguments, which means that > >> infinity (+/-oo) cannot be in the domain set of a function, and thus > >> can never be an argument to a function. > > Brian Chandler wrote: > > Right. Well see above. Isn't that just a more carefully written > > version of what Tony said? > > No. Tony says that real-valued functions can return infinite > values. I said that they can't. Lie. I never said that. I corrected Virgil's misstatement on that matter, which correction you made fun of. Don't put words in my mouth. You're both being idiots. I said that real functions can be extended under certain conditions to the infinite to produce functional order relationships using ICI. > > It's a subtle distinction, but Tony is saying that whenever a > function returns an infinite value for a particular argument, > "standard" mathematics declares that function to be > "undefined" for that value. He states it like it's a cop-out by > standard mathematics. No, not exactly. It would be nice to have some definition for them, or at least be able to call them infinite rather than "undefined", but I have tried to stick with limit definitions when discussing ICI as you well know, so can it. > > Remember, Tony is trying to posit that real functions are > allowed to operate on non-real infinite values (assuming a > few criteria are met) just as if those infinite value acted like > real values. The real function f(x) = x+1 works on all real x, > but Tony claims that it also works on infinite x in exactly the > same way. Yes, I do. > > It's like he's trying to eliminate the notion of an "undefined" > function value and just go ahead and replace it with "infinite > result". If infinities can be ordered quantitatively and distinguished, I see that as progress. > > David R Tribble wrote: > >> An example is the reciprocal function, f(x) = 1/x. The domain of f > >> is R \ {0}, i.e., all real values except 0. The function does not > >> "return infinity" for argument 0, rather, f is simply does not have > >> any result for it. It would be more correct to say that f(0) is > >> "nothing" or "the empty set". > > Brian Chandler wrote: > > I certainly don't think it would be correct to say that the reciprocal > > of zero is the empty set. > > Okay, but I think it makes sense to say that there are no > values of the reciprocal function for argument zero, or that there > is no reciprocal of zero. Which to me sounds a lot like saying > that the set of values for recip(0) is empty. > > We don't disagree, we just might have slightly different notions > of what "undefined" means for functions as sets. (And I never > claimed to be an expert.)- Hide quoted text - > > - Show quoted text -
From: MoeBlee on 14 Jun 2010 10:46
On Jun 13, 4:55 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 13, 4:37 am, Tony Orlow <t...(a)lightlink.com> wrote: > > "complete relation", set of all pairs of members from each set, > > For any sets S and T, we call that S X T. Sorry, CORRECTION: I meant (S u T) X (S u T). Hmm, it seems there should be a common term for that. Perhaps there is, but it's not coming to my mind now. Maybe it is 'complete relation', but I don't recall ever reading it. MoeBlee |