Uncountability of the Rationals?
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From: MoeBlee on 13 Jun 2010 18:05 On Jun 13, 4:52 am, Tony Orlow <t...(a)lightlink.com> wrote: > On Jun 11, 3:52 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > > On Jun 11, 2:33 pm, MoeBlee <jazzm...(a)hotmail.com> wrote:> On Jun 11, 2:07 pm, Tony Orlow <t...(a)lightlink.com> wrote: > > > 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. > > > Wait, I think I'm incorrect. > > > Okay for a given S, there is only one increasing function f on w that > > gives S as the range. > > > But I don't know why you refer to "floor" and "ceiling" the way you > > do. The values of g(a) and g(b) are NATURAL NUMBERS. > > I refer to floor() and celing() to account for one choosing any given > range, whether or not its extrema are inverse members of N. I think you don't need that. Look at my later formulation below (or, if I slipped up in it, just make obvious needed correction). > > I think what you mean is > > > card({xeS | x e [a b]}) > > = > > (max(range(g restricted to [a b])) - min(range(g restricted to [a b]))) > > +1 Right or not that this is equivalent to what you mean? It's true in the finite case. And, if {xeS | x e [a b]} is not finite, then, I'd like to know how you defined "-". > > Yes, that is just a very roundabout way of restating the cardinality > > of some finite subset of a closed real interval. > > > Now, what exactly is the rest of your proposal? And does it contradict > > ZFC or not? > > ICI contraidicts the model of the von Neumann ordinals, and extends > natural density through IFR. Contradictions are in STATEMENTS, not in models. As to ICI and IFR, now that we've gotten SOME progress in making those SOMEwhat more definite, why don't you just state your result and the proof from your IFR (axiom?) and ICI (axiom?)? MoeBlee \\\"model of the von Neumann ordinals"
From: MoeBlee on 13 Jun 2010 18:59 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. > > > > > > 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. > 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 'N' stands for the set of natural numbers (or perhaps the set of positive natural numbers) then of course N is countable. 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. 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. > I make no apologies. Oh, you're such a brave maverick you are! > > > 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. > 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? > > > 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. 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. > > 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. 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. 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). > > > > 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. > 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? > > > > 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? Unless you say otherwise, I take a model (to be mathematically strict per a certain definition) to be in the sense of mathematical logic. 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'. You're just RE-positing your conclusion again. I've already heard it from you a million times. 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. > 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. > > > > > > 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 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. > > > 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? "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. > > > 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. > > > 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 exists such > > an ordinal number. > > Uh huh. How so? You want me to prove for you what is in virtually every textbook in set theory, and is easy for anyone who even understands the basics of the matter. Moreover, I've as much as done it for you, more than once, in posts years ago. > > > Logically, you must concede, > > > neither statement proves that omega exists as a number. > > > There's really nothing to concede. It's never been at issue. > > Eyep, s'kinda central, ackchooley. No, it has never been an issue (central or otherwise) that the statements you mentioned don't prove that "omega exists as a number". > > > > I am not > > > obligated to imagine such a number, or give it any credibility. > > > Of course you're not. You're not obligated to imagine anything at all. > Nope > > > > When > > > it comes to the countably infinite, we can only look at this as some > > > kind of limit, > > > It is a "limit" in a certain special sense; it is a limit ordinal. So > > what? > > Ordinal schmordinal. In what OTHER sense is it a limit? Out of time. More later perhaps. MoeBlee
From: Brian Chandler on 14 Jun 2010 00:37 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. > > 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. So unpicking the blather and typos, Tony assumes Virgil refers functions whose domain and/or range is the reals, in which case what Tony says is true, if ill-expressed. (In real[ha ha!] mathematics, Tony, no-one "declares" things the way you imagine.) > Virgil has the advantage of knowing more math than you, Tony. > You should not pretend to be an expert when you're not. It makes > you look silly and pretentious. True. But this paragraph has now been written so many times it is unlikely to have any effect. > 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. Right. Well see above. Isn't that just a more carefully written version of what Tony said? > 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". I certainly don't think it would be correct to say that the reciprocal of zero is the empty set. Brian Chandler
From: Brian Chandler on 14 Jun 2010 00:47 David R Tribble wrote: > MoeBlee wrote: >> Tony Orlow wrote: >>>And Uncle Tom Cobbley wrote: > You need to stop beating around the bush and just go ahead > and state your primary axiom: > > Tony's Axiom. [TA1] > Given property P(x) for all real x, > P(w) for any non-finite ordinal w. No. Even Tony isn't quite that stupid. See, there's a property F called "finite". And everyone knows that even though F(x) is true for all (finite) real x, F(x) is false for all non-finite x. Tony's answer to this conundrum is a piece of verbiage I can never remember exactly, something about "limit", "zero", and "tending", but in practice it means that TA1 is qualified by a condition equivalent to "provided Tony can't see any immediately derived contradiction". See the long rambling thread about the "staircase limit". Tony's intuition says that a staircase is always a staircase, therefore the limit of the staircases is not a normal point set (which would have the expected length 1/root(2)) but an "infinite staircase", a new sort of point set in which alternate points are 90 degrees apart in orientation. So it is obvious there will never be a coherent version of TA1, since it's just part of Tony's Belief System, which is indefinitely elastic with respect to new discoveries. **Tony does not understand the concept of axiomatic mathematics, so you are wasting your time.** (HTH) Brian Chandler
From: Virgil on 14 Jun 2010 00:47
In article <b27319dd-1f53-49e5-a876-753b6301f008(a)g39g2000pri.googlegroups.com>, Brian Chandler <imaginatorium(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. > > > > 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. |