Uncountability of the Rationals?
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From: Transfer Principle on 16 Jun 2010 22:35 On Jun 15, 11:25 pm, Brian Chandler <imaginator...(a)despammed.com> wrote: > In a truly extraordinary moment, Transfer Principle wrote: > > On Jun 15, 9:18 pm, Brian Chandler <imaginator...(a)despammed.com> > > wrote: > <snip bit about Conway's ONAG> > > So yes, sqrt(omega) is a surreal. But the big difference > > between surreals and TO's numbers is that the latter is > > supposed to be a sort of set _size_ (Bigulosity). It > > makes no sense to state that the set of perfect squares > > has the surreal sqrt(omega) as its set size. > Have you gone barking mad? Don't you remember the Agreement? You > *cannot* say "makes no sense". You *must* say "... mumble mumble, > working in a different theory". Yes, I admit that Chandler has caught me red-handed. And so since I made an error regarding the "Agreement," I must admit and correct my error. It is possible that indeed Chandler is working in a different theory -- one in which surreals can be applied to set sizes. And so let us attempt to do exactly that. So far, we already have the following set sizes (since we really are discussing surreals here, we can use Conway's name for the unit infinity "omega," rather than Chandler's name "tav"). So far we have: {1,2,3,4,5,6,7,8,9,...} has set size omega. {2,3,4,5,6,7,8,9,10,...} has set size omega-1. {2,4,6,8,10,12,14,16,18,...} has set size omega/2. {1,4,9,16,25,36,49,64,81,...} has set size sqrt(omega). But what about {2,4,8,16,32,64,...}? I don't recall Conway saying anything about lg(omega) (or any other base of log). If lg(omega) is in No, then when exactly is the _birthday_ of lg(omega)? One might try the surreal: {1,2,3,4,5,... | ...,cbrt(omega),sqrt(omega),omega} But I think that this is the omega-th root of omega, not the logarithm of omega. Another question is, if we're allowing surreals, then what is the set size of a set whose elements are themselves surreals, such as: {1,2,3,4,5,...,omega,-omega,omega-1,omega/2,1/omega} Indeed, I'm not sure how TO would assign a Bigulosity to a set whose elements are themselves Bigulosities. One might note that using standard ZFC definitions of cardinality (and ordinality), the set of ordinals less than a given ordinal is order-isomorphic (under e) to the ordinal itself. So one might say that: {1,2,3,4,5,...,omega-2,omega-1,omega} has size omega. But even if we were to restrict the elements to the sur-integers (the analogs of the integers in No), this object is too large to be a set. There is no set large enough to contain all the surreals (or even the sur-integers). So this won't work. Until Chandler and TO state otherwise, we are only assigning surreals to subsets of standard N+. Beginning with omega, we can find subsets of N+ whose set size is a sur-integer, as follows: Birthday omega: {1,2,3,4,5,6,7,8,9,...} has set size omega. Birthday omega+1: {2,3,4,5,6,7,8,9,10,...} has set size omega-1 Birthday omega+2 {3,4,5,6,7,8,9,10,11,...} has set size omega-2 Birthday omega+3 {4,5,6,7,8,9,10,11,12,...} has set size omega-3 Birthday omega+4 {5,6,7,8,9,10,11,12,13,...} has set size omega-4 Birthday omega+omega {2,4,6,8,10,12,14,16,18,...} has set size omega/2 Birthday omega+omega+1 {4,6,8,10,12,14,16,18,20,...} has set size omega/2-1 {1,2,4,6,8,10,12,14,16,...} has set size omega/2+1 Birthday omega+omega+2 {6,8,10,12,14,16,18,20,22,...} has set size omega/2-2 {1,2,3,4,6,8,10,12,14,...} has set size omega/2+2 Birthday omega*3 {4,8,12,16,20,24,28,32,36,...} has set size omega/4 {2,3,4,6,7,8,10,11,12,...} has set size 3*omega/4 Birthday omega*omega {1,4,9,16,25,36,49,64,81,...} has set size sqrt(omega) At this point, we are trying to solve the problem of given a sur-integer at most omega (and with sufficiently early birthday), finding a canonical subset of N+ whose set size is that sur-integer. Since the surreals are defined by Conway in terms of left and right sets, what we are really saying is if we know the canonical left and right sets of a surreal, and sets whose set size is each surreal in the left and right sets, can we find a set whose set size is the given surreal? Let's see: {2,3,4,5,...} has size omega-1 = [1,2,3,... | omega} {3,4,5,6,...} has size omega-2 = {1,2,3,... | omega-1} {4,5,6,7,...} has size omega-3 = {1,2,3,... | omega-2} {5,6,7,8,...} has size omega-4 = {1,2,3,... | omega-3} So, if we have a lone surreal on the right, we remove the smallest element of the subset of N+. Since we have: {2,4,6,8,...} has size omega/2 = {1,2,3,... | ...omega-2,omega-1,omega} {4,8,12,16,...} has size omega/4 = {1,2,3,... | ...omega/2-2,omega/2-1,omega/2} having infinitely many surreals on the right means that we alternate elements of the subset of N+. Since we have: {1,2,4,6,8,...} has size omega/2+1 = {omega/2 | omega} {1,2,3,4,6,8,...} has size omega/2+2 = {omega/2+1 | omega} {2,3,4,6,7,8,10,...} has size 3*omega/4 = {omega/2,omega/2+1,omega/2+2,... | omega} having a lone surreal on the left means adding back a missing element of the subset of N+, and infinitely many suureals on the left means adding back alternating elements of the subset of N+. But then we have: {1,4,9,16,...} has size sqrt(omega) = {1,2,3,... | ...omega/8,omega/4,omega/2,omega} and now it's not obvious how the set {1,4,9,16,25,36,...} is even related to the surreals omega/2, omega/4, omega/8, and so on, since the canonical sets for these surreals don't even contain numbers like 1, 9, 25, and so on. Until this question is answered, we can't even begin to find a surreal for {2,4,8,16,32,64,...}. Still, I must apologize to Chandler for dismissing his idea, and investigate his idea a little further.
From: Transfer Principle on 16 Jun 2010 22:52 On Jun 16, 8:08 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Tony Orlow <t...(a)lightlink.com> writes: > > Actually, I meant a general formulaic relation, not necessarily > > algebraic, but with an inverse function that can be determined through > > algebra, not necessarily restricted to polynomials, but also including > > exponents and logs, etc. > That's not very explicit. How about this: An algebraic function is a real-valued function which is the composition of finitely many real-valued polynomial, radical, rational, exponential, and logarithmic functions, and whose inverse (or at least the real-valued branches thereof) is also the composition of finitely many polynomial, radical, rational, exponential, and logarithmic functions. The definition is similar to that of "elementary function," except that we want to exclude the trigonometric and inverse trigonometric functions. (Notice that I wrote _real-valued_, so there's no reason to mention that a trigonometic function is actually a complex-valued exponential function.) After all, TO writes nothing about cos(tav) or cos(zillion). We also wish to exclude elementary functions whose inverses aren't elementary, such as f(x) = xe^x, whose inverse is the Lambert-W function. After all, TO writes nothing about LambertW(tav) or LambertW(zillion).
From: Ross A. Finlayson on 16 Jun 2010 22:56 On Jun 15, 4:44 pm, Tony Orlow <t...(a)lightlink.com> wrote: > On Jun 15, 5:40 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > > > On Jun 14, 8:43 pm, David R Tribble <da...(a)tribble.com> wrote: > > > > Jesse F. Hughes wrote: > > > > It seems to me that he's stated his assumption, but he doesn't get quite > > > > how much it assumes. > > > > Given any real valued functions f and g, if lim (f - g) > 0, then f(x) > > > > > g(x) where x is any infinite number. > > > > As a consequence of this, I guess, it follows that f and g are defined > > > > on infinite numbers, though we don't know anything about their values > > > > aside from the fact that f(x) > g(x). > > > It's obvious that Tony can't and won't answer any of these > > > questions, and no one here (except Walker) really sees him > > > as capable of going any further in any meaningful sense. > > > So except for deriving some entertainment value and learning > > > a few new things, I don't see the point in indulging him any more. > > > Teaching pigs to sing, and all that. > > > I've stood back and watch this thread grow, allowing TO to > > discuss his ideas in more detail before posting again. I > > can't respond to every post to which I want to respond, but > > let me start here since my name is mentioned here. > > I was wondering where you were. ;) > > > > > > > As expected, I don't believe that working with theories > > other than ZFC or set sizes other than standard cardinality > > is analogous to "teaching pigs to sing." I definitely > > prefer to believe that there is a theory in which infinite > > sets work differently from how they work under ZFC, and > > perhaps working as TO or another poster would like them to. > > > According to Tribble, there are many questions which TO > > can't and won't answer about his theory. One of these > > questions (asked IIRC by MoeBlee) is to which of the axioms > > of ZFC does TO object? But to me, the answer to this > > question is obvious. If a poster disagrees with how the > > infinite sets work under ZFC, then they reject the axiom > > which guarantees their existence -- and that axiom is, of > > course, the Axiom of Infinity. > > I rather think not. The Axiom of Infinity declares something like N a > set, when it's really a sequence, but asserting the existence of this > object doesn't lead to all the confusion. The Axiom of Choice, on the > face of it, doesn't seem exactly wrong, but it seems to, perhaps in > combination with assumptions of truth about the ordinals, lead to > bizarre conclusions, being interpreted strangely. That there exist > countably infinite sets seems obvious to me. > > > > > I believe that among the sci.math posters who argue against > > ZFC, the most common axioms to reject are Infinity and > > Choice, followed by Powerset (if they accept infinite but > > not uncountable sets). > > That's generally the Anti-Cantorians, as opposed to us here Post- > Cantorians. See? > > > > > And thus, a good starting point is to start with ZF and > > replace Infinity with a new axiom. This new axiom won't > > merely be ~Infinity, since we aren't just trying to get rid > > of infinite sets but replace them with new objects that > > work differently from the infinite sets under ZF. > > My problem's with von Neumann. He's gonna pay, I tell ya... > > > > > In another thread, I mentioned how Infinity is used to > > prove that every set has a transitive closure. Therefore, > > we can take ZF, drop Infinity, and add a new axiom: > > > There exists a set with no transitive closure. > > > We notice that in this theory, ~Infinity would be a theorem > > (proved via Deduction Theorem/contrapositives). > > Okay. Do you agree wth that axiom, or are you just trying it on for > Halloween? ;) > > > > > But the problem here is that it isn't obvious how this > > theory matches the intuition of any sci.math poster. Also, > > it's not evident how this theory is related to math for the > > sciences, either. Until those objections are addressed, no > > one is going to accept this theory. > > Nah, it's less the axioms than the models, I believe. > > > > > It's doubtful that any textbook discusses sets that have no > > transitive closure, since most textbooks are grounded in ZF, > > which proves that every set does have one. A good starting > > point might be old zuhair threads, since zuhair mentioned > > transitive closures in his theories all the time. > > The H-riffics have transitive closure, after some uncountable number > of iterations. Does that count? > > :) TOny Well yeah post-Cantorianism started with Zermelo and Fraenkel and well basically post-Cantorianism as it was referred to in the literature is that of the development of set theory after the publication of Cantor's results around 1900. Then, this modern post-Cantorianism, where post-Cantorianism is modern mathematics, to which Tony refers, this modern post-Cantorianism is a reaction to that all the asymptotic methods of numbers still work with the uncountable irrelevant to spatial density in the finite, for each finite, in the fixed, complete in the countable. That's more about the direction where, surely, all the commonly held notions about the simple case asymptotics hold true, to really care about the uncountable just requires honoring the space, it's a way to require that. (Completeness theorems.) Tony are your H-riffics the same as the H numbers? If you use the H numbers to derive that fact or Tribble's Eta for example, that has to go in the definition. Here by H numbers I mean the constructive hyperreals (constructable). You indicate that as sets the elements have a particular construct that is the same for all consequences in all models, that the numbers have set-theoretic structure, to use them as sets. Then, in terms of using one of these numbers in an algorithm, what's the most convenient way to write it? Here that would indicate a reasonable guess at what any reasonably structured value could efficiently use as its form. In set theory, everything's a set. Tony, just add all the rules that build up on the von Neumann ordinals into new ordinals, they're still the same ordinals with the rules attached, in the resulting system that knows what they are as ordinals, which are very useful in many cases as numbers. (Von Neumann ordinals are small sets, with basically two elements one containing the previous ordinal the other empty, there are various representations of the von Neumann ordinals.) As you can see, for the convenience of maintaining the rules of all the completions of the ordinals in various systems, that is or can be simply accorded with information theoretic terms in space (mathematical space). Have a good one, warm regards, Ross Finlayson
From: Jesse F. Hughes on 16 Jun 2010 23:00 Transfer Principle <lwalke3(a)lausd.net> writes: > On Jun 16, 8:08 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Tony Orlow <t...(a)lightlink.com> writes: >> > Actually, I meant a general formulaic relation, not necessarily >> > algebraic, but with an inverse function that can be determined through >> > algebra, not necessarily restricted to polynomials, but also including >> > exponents and logs, etc. >> That's not very explicit. > > How about this: > > An algebraic function is a real-valued function which is the > composition of finitely many real-valued polynomial, radical, > rational, exponential, and logarithmic functions, and whose > inverse (or at least the real-valued branches thereof) is > also the composition of finitely many polynomial, radical, > rational, exponential, and logarithmic functions. Yes, that's explicit. And, who knows, it might be what Tony meant. Perhaps he'll say so. -- "Am I am [sic] misanthrope? I would say no, for honestly I never heard of this word until about 1994 or thereabouts on the Internet reading a post from someone who called someone a misanthrope." -- Archimedes Plutonium
From: Transfer Principle on 16 Jun 2010 23:31
On Jun 14, 10:50 pm, Brian Chandler <imaginator...(a)despammed.com> wrote: > > Don't you make any connection between a set of pairs and, say, spatial > > coordinates? When you talk about points in n dimensional space, do you > > not define them as unique n-tuples in the spatial set of points? > Look, let's not use "omega" to refer to any of your stuff. It's an > abuse of respected terminology. As I understand it, there's a stage in > which we "declare a unit infinity", so I declare mine to be Tav. OK? If we are specifically trying to make TO's theory more rigorous, than I agree with Chandler, to some extent. So let's start with ZF and add a new primitive symbol to denote this "unit infinity." We could call it "tav" as per Chandler. So we drop the Axiom of Infinity from ZF. Why? It's because TO wants all infinite sets in his theory to adhere to his rules, yet Infinity proves the existence of a set, namely standard omega, that doesn't adhere to TO's rules. So we must drop Infinity. It is often pointed out in this type of thread that Infinity proves the existence of a (nonempty) successor inductive set, which is strictly stronger than mere existence of an infinite set. It might be possible to take advantage of this fact, and see whether it's possible for tav to be an infinite set whose existence does _not_ imply (in ZF-Infinity) the existence of a successor inductive set. TO points out that the cornerstone to his theory is something called ICI, Infinite Case Induction. Since "induction" sounds like a schema, we can add to our theory a schema for ICI. But how can we state the ICI Schema in a manner that's rigorous enough for Chandler and others? We notice that the schemata labeled TA1 and TA2 are stated in terms of real numbers, leading posters like MoeBlee to ask for the definition of "<" and other symbols. For after all, in standard theory, we start with omega and define (rational numbers, then) real numbers, but here there is no omega, so there aren't any reals yet. Since our language contains the primitives "e" and "tav" it would be preferable to state ICI in terms of these primitives, not reals or the "<" relation. In another thread, I once posted the following schema, which is a rewriting of TA2 using these primitives: If phi doesn't contain "tav" then all closures of: (phi(0) & Ax (phi(x) -> phi(xu{x})) -> phi(tav) are axioms. But then Hughes let phi be the statement "x is finite" (using a standard definition of finite) to prove that tav is finite, yet TO wants tav to be infinite. Furthermore, if we let phi be the statement "0ex or 0=x," we can prove that either 0etav or 0=tav. At this point, I suspect that tav is actually the empty set -- which is definitely not what TO intends tav to be. So let's change the schema to the following: If phi doesn't contain "tav" then all closures of: (En (n fin. nat. & phi(n)) & Ax (phi(x) -> phi(xu{x})) -> phi(tav) are axioms. Here "fin. nat." refers to the (standard) definition of finite natural. This new schema reflects TO's comments about how he can prove 2^tav > tav^2. We can let phi be the statement "2^x > x^2" and n be 5. But we still have problems. Notice that by letting phi be the statement "x is finite," we still have tav finite -- yet by letting phi be the statement "mex" for each natural number m in turn (and letting n = m+1, of course), we can prove "me(tav)" for _every_ natural number m. So somehow tav is finite, yet for every natural number m we can prove that tav has m as an element? At first this sounds like a contradiction, but it was mentioned in earlier threads that: For every natural number n, it is provable that tav contains n. and It is provable that for every natural number n, tav contains n. are distinct (in first-order logic at least -- I'm not sure about second-order logic). But even so, this would make tav a really strange set to contain every natural number as an element, yet still somehow be "finite" -- and it certainly doesn't describe tav as intended by TO. Furthermore, since we have all the other axioms of ZF besides Infinity, we do have the Separation Schema, and by this schema, the set: w = {me(tav)} | m (standard) finite natural} must exist. But what set is w? It would appear to equal omega, yet is still a "finite" set? What we really need to avoid this contradiction/paradox is to allow some instances of the ICI schema and not others, but MoeBlee uses the word "oracle" to describe this. Indeed, back when TO was describing ICI, I even _warned_ him that his schema might be described as "oracular," since "oracle" is a buzzword commonly used by MoeBlee to object to theories (just as Hughes uses the phrase "ad hoc"). So in order to find a schema acceptable to MoeBlee, we must avoid these "oracles" at all costs. Note that, since we are trying to avoid the existence of omega, we can't allow for a set to which the Separation Schema can be applied to it to recover omega. This is why, in a post that I wrote yesterday, I wish to consider sets without a transitive closure, since one _can't_ apply Separation to such a set to recover omega. But then again, it's not obvious how such sets can describe anything that TO, or any other poster (save perhaps zuhair), is writing. |