Uncountability of the Rationals?
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From: Tony Orlow on 18 Jun 2010 08:30 On Jun 16, 10:35 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > 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. Hi Transfer - Your input is always welcome. Of course the theory is mine, as it has been forever, and not Brian's, although he's a worth and welcome adversary. > > 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} What is the fomula for this sequence? > > Indeed, I'm not sure how TO would assign a Bigulosity > to a set whose elements are themselves Bigulosities. That might depend. > > 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. One might... > > 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. If you say so. I call them collections, and assert that they exist, and then it can bet deteermined what is a set vs. a class. > > Until Chandler and TO state otherwise, we are only > assigning surreals to subsets of standard N+. Then it's a countable set. > > 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. mapped to N+ by g(x)=x > > Birthday omega+1: > {2,3,4,5,6,7,8,9,10,...} has set size omega-1 mapped to N+ by g(x)=x-1 > > Birthday omega+2 > {3,4,5,6,7,8,9,10,11,...} has set size omega-2 mapped to N+ by g(x)=x-2 > > Birthday omega+3 > {4,5,6,7,8,9,10,11,12,...} has set size omega-3 mapped to N+ by g(x)=x-3 > > Birthday omega+4 > {5,6,7,8,9,10,11,12,13,...} has set size omega-4 mapped to N+ by g(x)=x-4 > > Birthday omega+omega > {2,4,6,8,10,12,14,16,18,...} has set size omega/2 mapped to N+ by g(x)=x/2 > > Birthday omega+omega+1 > {4,6,8,10,12,14,16,18,20,...} has set size omega/2-1 mapped to N+ by g(x)=x/2-1 > {1,2,4,6,8,10,12,14,16,...} has set size omega/2+1 {1} u {x: g(x)eN+} mapped to N+ by g(x)=floor(x/2) = 1 + ceiling(g(omega))-floor(g(1))+1 =1+omega/2-0+1 =omega/2+2 > > Birthday omega+omega+2 > {6,8,10,12,14,16,18,20,22,...} has set size omega/2-2 Mapped from N+ using f(n)=2n+4, g(x)=x/2-2 size = floor(g(omega))-ceiling(g(1))+1 = (omega/2-2)- (-1) + 1 = omega/2-2 (notice it's the evens sans {2,4} ) > {1,2,3,4,6,8,10,12,14,...} has set size omega/2+2 Yes, that's the above plus {1,2,3,4} so omega/2-2+2 = omega/2+2 > > Birthday omega*3 > {4,8,12,16,20,24,28,32,36,...} has set size omega/4 Yes, that's f(n)=4n > {2,3,4,6,7,8,10,11,12,...} has set size 3*omega/4 Seems right. > > Birthday omega*omega > {1,4,9,16,25,36,49,64,81,...} has set size sqrt(omega) Right > > 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. Not quite sure how you get the series of surreals on the left for this function. > > Until this question is answered, we can't even begin to > find a surreal for {2,4,8,16,32,64,...}. Okay, perhaps not in Conway's system, but it still works in IFR with ICI. > > Still, I must apologize to Chandler for dismissing his > idea, and investigate his idea a little further. Interesting thoughts as always, but Chandler's idea is but a reflection of his comprehension of mine, as you know. Just ask. Tony
From: Tony Orlow on 18 Jun 2010 08:38 On Jun 16, 10:56 pm, "Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote: > 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- Hide quoted text - > > - Show quoted text - Hi Ross - Nice to see you're still kicking. :) No, the H-riffics are nothing like the Hyperreals. They are supposed to be a well ordering of the reals (I dunno if you remember that thread, years ago), though there remain questions about whether the structure is totally connected, and therefore whether it can have total order. I'll have to play with that some more. However, the talk here about hyperreals and surreals is related to specifically ordered different countably infinite set sizes, and whether those produced by the Inverse Function Rule (IFR) and Infinite Case Induction (ICI) can be classified as necessarily surreals as Conway defined them. It's possible they don't, and may constitute a superset of his surreals. Dunno yet. Take care, TOny
From: Tony Orlow on 18 Jun 2010 08:40 On Jun 16, 11:00 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Transfer Principle <lwal...(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. That list probably covers the gamut, at least for now. So, I guess I didn't misuse "algebraic" after all. > > -- > "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- Hide quoted text - > I don't know the meaning of the word "crank". Therefore I cannot be one. QED. ;) Tony
From: Jesse F. Hughes on 18 Jun 2010 08:52 Tony Orlow <tony(a)lightlink.com> writes: > On Jun 16, 11:00 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Transfer Principle <lwal...(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. > > That list probably covers the gamut, at least for now. So, I guess I > didn't misuse "algebraic" after all. Yes, you did misuse "algebraic". In my experience, an algebraic function is one which preserves certain algebraic structure. But no matter. We'll assume that Walker's definition of "algebraic bijection" is what you "probably" (probably?) meant. I guess it will follow that the set P of primes has no size, since there is no algebraic bijections between P and N+? >> "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 > I don't know the meaning of the word "crank". Therefore I cannot be > one. QED. ;) You're not really required to comment on every .sig quote. Feel free, I suppose, but don't feel compelled. -- Jesse F. Hughes "I guess it's a passable day to die." -- Lt. Dwarf, /Star Wreck:In the Pirkinning"
From: Tony Orlow on 18 Jun 2010 10:13
On Jun 17, 9:31 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Tony Orlow <t...(a)lightlink.com> writes: > >> And, yes! subset is DEFINABLE from 'e'. > > >> Just as 'is an ordinal' is DEFINABLE from the mere primitives. > > > Is it? Demonstrate as I did for you just now. > > x is an ordinal iff (Ay,z in x)(y in z or z in y) & > (Aw,y,z in x)((w in y & y in z) -> w in z) & > (A z)( ((Ay)( y in z -> y in x ) & (Ey)( y in z )) -> > (Ey in z)(Aw in z)( y in w ) ) & > (A y in x)(A z in y)( z in x ). > > Not so hard. The first two clauses assert that elementhood is a > linear order on x (anti-reflexivity is trivial, assuming Regularity), > the next clause that it is a well-ordering and the final clause that x > is transitive. Someone else will surely correct any errors in the > above. > > Tell me, Tony, did you *really* think it was impossible? Did you > *really* think that "x is an ordinal" is literally undefinable in the > language of ZFC? And that generations of mathematicians have been > pulling one over on the public? You miss the point. I never said there was no deifnition. I said the definition does not "floow from the axioms" in the sense that it is not a theorem of the axioms. It may be a statement "consistent" in that there is verifiably no contradiction, however, that does not mean that deifnition is entailed by the axioms in any way. I rather think my definitions for IFR, IC, and the combination are considerably more succint and therefore matheatically beautiful. Just my opinion.... > > -- > Quincy (age 5): Baba, play some [computer games]. > Mama: Quincy, if you want [Baba] to live, don't make those > suggestions. > Quincy: Make those suggestions. Got it. Tony |