Uncountability of the Rationals?
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From: David R Tribble on 10 Jun 2010 14:43 Transfer Principle (LWalker) wrote: >> Later on, Tribble points out that it is similar in >> some ways to the AP-reals. A key difference between >> the AP-reals and the T-riffics is that the former >> is decimal, while the latter is binary. > Tony Orlow wrote: > No, the T-riffics can be expressed in any base, as can the H-rifiics. > As a neo-taoist I tend to like binary. 0 and 1 are rather elemental. Likewise, the AP-reals can be expressed in any base. AP has shown examples in both decimal and binary. AP' says that (his) 999...999 (decimal) is equal to 111...111 (binary), and calls it the "largest number in the world". I asked him and he confirmed that 999...999 is evenly divisible by 9. When I asked him if likewise 111...111 (binary) was also evenly divisible by 9 (1001 binary), he insisted that it was, but he wasn't very forthcoming with the coherent arithmetic to show this. Another similarity is that AP uses the notation 1:000...000 to represent the "next" value beyond 999...999, which looks a lot like TO's 1:000...000 notation. The difference is that AP's number system "wraps around", being inherently based on points on the surface of a (number) sphere. TO's numbers are based on a (nested?) linear number line. One major difference is that AP has never adequately explained what the "..." ellipsis in "999...999" actually means. He has variously said it's an "infinite" sequence of digits, but he has also said that any number larger than 10^500 is "infinite". So, who can tell? However, AP does say that starting with 000...001 and successively adding 1 eventually gets you to 999...999. I think that's the same with TO's H-riffics (?).
From: Tony Orlow on 10 Jun 2010 15:03 On Jun 10, 2:18 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jun 7, 8:36 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > On Jun 5, 11:36 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > The H-riffics of a given base b are based on the idea that: > > 1 is a number > > if x is a number then both b^x and b^-x are both numbers. > > So, for instance, given b=2, we have the follwing numbers: > > 1 > > 2,1/2 > > 4,1/4,sqrt(2),sqrt(2)/2 > > 16,1/16,... > > etc. > > OK, I see what's going on here. I figure you probably would. :) > > Tribble describes the H-riffics as a binary tree. To this > end, it is similar to WM's binary tree, but there is a > key difference between TO's tree and WM's. > > > If we denote each number by a string of 0's and 1's each representing > > the expontiation by x and -x, respectively, certainly 3 cannot be > > expressed in any finite string using H-riffic base 2, but I think is > > represented as an infinite string, as someone years ago confirmed. I > > cannot remember if he said it was 001001001... but it was something > > like that. > > And thus lies the difference between the trees of > TO and WM. I have supported Meuckenheim in his arguments at times because they made sense, but he is one of those ... Anti-Cantorians. I don't agree with his conclusions. But, no matter.... :) > > Suppose we want to find 3 on this tree. So we > start at the root, which is 1. Now should we take > the 2^x branch or the 2^-x branch? Since 2^1 = 2, > 2^-1 = 1/2, and 3 is closer to 2 than to 1/2, one > might think that we should start with 2^x. > > Indeed, one might think that if we want to find a > number less than 1, we should start with the 2^x > branch, and if we want a number less than 1, we > should start with 2^-x branch. > > But this is clearly wrong, since if we start at > the 2^-x branch, we reach 1/2, then we can proceed > with 2^x to obtain sqrt(2), 2^sqrt(2), and even > 2^2^sqrt(2) -- and this last number is already > larger than 3. Thus, we can't even tell which > branch to _start_ with when looking for 3. Yes, that's what makes them rather horrific. Still... > > On the other hand, it's easy to find the _last_ > branch that will take us to the 3. Since all the > numbers on the tree are positive, and 2^x > 1 for > positive real x, while 2^-x < -1 for all positive > real x, we know that the _last_ step to obtain 3 > is to take the 2^x branch from lg(3) (i.e., > log_2(3)) to 3. > > Since lg(3) = 1.584962501... > 1, the penultimate > step must have been to take the 2^x branch from > lg(lg(3)) to lg(3). > > Since lg(lg(3)) = .6644487075 < 1, the > antepenultimate step must have been to take the > 2^-x branch from lg(-lg(lg(3))) to lg(lg(3)). > > And so on. And so in TO's binary notation, we can > see that the pattern for 3 ends in: > > ...0111100101101010100101001101010111100 > > These bits don't form a sequence order-isomorphic > to omega. Instead, the order is *omega (omega^op), > which isn't even a wellorder. So any attempt to > wellorder R using H-riffics is doomed. I think your backwards extrapolation of the bit sequence is correct, so what someone told me years ago probably wasn't. But, you just lost me. What is *omega, and does (omega^op) mean some operation applied to omega? I never quite understood why it didn't work, and would love to someday. Could you give more detail? Thanx. > > The big difference between TO's and WM's trees is > that the paths of WM form Cauchy sequences, and > we can tell to which real such C-sequences will > eventually converge. Thus, to find 1/3 on the > WM tree, we begin at the root 1/2. Since 1/4 is > closer to 1/3 than 3/4 is, we choose the path > leading to 1/4. Since 3/8 is closer to 1/3 than > 1/8 is, we take the 3/8 path. Since 5/16 is closer > to 1/3 than 7/16 is, we take the 5/16 path. This > gives us the C-sequence: > > {1/2, 1/4, 3/8, 5/16, 11/32, 21/64, ...} > > and the limit of this C-sequence is 1/3. Thus, > this is the correct path. > > But the paths of TO's tree aren't C-sequences (at > least not using the standard metric). A typical > path on TO's tree looks like: > > {1, 1/2, sqrt(2), 2^sqrt(2), 2^-(2^sqrt(2))...} > > which evidently doesn't converge at all. Indeed, > most paths will oscillate between values less > than 1 and values more than 1! No wonder we can't > find the path leading to 3. Yep, they suck in that respect, don't they? And yet, for every real there is a unique parent, and two unique chldren. Thus there exists a sort of circular descendency, except it's uncountable. Am I wrong on this? > > > In any case, it becomes strange at this point, because > > after this infinite string that equals 3 lie more numbers: > > 3 > > 8,1/8 > > 256,1/256,8th-root(2),1/8th-root(2) > > etc. > > And now we can see why. If 3 has the bit pattern: > > ...0111100101101010100101001101010111100 > > then 8 has the bit pattern: > > ...01111001011010101001010011010101111000 > > while 1/8 has the bit pattern: > > ...01111001011010101001010011010101111001 > > Then 256 has the bit pattern: > > ...011110010110101010010100110101011110000 > > and so on. > > So we see that TO's "binary tree" can hardly be > considered a "tree," at least not using the > standard definition of tree. It is not countable as a structure, and yet it is discrete. A strange beast, no? Can I be the first to come up with something so very distasteful? heh. > > > They're certainly > > somewhat horrific to try to calculate, a superset of tetrations > > apparently. > > Ah yes, tetration, one of my favorite subjects > in mathematics. Yes, this is definitely related > to tetration, since: > > 2^^0 = 1 > 2^^1 = 2 (bit pattern 0) > 2^^2 = 4 (bit pattern 00) > 2^^3 = 16 (bit pattern 000) > 2^^4 = 65536 (bit pattern 0000) Somehow I'm not surprised O Transfer. > > Years ago, a poster named Lode Vandevenne once > tried to define tetration using some of the > patterns that I mentioned in this thread, but > using sqrt(2) rather than 2 as the base. But that > attempt failed. One would have to use some inverse of the exponent I guess to achieve any such thing. Thanks so much. Tony :)
From: Jesse F. Hughes on 10 Jun 2010 15:05 Transfer Principle <lwalke3(a)lausd.net> writes: > To me, the difference between ZFC and non-ZFC theories > is more analogous to the difference between a flight > on American Airlines from JFK to LAX, and a flight on > Virgin America from JFK to LAX. In this analogy, > MoeBlee is forcing everyone to fly only on American > Airlines and not Virgin or any other airline. No, he's not. Moe has *never* said that ZFC is the only acceptable theory. > In another thread, he claims that he doesn't consider ZFC to be the > best theory, but the fact that he compares ZFC to an airliner and > other theories to rubber balls speaks for itself. The fact is that Tony, AP, etc., have *not* offered any coherent mathematical theory at all *and you know it*. Thus, if Moe criticizes their blatherings, then it is extraordinarily disingenuous to claim this as evidence that he accepts only ZFC. -- Jesse F. Hughes "Really, I'm not out to destroy Microsoft. That will just be a completely unintentional side effect." -- Linus Torvalds
From: Tony Orlow on 10 Jun 2010 15:19 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. > > MoeBlee Please cite an instance recently where I've used any circular logic. Thanks, Tony
From: Tony Orlow on 10 Jun 2010 15:27
On Jun 10, 2:43 pm, David R Tribble <da...(a)tribble.com> wrote: > Transfer Principle (LWalker) wrote: > >> Later on, Tribble points out that it is similar in > >> some ways to the AP-reals. A key difference between > >> the AP-reals and the T-riffics is that the former > >> is decimal, while the latter is binary. > > Tony Orlow wrote: > > No, the T-riffics can be expressed in any base, as can the H-rifiics. > > As a neo-taoist I tend to like binary. 0 and 1 are rather elemental. > > Likewise, the AP-reals can be expressed in any base. AP has > shown examples in both decimal and binary. > > AP' says that (his) 999...999 (decimal) is equal to 111...111 > (binary), > and calls it the "largest number in the world". I asked him and he > confirmed that 999...999 is evenly divisible by 9. When I asked him > if likewise 111...111 (binary) was also evenly divisible by 9 (1001 > binary), > he insisted that it was, but he wasn't very forthcoming with the > coherent arithmetic to show this. I recall this hole in his number system. I was of some assistance in concocting mine. > > Another similarity is that AP uses the notation 1:000...000 to > represent the "next" value beyond 999...999, which looks a lot > like TO's 1:000...000 notation. The difference is that AP's > number system "wraps around", being inherently based on > points on the surface of a (number) sphere. TO's numbers > are based on a (nested?) linear number line. To some extent. I'm not sure I would use the term "nested". I've discussed the number circle a little, but the T-riffics don't "entertain" idea. They extend indefinitely, uncountably, in either direction. The number circle, however, is nothing to coff at unnecessarily. > > One major difference is that AP has never adequately explained > what the "..." ellipsis in "999...999" actually means. He has > variously said it's an "infinite" sequence of digits, but he has > also said that any number larger than 10^500 is "infinite". > So, who can tell? Well, I have stated specifically that digital points may be placed anywhere infinitely from the 0-place digital point, and that they must be specified formulaically in order to use any kind of arithmetic on them. That's a major difference. > > However, AP does say that starting with 000...001 and successively > adding 1 eventually gets you to 999...999. I think that's the same > with TO's H-riffics (?). Um, sure, after an uncountable number of iterations, which doesn't make sense in standard transfinitology. That doesn't bother me. If you want to consider a new theory, then you must put the old aside for a little bit. There can be such a thing as an uncountable sequence without limit ordinals, although like the reals, it will be impossible to identify every step in the process. Most of the transcendentals will elude us all, at least for a time. That's okay. What's the relationship between pi and e? Much Love, TOny |