Uncountability of the Rationals?
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From: Transfer Principle on 10 Jun 2010 13:36 On Jun 10, 9:17 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 10, 7:17 am, Tony Orlow <t...(a)lightlink.com> wrote: > MoeBlee: But the airplane carries 300 people at a time from New York > to L.A. in five hours. Even though your ball may be nice to you > because it doesn't have schlocky wings, it can't carry 300 people at a > time from New York to L.A. All it carries is the dust on it for a > distance of about five feet. So you can't compare the airplane, which > accomplishes a certain purpose, with your ball, which does not > accomplish that purpose. > ToeKnee: You just don't get it. Sigh. > ChanceFor PrinceOfPull: There goes MoeBlee again silencing alternative > theories. 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. 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. I strongly disagree that any theory other than ZFC is like trying to roll a ball across the country. What theory, then, would I say corresponds to the ball being rolled across the table? How about that infamous theory whose lone axiom is "Axy (x=y)"? I can't see how one can derive even arithmetic from that theory. Yes, that theory definitely drops the ball. I like to believe that a wonderful theory can come out of allowing proper subsets to have distinct sizes, but MoeBlee won't even let us _build_ the plane, much less fly it anywhere.
From: Transfer Principle on 10 Jun 2010 14:18 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. 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. 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. 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. 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. > 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. > 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) 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.
From: Tony Orlow on 10 Jun 2010 14:35 On Jun 10, 1:36 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jun 10, 9:17 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jun 10, 7:17 am, Tony Orlow <t...(a)lightlink.com> wrote: > > MoeBlee: But the airplane carries 300 people at a time from New York > > to L.A. in five hours. Even though your ball may be nice to you > > because it doesn't have schlocky wings, it can't carry 300 people at a > > time from New York to L.A. All it carries is the dust on it for a > > distance of about five feet. So you can't compare the airplane, which > > accomplishes a certain purpose, with your ball, which does not > > accomplish that purpose. > > ToeKnee: You just don't get it. Sigh. > > ChanceFor PrinceOfPull: There goes MoeBlee again silencing alternative > > theories. > > 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. 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. > > I strongly disagree that any theory other than ZFC is > like trying to roll a ball across the country. What > theory, then, would I say corresponds to the ball being > rolled across the table? How about that infamous theory > whose lone axiom is "Axy (x=y)"? I can't see how one > can derive even arithmetic from that theory. Yes, that > theory definitely drops the ball. > > I like to believe that a wonderful theory can come out > of allowing proper subsets to have distinct sizes, but > MoeBlee won't even let us _build_ the plane, much less > fly it anywhere. Thank you Transfer. Unfortunately it get much ugier sometimes, and I have to retreat, if only for my sense of self-worth. One time, he actually hid a bunch of anvils inside the back hatch, and I almost crashed. ;) Peace, Tony
From: MoeBlee on 10 Jun 2010 14:42 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
From: Tony Orlow on 10 Jun 2010 14:42
On Jun 10, 1:36 pm, Transfer Principle <lwal...(a)lausd.net> wrote: <snippasuarus rex pro brevitatis> Hi Transfer - I was wondering what thoughts you had on the countably infinitely long complete list of digital strings, if anything. Or, should I start a thread on that, particularly? :) Have a nice day, Tony |