Uncountability of the Rationals?
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From: David R Tribble on 11 Jun 2010 00:45 Tony Orlow wrote: >> Consider f(x) and g(x) to be any two given arithmetic formulae on x. If: > David R Tribble writes: >> How do you get from any finite x to any non-finite x? >> [...] >> Now explain how this function retains its arithmetic meaning >> when extended to non-real (infinite) x. Or even how it can >> be extended in this way at all. > Jesse F. Hughes wrote: > Blah, blah, blah. > You're just jealous because Tony can prove that > sqrt(aleph_1) > ln(aleph_1) > cos(aleph_1) > and you can't. Ouch! You've discovered my dirty little secret. Actually, I secretly crave Tony's fascinating mathematical prose, so I am compelled beyond my will to keep asking him for results from his insightful math. But he's crafty, teasing me like that, never giving me any real answers I can sink my teeth into. I wish I could quit...
From: David R Tribble on 11 Jun 2010 00:55 Tony Orlow wrote: >> E 0 >> E 1 >> 0<1 >> x<y -> E z: x<z<y >> This leads to some level of infinites between counting numbers. > David R Tribble wrote: >> That's hard to believe. The only thing that your definitions do >> is define two numbers, 0 and 1, and an order relation for them. >> Where are all of the rest of the reals? Or even the naturals? > Brian Chandler wrote: > Well, his last line seems to say that there is also z, such > that 0 < z < 1. Yeah, see, that was the part that really threw me. Given his theory of (0, 1, <), I was having a really hard time finding any z that satisfied 0 < z < 1. I think it's supposed to be one of those "counting numbers" he mentions. > For example, the binary fractions would surely fit his definition. Surely they would, but Tony seems to be missing an axiom or two for defining anything beyond 0, 1, and <. Maybe I just missed it in the first reading. I'll check again. Maybe I can also find those counting numbers and levels of infinities he mentions.
From: Brian Chandler on 11 Jun 2010 01:02 David R Tribble wrote: > David R Tribble writes: > >> So then what is the *general* case for IFR? > > > > Jesse F. Hughes wrote: > > Could someone remind me what IFR stands for? > > I don't remember what Tony's "IFR" stands for, either. Something > about a "formulaic ratio" or some such. Me! Me! "Inverse function ratio". I think. (Buried here is another of Tony's problems, which he shares with many other cranks: he hasn't grasped the idea of a generic mapping, not necessarily represented by an elementary algebraic expression.) Brian Chandler
From: Brian Chandler on 11 Jun 2010 01:11 David R Tribble wrote: > Tony Orlow wrote: > >> E 0 > >> E 1 > >> 0<1 > >> x<y -> E z: x<z<y > >> This leads to some level of infinites between counting numbers. <snip> > > For example, the binary fractions would surely fit his definition. > > Surely they would, but Tony seems to be missing an axiom or > two for defining anything beyond 0, 1, and <. Maybe I just missed > it in the first reading. I'll check again. Maybe I can also find those > counting numbers and levels of infinities he mentions. I think you are being *very* slightly unfair. Tony's writing style is a bit strange, but surely the above can be seen as a definition of UI, the Unit Interval. UI is a set of elements, with a total ordering < such that: UI includes an element 0 UI includes an element 1 If x and y are elements of UI and x < y, then there exists an element p of UI such that x < p < y. My _guess_ is that Tony thinks this gives him something whose bigulosity he can "declare", despite the fact that there are any number of sets of different cardinalities even which match this definition. I believe he also "declares" that the bigulosity of UI is in some simple relation to the bigulosity of the TNaturals, or something similar. But this has become sadly repetitive. Reading the old threads is just as entertaining, and less work. Brian Chandler
From: Brian Chandler on 11 Jun 2010 02:28
Brian Chandler wrote: > David R Tribble wrote: > > David R Tribble writes: > > >> So then what is the *general* case for IFR? > > Jesse F. Hughes wrote: > > > Could someone remind me what IFR stands for? Right, here it is: http://groups.google.com/group/de.sci.mathematik/msg/7ef87cda529e254f?&q=orlow+ifr+squares The number of natural squares is (obvious really!) sqrt(Bigun), where we use "Bigun" for the "size" of the Tnats. [quote] IFR = Inverse function rule. If f(x) gives the values for a monotonically increasing function on N, then the inverse produces those natural values from the set produced by f(x), and subtracting the least from the greatest and adding 1 gives the number of naturals in that range. [end quote] So if S = { x^2 | 1 <= x <= Bigun } the function f(x) is x^2. This is a monotonically increasing function: OK! Now "the inverse function" is (obviously!?) sqrt(x), so the number of squares is sqrt(Bigun) - sqrt(1) + 1 = sqrt(Bigun) And so on. I mean, the number of primes, for example, is found immediately using the inverse prime function, and um Brian Chandler |