Uncountability of the Rationals?
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From: David R Tribble on 10 Jun 2010 22:25 Transfer Principle (LWalker) wrote: >> In searching for a way to make the T-riffics more >> rigorous, we can start by letting z, the symbol for >> "zillion," be a primitive and go from there. > Tony Orlow wrote: > Yes. > E 0 > E 1 > 0<1 > x<y -> E z: x<z<y > > This leads to some level of infinites between counting numbers. 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?
From: David R Tribble on 10 Jun 2010 22:51 Tony Orlow wrote: > Consider f(x) and g(x) to be any two given arithmetic formulae on x. If: > > (1) It can be classically inductively proven that f(x)>g(x) for every > real value greater than some finite x (the base case), and > > (2) The difference between the two terms which establishes the > inequality does not have a limit of zero as x approaches infinity, > then > > (3) If omega is greater than any finite counting number, or c or > aleph_1 (whether they are the same or not), then the same rule also > applies to such large numbers. > > Please let me know which step seems vague. How do you get from any finite x to any non-finite x? Your assumption is, by fiat, that this is the case. However, you have given no arithmetic justification for this. Specifically, given an arithmetic formula f(x) on all real x, on what basis can you assert that it automatically applies to any non-real x? Explaining how this is the case for a simple example would help immensely. For example, take a simple real function: f(x) = x + 1 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.
From: FredJeffries on 10 Jun 2010 23:20 On Jun 9, 6:51 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > http://en.wikipedia.org/wiki/Schnirelmann_density > > This looks a little heady to wade through right now but, "the > Schnirelmann densities of the even numbers and the odd numbers, which > one might expect to agree, are 0 and 1/2 respectively". That's nothing > like Bigulosity. > If I may please ask, according to Bigulosity, are there the same number of even natural numbers as odd, which would seem to be indicated by the figure: 0 <--> 1 2 <--> 3 4 <--> 5 . . . OR is there one more even number than odd, which the following seems to indicate: 0 2 <--> 1 4 <--> 3 6 <--> 5 . . . ?
From: Jesse F. Hughes on 10 Jun 2010 23:29 David R Tribble <david(a)tribble.com> writes: > Tony Orlow wrote: >> Consider f(x) and g(x) to be any two given arithmetic formulae on x. If: >> >> (1) It can be classically inductively proven that f(x)>g(x) for every >> real value greater than some finite x (the base case), and >> >> (2) The difference between the two terms which establishes the >> inequality does not have a limit of zero as x approaches infinity, >> then >> >> (3) If omega is greater than any finite counting number, or c or >> aleph_1 (whether they are the same or not), then the same rule also >> applies to such large numbers. >> >> Please let me know which step seems vague. > > How do you get from any finite x to any non-finite x? > > Your assumption is, by fiat, that this is the case. However, > you have given no arithmetic justification for this. > > Specifically, given an arithmetic formula f(x) on all real x, > on what basis can you assert that it automatically applies > to any non-real x? > > Explaining how this is the case for a simple example would > help immensely. For example, take a simple real function: > f(x) = x + 1 > 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. 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. -- Jesse F. Hughes "The American people would have been incredibly proud of watching our military folks dispense with basic health care needs to people who needed help." --George W. Bush, March 13, 2007
From: David R Tribble on 10 Jun 2010 23:35
MoeBlee wrote: >> MoeBlee: 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. > Transfer Principle (L Walker) wrote: > [...] In this analogy, > MoeBlee is forcing everyone to fly only on American > Airlines and not Virgin or any other airline. > 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. Bad, bad, naughty MoeBlee, preventing you from considering other theories like that. Sorry, but how exactly is he forcing you to use only ZFC? Perhaps you have some secret knowledge about MoeBlee's hidden powers over the Internet, as but far as the rest of us are concerned, this is a free forum for suggesting new ideas. It's Tony who keeps claiming that his bigulosity, T-riffics, IFC, and all the rest fit well within (or as simple extensions of) standard theory, in spite of all continuing evidence to the contrary. Since you're so quick to defend Tony's ideas from sinister forces like MoeBlee, it's apparent that you possess a deeper understanding of them than the rest of us. So perhaps you can elucidate these ideas in a more formal way, since Tony seems to be having such a problem doing that? |