Uncountability of the Rationals?
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From: Jesse F. Hughes on 11 Jun 2010 19:08 Tony Orlow <tony(a)lightlink.com> writes: >> f(g(x)) cannot equal g(f(x)), can it? The functions are f:N -> S and >> g:S -> N, so f(g(x)) is in S, while g(f(x)) is in N. > > Yes, it most certainly can. If x is in S, then g maps to a member of N > which f maps back to the same member of S. If x is in N, then f maps x > to a member of S which g maps back to the same member of N. You wrote f(g(x)) = g(f(x)) = x. This is nonsense. g(f(x)) is in S and f(g(x)) is in N, so f(g(x)) != g(f(x)). What you meant was f(g(x)) = x and g(f(x)) = x but in the former case, x is in N and the latter x is in S. It's not a deep issue, but what you wrote was mistaken. -- Jesse F. Hughes "The people who made up the words could have said 'newspaper' is 'trees'." -- Quincy P. Hughes, five-year-old Wittgensteinian (This comment came out of the blue at breakfast.)
From: Jesse F. Hughes on 11 Jun 2010 19:14 Tony Orlow <tony(a)lightlink.com> writes: > On Jun 11, 12:44 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Tony Orlow <t...(a)lightlink.com> writes: >> > On Jun 11, 12:21 am, David R Tribble <da...(a)tribble.com> wrote: >> >> Tony Orlow wrote: >> >> > For, "none >> >> > shall drive us from the Garden which Cantor has created for us". If it >> >> > doesn't produce fruit, it's time to plant a new bed, or at least >> >> > fertilize. >> >> >> Well, it's done pretty well for the last 100+ years. Have any >> >> of your ideas produced any results of consequence yet? >> >> >> I'm still waiting to hear what bigulosity/IFR says about the size >> >> of the set of natural squares. Any results yet? >> >> > I already did that. According to IFR it's sqrt(omega), which according >> > to ICI is less than omega. >> >> So, it is a(n ordinal) number x such that x * x = omega, is that right? > > Ordinals have nothing to do with my theory. I've called the von > Neumann ordinals "schlock" for years. You should know that, at the > very least. > >> >> I'm not clever like you, but I'd wager that one can prove no such >> ordinal exists, when * stands for ordinal multiplication. Does that >> bother you? > > Not much, nope, not at all. > >> >> As well, one can prove that omega is the least infinite ordinal (where >> "least" is defined in terms of cardinality, of course). That is, if >> alpha is an infinite ordinal, then there is an injection from omega to >> alpha. Is this contrary to your claims? You sometimes say that you >> don't think omega is the smallest infinite ordinal, but I'm not sure >> what you mean when you say that. > > I don't use the word "ordinal" in my arguments (sure, go find one > mention from 1996 or whatever). I say there are a wide spectrum of > countable and uncountable infiniites, given the right techniques. But you do think that there is a "number" x such that x * x = omega? Yes or no? Anyway, you have said, if I recall correctly, that you're not using ZFC, so perhaps this result is not surprising. But what theory of sets (and of numbers) do you have in mind? What are the axioms? If you can't state the axioms, then don't you think it's a bit premature to state theorems? -- Jesse F. Hughes "To your limited perspective it looks like nothing is happening, while already I have more impact on the math world with some posts here or on my blog than just about any other human being on the planet." JSH
From: Jesse F. Hughes on 11 Jun 2010 19:09 Tony Orlow <tony(a)lightlink.com> writes: >> > Try it on any finite set. Now imagine applying it to the range >> > [0,omega]. >> >> > I hope that is clear enough for you. >> >> I'm not sure. Let's try. Let S = {1/4, 1/16, 1/64}. That's a finite >> set. > > Stop. > > You have listed three elements. You have not defined how they are > mapped from N. You said try it on a finite set, so I tried a function with domain {1,2,3}. I misunderstood what you meant. -- Scissors and string, scissors and string, When a man's single, he lives like a king. Needles and pins, needles and pins, When a man marries, his trouble begins. --- Mother Goose
From: Jesse F. Hughes on 11 Jun 2010 19:10 Tony Orlow <tony(a)lightlink.com> writes: >> "So now, The Hammer is here, and with it, the end of days. The world will be >> destroyed, and then remade, as foretold. You will be lost, with your >> children, and then there will be others, and one day they will be tested, and >> will pass, but that is another story." --James S Harris gets a bit excited.- Hide quoted text - >> > > PS - If you wouldn't mind laying off, the JSH references get a tad > tiring. I never overrule the random sig picker. -- "People make mistakes. Better to live today and learn the truth, than to be one of those poor saps who died deluded, thinking they knew certain things that they just didn't. Thinking they had proofs that they didn't." --James S. Harris, almost too sad for a .sig
From: MoeBlee on 11 Jun 2010 19:26
On Jun 11, 12:59 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 11, 8:46 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > omega/2 > > What operation is '/' where <w 2> are in the domain? > > Ordinal division? > > It's not cardinal division, until you DEFINE such a thing. I'm still interested in the above question. MoeBlee |