Cantor Confusion
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From: G. Frege on 24 Jan 2007 08:14 On Wed, 24 Jan 2007 12:56:39 GMT, Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: > > [...] But the binary representation of a real is [or may be] > infinite (without end) - and that is the point? > Yes, that is the point. F. -- E-mail: info<at>simple-line<dot>de
From: Tez on 24 Jan 2007 08:20 imaginatorium(a)despammed.com wrote: [snip] > I've been watching various cranks here for some time now - in a > gruesome sort of way, it's actually fascinating. I think one of the > commonest themes running through what they say is a (mis)conception > that "infinity" is a sort of number, very very big number, bigger than > any number you'd ordinarily think of as a number, but anyway it's at > the end of (e.g.) the list of natural numbers. One or two of the less > coherent [if that's not an antoximoron] of the cranks just mumble that > "actual/completed infinity" is nonsense - which is true, I suppose; but > this is not a variety of "infinity" that mathematicians talks about. If > you can force yourself to talk only of (e.g.) bitstrings with two ends, > or lists of numbers that are unending, you may well clear up your own > confusion. Yes, it's bizarre. I recall one of the first things said to us in my analysis course was that infinity is not a real or natural number. "Lim f(n), n: 1->inf " is simply an abuse of notation (with a well-defined meaning). What I don't understand is why this doesn't appeal to cranks' intuitions: Consider a sequence a_n where a_n = 1 for all n e N. What's the "length" of this sequence? You'd hope everyone would agree it's (plain ol', "actually", "potentially") infinite. Since it has infinite length, one of these 1s (according to the usual crank reasoning) must be infinite! Which one? Or perhaps I should just claim that I've now found a contradiction. ZFC has crumbled at my feet! > Brian Chandler > http://imaginatorium.org -Tez
From: Dave Seaman on 24 Jan 2007 08:23 On Wed, 24 Jan 2007 13:07:52 GMT, Andy Smith wrote: > In message <ep7kn4$79j$2(a)mailhub227.itcs.purdue.edu>, Dave Seaman ><dseaman(a)no.such.host> writes >>Your original statement was that there are not enough integers to define >>even one real. That's patently false. Were you assuming that we are >>allowed to use only a single integer in the definition? >> >> > You misunderstand what I meant, I think, or at any rate I didn't express > myself clearly. > If you have a transcendental, you need to specify an infinite number of > bits to distinguish it from the set of all alternative transcendentals. > You specify reals as a Cauchy sequence, which unambiguously points > towards the point, but the point itself needs an infinite number of bit > positions. But you can't label an infinite number of bit positions - you > need to have all of the bits as a completed set to define the real - and > that is not a finite number. The reason for my objection is this: we define the rationals from the integers, and we define the reals from the rationals. That certainly implies that there are enough integers to define, not just one real, but all of the reals. Perhaps you meant to say something different, but you didn't. You don't seem to have considered the possibility that we might use more than one integer in defining a real number. -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. <http://www.mumia2000.org/>
From: G. Frege on 24 Jan 2007 08:23 On 24 Jan 2007 05:20:07 -0800, "Tez" <terence.hoosen(a)gmail.com> wrote: > > What I don't understand is why this doesn't appeal to cranks' > intuitions: Consider a sequence a_n where a_n = 1 for all n e N. > > What's the "length" of this sequence? You'd hope everyone would agree > it's [...] infinite. [If] it has infinite length, one of these 1s > (according to the usual crank reasoning) must be infinite! > No, you got that wrong. Not one of the 1's has to be infinite, but one of the indices! (Yes, I know, that's nonsense.) F. -- E-mail: info<at>simple-line<dot>de
From: Andy Smith on 24 Jan 2007 08:34
In message <1169644806.887614.311180(a)13g2000cwe.googlegroups.com>, Tez <terence.hoosen(a)gmail.com> writes > >imaginatorium(a)despammed.com wrote: >[snip] >> I've been watching various cranks here for some time now - in a >> gruesome sort of way, it's actually fascinating. I think one of the >> commonest themes running through what they say is a (mis)conception >> that "infinity" is a sort of number, very very big number, bigger than >> any number you'd ordinarily think of as a number, but anyway it's at >> the end of (e.g.) the list of natural numbers. One or two of the less >> coherent [if that's not an antoximoron] of the cranks just mumble that >> "actual/completed infinity" is nonsense - which is true, I suppose; but >> this is not a variety of "infinity" that mathematicians talks about. If >> you can force yourself to talk only of (e.g.) bitstrings with two ends, >> or lists of numbers that are unending, you may well clear up your own >> confusion. > >Yes, it's bizarre. I recall one of the first things said to us in my >analysis course was that infinity is not a real or natural number. "Lim >f(n), n: 1->inf " is simply an abuse of notation (with a well-defined >meaning). > >What I don't understand is why this doesn't appeal to cranks' >intuitions: >Consider a sequence a_n where a_n = 1 for all n e N. > >What's the "length" of this sequence? You'd hope everyone would agree >it's (plain ol', "actually", "potentially") infinite. Since it has >infinite length, one of these 1s (according to the usual crank >reasoning) must be infinite! Which one? Or perhaps I should just >claim that I've now found a contradiction. ZFC has crumbled at my >feet! > This is probably aimed at me, fair enough. Consider a bit sequence b_n where b_0 is the least significant bit and b_n = 1 A n e N and the number defined by m = b_0 +2*b_1 + ... 2^nb_n ... Is m a natural number? (I would say not) -- Andy Smith |