Cantor Confusion
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From: David Marcus on 23 Jan 2007 21:55 David R Tribble wrote: > Andy Smith wrote: > >>Since the integers are finite, you cannot represent a real requiring an > >>actually infinite number of bits, is what I meant. Maybe that is too > >>simplistic? > >> > >> To be more explicit, to represent (= address) all the reals in say > >> [0,1] you would need as many bits for your integers as the reals occupy. > >> But that would require integers with an actually infinite bit length > >> e.g. say, the reflection of the reals about the decimal point to give > >> "numbers" like ...1101 > >> > >> Which is where I came in (with what's the problem with enumerating the > >> reals?) but I am now better informed - . If all members of N are > >> finite, there is no prospect of addressing the reals, so no surprise > >> there.... > > Andy Smith wrote: > > Are you sure you are better informed? What you wrote is nonsense. Every > > real number has a binary expansion (and a decimal expansion). So, what > > in the world are you trying to say? > > No, I think he's almost got it. I think he's trying to say (in > computer programming terms) that any given real in [0,1] > requires a countably infinite number of bits to represent > as a binary fraction (bitstring), which is correct. And that > those infinite bitstrings cannot be mapped to finite naturals > (e.g., by reflecting the digits about the binary point), > because you'd end up with infinite-length binary integers, > which are not naturals. > > So I think he's reached the (correct) conclusion that you > can't denumerate the reals (in [0,1]) using naturals, > albeit in a somewhat clumsy way of saying it. If that's what he means, then I'll agree he could be close. It isn't a proof, but as a heuristic it is OK. -- David Marcus
From: Dik T. Winter on 23 Jan 2007 22:12 In article <mF$aeDY+cetFFwj+(a)phoenixsystems.demon.co.uk> Andy Smith <Andy(a)phoenixsystems.co.uk> writes: .... > Cantor's hypothetical numbered list of the reals is also finite ? No. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 23 Jan 2007 22:36 In article <1169578564.772090.181560(a)v45g2000cwv.googlegroups.com> imaginatorium(a)despammed.com writes: .... About Wolfgang M�ckenheim. > You've escaped noticing? All around you in this thread is the endless > spewing of our German friend who, inexplicably, is said to _teach_ > mathematics at some sort of college. He's just reponded and "confirmed" > that your confusion is "correct". (But all in undefined verbiage.) Actually I am getting the impression that what he presents here is not what he presents in the mathematics courses at his University. (A Fachhochschule is called a University in the Netherlands.) I received his book and I found no errors in the first four chapters I did read (of the ten in all). I am only a bit unlucky about his distinction between (indeed) actual and potential infinity, but that can be clarified later. Also his statement (in chapter 3) that irrational numbers only exist due to actual infinity is based on the representation of numbers in some integral base. That appears (to me) a bit shortsighted (*). On the other hand, the introduction of rationals, irrationals, algebraic numbers and whatever is clear (not strict, but clear enough for the intended audience). Also his explanation of the Peano axioms is correct. So unlike some crank books, this book does not contain serious errors in the first four chapters. That may come as a surprise to some (it did to me). (*): The reason appears to be that irrationals can only be given by a rule about how to compute it. But I think that: 0.142857142857... is also nothing more than a rule how to compute it. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: G. Frege on 24 Jan 2007 02:20 On Tue, 23 Jan 2007 19:49:09 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >> >> OK, some numbers like the rationals and algebraic >> numbers have compact representations, but they are 0% >> of the total ... >> > What does "zero percent" mean in this context? > Measure 0? F. -- E-mail: info<at>simple-line<dot>de
From: G. Frege on 24 Jan 2007 02:31
On Tue, 23 Jan 2007 23:34:03 GMT, Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: > > I meant that the infinite non-repeating irrational binary expansion of > sqrt(2) requires an [...] infinite set of numbers to define its > location on the line... > Right. > > Since the integers are finite, ... > Non sequitur. Yes, the integers are finite, but there are infinitely many of them. To be precise: countable infinite many of them. And that's actually (sic!) enough to represent each and any real number. F. P.S. Could you PLEASE stop talking about "actually" infinite this and "actually" infinite that?! I already told you that this is not sensible in the present context. Do you want to _prove_ that you are a bonehead? A candidate for crankhood? Or what? -- E-mail: info<at>simple-line<dot>de |