Cantor Confusion
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From: Andy Smith on 24 Jan 2007 04:27 In message <1169599939.569716.222730(a)k78g2000cwa.googlegroups.com>, David R Tribble <david(a)tribble.com> writes >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. > Yes!! Thank you. -- Andy Smith
From: Andy Smith on 24 Jan 2007 04:38 In message <4f3er2199791dterims6m7t9u35j4o1dap(a)4ax.com>, G. Frege <nomail(a)invalid.?.invalid> writes >On Tue, 23 Jan 2007 21:55:14 -0500, David Marcus ><DavidMarcus(a)alumdotmit.edu> wrote: > >> >> 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. >> >Though by using (almost) the same heuristic he might conclude that >rational numbers aren't countable too. Well... > > >F. > No I wouldn't because it is clear that you don't need the same address space for the rationals as the reals. You can get one unique mapping by a bit shift left to the end of the first repeating sequence of bits. So e.g. 0.10101... becomes 101; 0.101011111.. becomes 10101 etc. (I think that's unique, but anyway, happy with the standard method of mapping the rationals). -- Andy Smith
From: Andy Smith on 24 Jan 2007 04:48 In message <9b2er2p5taadlea28n6fb3g4nuvmqeijhs(a)4ax.com>, G. Frege <nomail(a)invalid.?.invalid> writes >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. > Any systematic scheme for mapping the reals to integers will be the same, subject only to permutations of the bit positions. So a systematic scheme is, start with bit 1, then bit 2, then bit 3 etc. This corresponds to a reflection of the possible set of numbers in n bits about the binary point. Any real number has an infinite binary expansion, and its corresponding mapping integer is infinite (=NaN) > >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? > OK, sorry. -- Andy Smith Phoenix Systems Mobile: +44 780 33 97 216 Tel: +44 208 549 8878 Fax: +44 208 287 9968 60 St Albans Road Kingston-upon-Thames Surrey KT2 5HH United Kingdom
From: G. Frege on 24 Jan 2007 04:47 On Wed, 24 Jan 2007 09:38:00 GMT, Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: >> >> To represent a real number between 0 and 1, you need 1 bit for >> each positive integer. >> >> x = b1*(1/2)^1 + b2*(1/2)^2 + b3*(1/2)^3 + b4*(1/2)^4 .... >> >> Why do you think this requires an "actually infinite" integer? >> > Because you need binf to complete the sum; inf is not a natural number > and you need an actual infinity of bits to describe it. > You are talking nonsense, again. Aren't you able to understand the difference between an (a) infinite integer and (b) infinitely many integers ? We do not need "binf" (?) to "complete the sum" because this sum is _never_ "completed". > > If you systematically try to address (map) the reals you need integers > with as many bits as the reals; NaN. > Bla bla. Seems that you are desperately striving for a career as a crank here. Go ahead, I think you will succeed! :-) F. -- E-mail: info<at>simple-line<dot>de
From: Andy Smith on 24 Jan 2007 04:58
In message <6e5er2ddal5t4ctrnt65dk8nktd40031pi(a)4ax.com>, G. Frege <nomail(a)invalid.?.invalid> writes >On Tue, 23 Jan 2007 19:45:39 -0500, David Marcus ><DavidMarcus(a)alumdotmit.edu> wrote: > >>> >>> People cannot conceive of an infinite past, [...] >>> >> Why not? I believe that was the usual assumption before the Big Bang >> was discovered. >> >Not really... Remember? > >"In the beginning God created the heavens and the earth. Now the earth >was formless and empty, darkness was over the surface of the deep, and >the Spirit of God was hovering over the waters." > >This happened about 6000 years before Christ's birth, or so. > A long time before Aristotle then... -- Andy Smith |