Cantor Confusion
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From: G. Frege on 24 Jan 2007 04:56 On Wed, 24 Jan 2007 09:48:01 GMT, Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: >> >> Why in the world do we need infinite integers to have no beginning? The >> real numbers provide a fine model for time that extends indefinitely >> into the past [and future]. >> > If you found an eternal clock ticking [...] e.g. once a second, wouldn't > you say that it cannot have been ticking for all of negative time, > otherwise it would have already made an infinite [...] number of ticks? > Sure it will have made an infinite number of ticks already, you are completely right here. So what? <----- Past ----->Present<---- Future ----> ... -|---|---|---|---|---|---|---|---|--- ... -4 -3 -2 -1 Now +1 +2 +3 +4 (seconds) (Note that there -in this model- is no beginning in time.) F. -- E-mail: info<at>simple-line<dot>de
From: G. Frege on 24 Jan 2007 05:03 On Wed, 24 Jan 2007 09:48:01 GMT, Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: > > I won't use the adjective "actual" here to avoid > upsetting Mr Frege > Well if we are talking about _mathematical_ questions, the attribut "actual" is not sensible. On the other hand if we are talking about _philosophical_ questions it certainly might be useful to consider the distinction between "actual" and "potential" infinite. (On the other hand, if -in contrary to Aristotle- we allow for actual infinities, there's no need any more to separate those two notions). F. -- E-mail: info<at>simple-line<dot>de
From: Andy Smith on 24 Jan 2007 05:13 In message <1baer212phrkloe4i9r6ghiq9c451h08fm(a)4ax.com>, G. Frege <nomail(a)invalid.?.invalid> writes >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 >? Yes, although my use of your terminology probably goes adrift. > >We do not need "binf" (?) to "complete the sum" because this sum is >_never_ "completed". > Exactly so. A transcendental requires an infinite number of bits to represent it (so as to distinguish it from the rest of the infinite set of other real numbers). >> >> 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. > OK, I will take the personal abuse, probably deserved. Shouldn't put my head above the parapet when I'm not qualified to do so. All that I was trying to observe was that all systematic numbering schemes of the reals are equivalent, corresponding to permutations of bit positions. And if you adopt one systematic scheme, such as starting with bit 0, then bit 1, etc, because the reals have an address space of an infinite number of bits, the corresponding numbers/indices must also require an infinite number of bits, and no natural number is infinite. So you can't do it - you don't have a big enough address space. Of course that may well be a load of bollocks Didn't use the word actually once :) -- Andy Smith
From: G. Frege on 24 Jan 2007 05:11 On Wed, 24 Jan 2007 09:27:59 GMT, Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: >> >> 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. > *sigh* Yes, you reached a correct "conclusion"; but by a faulty reasoning. That's certainly not something to be proud of (at least not in mathematics). F. -- E-mail: info<at>simple-line<dot>de
From: Andy Smith on 24 Jan 2007 05:16
In message <nsaer2drkb82nke1rr3uana5g5q4qd3uks(a)4ax.com>, G. Frege <nomail(a)invalid.?.invalid> writes >On Wed, 24 Jan 2007 09:48:01 GMT, Andy Smith ><Andy(a)phoenixsystems.co.uk> wrote: > >>> >>> Why in the world do we need infinite integers to have no beginning? The >>> real numbers provide a fine model for time that extends indefinitely >>> into the past [and future]. >>> >> If you found an eternal clock ticking [...] e.g. once a second, wouldn't >> you say that it cannot have been ticking for all of negative time, >> otherwise it would have already made an infinite [...] number of ticks? >> >Sure it will have made an infinite number of ticks already, you are >completely right here. So what? Doesn't that imply that you can have an infinite integer? It has done an infinite number of ticks, then infinite plus 1, plus 2 etc , all distinguishable infinite integers? > > <----- Past ----->Present<---- Future ----> > > ... -|---|---|---|---|---|---|---|---|--- ... > > -4 -3 -2 -1 Now +1 +2 +3 +4 (seconds) > >(Note that there -in this model- is no beginning in time.) > > >F. > -- 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 |