PS. (this is a Pre-script, just for variety). I've always liked: While marking their work, the teacher noted that John had written "had", whereas Jim had had "had had". "Had had" had had the teacher's approval.
Just to wind up this one, from way back, Tony Firshman wrote: > One very common one now is to use 'less' for everything, where 'fewer' > should be used. "Less people" for instance. The rule is *so* simple. > If one can count the noun (ie discrete items) then it is 'fewer'. > > http://www.askoxford.com/asktheexperts/faq/aboutgrammar/lessfewer?view=uk > http://www.gcse.com/english/less.htm > > Being me, I took TF at his word, that the rule should be "countable", and went onto my "infinity" theme. Firstly, I asked another friend about this, and he instantly said that the rule is just "fewer with a plural and less if not" (I think that makes sense). The plural does not have to be countable, hence there are infinitely fewer calculatable numbers than there are points on a straight line. Secondly, you might like to read the first chapter of "The Emperor's New Mind", by Roger Penrose. It could give you a feel for why the number of computable numbers is countable, versus points on a line, which aren't. Thirdly, the IEEE spec actually allows a choice of one (projective) or two (affine) infinities. The projective one is an unsigned infinity, and is what happens when cartographers project the globe onto a plane. You sorta have the south pole at the center of the plane and the north pole is projected off to infinity, all round the plane. The IEEE spec does allow a few operations involving infinities. It will happily let you add anything finite to an infinity with a quibble - the infinity is not affected. It will even allow two similarly signed infinities to be added together, and remain the same, and so on. Fourthly, the main sort of mathematical infinities that crop up are the "Aleph" sequence (Hebrew alphabet - or should that say alephbeth?). Aleph-null is the countable infinity, aleph-one is the first non-countable infinity. Have a look on Wikipedia <http://en.wikipedia.org/wiki/Aleph_number>. Fifthly, there are other "number systems" that can do sums just like the conventional stuff(sic), but can handle "infinity plus one" as a distinct "number", which you can add one to and get "infinity plus 2", and so on. <language head="need aspirin handy"> The Penrose argument about computable/countable goes roughly like this: A Turing machine can perform any function that any more complex computer (a Super-Nano-Multi-Cray, say) can do. In effect, we define a number as calculatable if there exists some Turing machine that can churn out any *specified* finite number of digits of its decimal expansion, in a finite time, and then stop. It is provable (but to spare you a little, I won't go into detail) that there exist only a countable number of Turing machines (i,e, you can give every one a "serial number"). Thus, at this point, which is really a lot harder than I've let on, we've got "computable numbers are countable". The number of points on a line are not countable. This is most prettily proved by the "diagonal slash", a "reductio ad absurdum" argument. Say one *could" count the points on a line (say from 0 to 1). In that case, you could write them down in order: Point number 1, decimal expansion: 0.445195495... and so on... Point number 2, decimal expansion: 0.724678748... and so on... Point number 3, decimal expansion: 0.256566745... and so on... Point number 4, decimal expansion: 0.154557676... and so on... Point number 5, decimal expansion: 0.786847688... and so on... Point number 6, decimal expansion: 0.959689689... and so on... Point number 7, decimal expansion: 0.747467476... and so on... Point number 8, decimal expansion: 0.990898997... and so on... Point number 9, decimal expansion: 0.265656565... and so on... ... and so on, to countable infinity. Along comes the irritating mathematician (me) and says "Hey, you've missed one out! Where's the one that goes 0.537650506....". I've given the first digit as one more than the first digit of point 1, the second digit one more than the second digit of point 2, and so on (wrapping 9's back to 0's). "Oh dear!" you say, and disappear in a puff of logic. </language head="normal"> -- Lau http://www.bergbland.info callto://LauReeves (see http:www.skype.com) Get a domain from http://oneandone.co.uk/xml/init?k_id=5165217 and I'll get the commission! _______________________________________________ QL-Users Mailing List http://www.q-v-d.demon.co.uk/smsqe.htm
