Obections to Cantor's Theory (Wikipedia article)
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From: malbrain on 25 Jul 2005 20:34 Virgil wrote: > In article <MPG.1d4ef2d3df7d5616989f71(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > Is there a 1-1 correspondence between digital strings and whole numbers? > > Depends on what one means strings and by whole numbers. TO includes > among his "strings" things which have infinitely many characters > including a first and last, yet are supposed to be serially ordered > with, except for the first, each having both an immediate predecessor, > and, except for the last, an immediate successor. > > > > Can you have an infinite set of strings without infinite lengths? > > Everyone but TO can. But TO handicaps himself. > > > > How can you have an infinite set of digital whole numbers, without having > > infinitely long strings representing infinite values? > > By allowing those strings to get arbitrarily long (no limit on length). That's what Tony is illustrating. Each of his infinite strings has a finite value. karl m
From: malbrain on 25 Jul 2005 20:43 Lee Rudolph wrote: > Robert Low <mtx014(a)coventry.ac.uk> writes: > > >Virgil wrote: > >[re TO] > >> If he wants to make up his own game, fine, but he will not find many > >> willing to play by his rules. > > > >He'll have to tell us what the rules are before we can > >decide if the game is interesting. > > At this point (and I think long before this point), we have a lot of > evidence that he is one or more of (a) irremediably stupid, (2) hopelessly > mad, (iii) successfully trolling. In each case, it's unlikely that > the game is going to be interesting; except, indeed, to those (who > seem to be many) who enjoy spending their summers (if in the Northern > Hemisphere) dealing with members of one or more of those classes. And you are, a member of the American Mathematical Society? Or do you have some other venue to take responsibility for expanding the craft of mathematics? karl m
From: malbrain on 25 Jul 2005 20:44 Barb Knox wrote: > In article <MPG.1d4ecd45545679a8989f6b(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > [snip] > > >keep in mind that > >inductive proof IS an infinite loop, so that incrementing in the loop creates > >infinite values, and the quality of finiteness is not maintained over those > >infinite iterations of the loop. > > Using your computational view, consider the following infinite loop > (using some unbounded-precision arithmetic system similar to > java.math.BigInteger): > > for (i = 0; ; i++) { > println(i); > } > > Now, although this is an INFINITE loop, every value printed will be > FINITE. Right? Not so fast. The behaviour of incrementing i after it reaches INT_MAX is undefined. karl m
From: malbrain on 25 Jul 2005 20:46 Robert Low wrote: > Virgil wrote: > [re TO] > > If he wants to make up his own game, fine, but he will not find many > > willing to play by his rules. > > He'll have to tell us what the rules are before we can > decide if the game is interesting. The game is interesting only after we can enforce the rules. karl m
From: malbrain on 25 Jul 2005 21:02
Chris Menzel wrote: > On Mon, 25 Jul 2005 15:27:28 -0400, Tony Orlow <aeo6(a)cornell.edu> said: > > I don't need a favor. > > Hey, have it your way. Sorry, but out here, when you run across a man in the desert without enough water to carry out his plan, you disabuse him of his notions by force if necessary. karl m |