Cantor Confusion
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From: G. Frege on 23 Jan 2007 16:07 On Tue, 23 Jan 2007 20:22:35 GMT, Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: > > Actually you miss the point. I wasn't saying that this was wrong, just > that you (i.e. mathematicians in general) had achieved a mental > perspective not commonly shared, and that this perspective was not an > intuitive one ... > So what? F. -- E-mail: info<at>simple-line<dot>de
From: G. Frege on 23 Jan 2007 16:10 On Tue, 23 Jan 2007 20:32:35 GMT, Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: > > Too much Aristotle. [...] > Well, ... maybe Aristotle didn't say the last word concerning certain topics? (Ever considered that possibility?) :-) F. -- E-mail: info<at>simple-line<dot>de
From: Virgil on 23 Jan 2007 16:16 In article <1169553709.149880.6320(a)s48g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Unless there is an infinite number the number of > numbers, with difference 1, cannot be infinite. But there cannot be an > infinite natural (= finite) number. This is another instance of Quantifier dyslexia: "(A x e N) (E y e N) ( y > x )" is a true statement. "(E z e N) (A x e N) ( y > x )" is a false statement. To argue that the first implies the second is the fallacy we call quantifier dyslexia (QD),which seems endemic among the kooks who post here.
From: Virgil on 23 Jan 2007 16:21 In article <mq+qF3d7BgtFFwhY(a)phoenixsystems.demon.co.uk>, Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: > So Cantor's argument doesn't rely on locating an actually infinite > member of his list as some limit process n->oo, he just posits that an > actually infinite list can exist and is complete, and then shows it > can't be (or possibly alternatively that an actually infinite list can't > exist?) Actually, Cantor's argument does not ever posit that any list is "complete". he merely posits a (not necessarily complete) list exists and then shows any such list to be incomplete. There are some who have tried to make Cantor's more direct proof into an indirect one when copying it, but the original was direct. > > OK, well I do see the argument better now, but if that was an argument > that I had suggested for the first time, you would be on me like wolves > ... > > Cheers
From: David Marcus on 23 Jan 2007 16:31
Andy Smith wrote: > G. Frege <nomail(a)invalid.?.invalid> writes > I am just observing that all of this is not intuitively obvious (i.e > definitely not in the same way as e.g. Euclid, or even basic calculus > is). Maybe not intuitively obvious to you. But, you shouldn't generalize. > >You might try to get a copy of > > > > Rudy Rucker, Infinity and the mind. The science and > > philosophy of the infinite. > > > >A very nice book. > > I bought a copy 25 years ago, and am just re-reading it now. But my > impression is that a lot of people on this NG would take issue with > Rudy Rucker particularly in respect of achieving actual infinity, not to > mention his Zeno based speed-up scheme to count to infinity and beyond > (David Marcus please comment?) I've read some of Rucker's stuff, but I forget if I read that book. I think Rucker knows his math. What does "achieving actual infinity" mean? > Well me to, struggling with the validity or otherwise of Cantor's > diagonalisation logic. And, arguing about the infinite gets contaminated > with perceptions/lines of argument easily derived from the finite, and > it is still not clear to me (short of taking a course in axiomatic set > theory) quite what inferences can be legitimately made concerning > infinite as opposed to finite situations. Taking a course in axiomatic set theory is a terrible way for a beginner to learn what inferences are valid. We normally teach this to math majors in an analysis course. "Calculus" by Spivak is probably the best book to learn this from. > But, to cut to the chase, if you don't have enough integers to define > even one real, what chance of counting them all? What in the world do you mean by "don't have enough integers to define even one real"? Do you have some problem defining sqrt(2)? -- David Marcus |