Cantor Confusion
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From: G. Frege on 23 Jan 2007 16:34 On Tue, 23 Jan 2007 19:32:32 GMT, Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: > > 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). > I'm sorry. Can't agree. Neither Euclid nor Calculus is "intuitive" (in your sense). Ever seen a (geometrical) _point_? :-) You should get that straight: the intuition of a mathematician (concerning mathematical questions) is not comparable with the intuition of a layman. Mathematical intuition _needs_ training! >> >> 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. > Great! > > But my impression is that a lot of people on this NG would take issue > with Rudy Rucker. > I don't think so. Of course, the book is _not_ a mathematical textbook. But it's not meant to be one. >> >> A quote from Herb Enderton's Elements of Set Theory: >> >> "Cantor's work was well received by some of the prominent >> mathematicians of his day, such as Richard Dedekind. But his >> willingness to regard infinite sets as objects to be treated in much >> the same way as finite sets was bitterly attacked by others, >> particularly Kronecker. There was no objection to a 'potential >> infinity' in the form of an unending process, but an 'actual infinity' >> in the form of a completed infinite set was harder to accept." >> > Yep, that is Aristotle's heritage. > Yes. >>> >>> If I was asked to sum it up, at present I would say that my >>> understanding is that you can't have an actually infinite integer, but >>> reals can be defined as having an [...] infinite binary >>> representation ... So no surprise that the reals are "uncountable". >>> >> Well, actually, it WAS a surprise. (Some still aren't able to get it. >> :-) >> > Well me to, struggling with the validity or otherwise of Cantor's > diagonalisation logic. > Strange thing. Can't imagine a more beautiful, elegant, and convincing proof/argument. :-) > > 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. > Actually, when dealing with sets you DON'T _have to_ differentiate between finite and infinite sets! The logic used for the reasoning is just standard (first order) logic. Ok, you have to know which "construction" are allowed (by the axioms of set theory). Otherwise you will (sooner or later) come up with nonsense. > > But, to cut to the chase, if you don't have enough integers to define > even one real, what chance of counting them all? > We have enough integers to define each and any real number (We can represent real numbers as infinite strings of digits.) It's just that we can't _count_ the reals. :-) F. -- E-mail: info<at>simple-line<dot>de
From: G. Frege on 23 Jan 2007 16:41 On Tue, 23 Jan 2007 16:31:47 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > I think Rucker knows his math. > I would think so too ... thought by Gaisi Takeuti (at the Institute for Advanced Study in Princeton, New Jersey) ... met G�del (there) ... Well, this guy really should know what he's talking about. :-) F. -- E-mail: info<at>simple-line<dot>de
From: Franziska Neugebauer on 23 Jan 2007 16:44 Virgil wrote: > In article <1169548709.607738.69810(a)s48g2000cws.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: >> Franziska Neugebauer schrieb: >> > mueckenh(a)rz.fh-augsburg.de wrote: >> > >> > > Franziska Neugebauer schrieb: >> > >> Even if we now use >> > >> >> > >> T(m) U T(n) := T(sup(m, n)). >> > >> (fin-u') >> > >> >> > >> and and try to define (inf-u) by >> > >> >> > >> T(1) U T(2) ... := T(omega) >> > >> >> > >> it is still left open what T(omega) shall mean. One has to take >> > >> special care for the notation: The union symbol "U" does not >> > >> mean the usual set theoretical union. >> > > >> > > It does. >> > >> > No, it does not. The usual set theoretical union is defined as >> > >> > A U B := { x | x e A v x e B } >> > >> > If A and B are trees {M_A, E_A} and {M_B, E_B} then the union is >> > >> > A U B = { x | x e {M_A, E_A} v x e {M_B, E_B} } >> > = { M_A, E_A, M_B, E_B } >> > >> > This union is not a tree. >> >> unless M_A, E_A and M_B, E_B have same elements at all levels they >> have in common. > > Actually, FN is not quite right about what a tree is. It is an > /ordered pair/ I had alread corrected the typos ("{...}" instead of "(...)") and wrong definition of pair nonetheless WM uses google which does not honor supersedes. You find the supersede in <45b4c935$0$97251$892e7fe2(a)authen.yellow.readfreenews.net> > of sets, (M,E), with the first set being a set of edges You mean vertices or nodes instead of edges here ^, do you? > and the second set, of edges, being a subset of the cartesian product, > MxM. > And a union of two or more ordered pairs is not, itself, an ordered > pair of the same type, so is not a tree at all. > > One realization of an ordered pair (X,Y) is as the set {X,{X,Y}}. Indeed this is the Kuratowski-pair. http://en.wikipedia.org/wiki/Ordered_pair > Then let A = (M_A, E_A) = {M_A,{M_A,E_A}}, where E_A \ss M_A x M_A, > and let B = (M_B, E_B) = {M_B,{M_B,E_B}}, where E_B \ss M_B x M_B, > then A U B = {M_A,{M_A,E_A},M_B,{M_B,E_B}}. > And is not itself an ordered pair unless A = B. True. F. N. -- xyz
From: G. Frege on 23 Jan 2007 16:46 On Tue, 23 Jan 2007 22:44:00 +0100, Franziska Neugebauer <Franziska-Neugebauer(a)neugeb.dnsalias.net> wrote: >> >> One realization of an ordered pair (X,Y) is as the set {X,{X,Y}}. >> > Indeed this is the Kuratowski-pair. > http://en.wikipedia.org/wiki/Ordered_pair > Not quite. Should read "... is as the set {{X},{X,Y}}", I guess. ~~~ F. -- E-mail: info<at>simple-line<dot>de
From: G. Frege on 23 Jan 2007 16:49
On Tue, 23 Jan 2007 15:38:34 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > But we disagree. We think it is very intuitive. [...] > Agree. On the other hand, imho the OP should realize that the intuition of a "mathematician" (concerning mathematical questions) is not comparable with the intuition of a (complete) layman. Mathematical intuition _needs_ training! F. -- E-mail: info<at>simple-line<dot>de |