Cantor Confusion
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From: Virgil on 24 Jan 2007 16:17 In article <1169642223.645196.70490(a)s48g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 23 Jan., 18:17, "MoeBlee" <jazzm...(a)hotmail.com> wrote: > > No, we do NOT have to suppose the list is of all real numbers. > > But you have to suppose that all enumerated lines and columns are > there. So you have to suppose the finished presence of an actually > infinishable set N. That is nonsense. > > Regards, WM An real infinite sequence is a perfectly well defined mathematical object, which is as "finished" as it need be for mathematical purposes. If WM has some philosophical objection to infinite sequences, or other functions with domains which are not finite, he should limit his objections to philosophical newsgroups.
From: Virgil on 24 Jan 2007 16:18 In article <ep7k43$79j$1(a)mailhub227.itcs.purdue.edu>, Dave Seaman <dseaman(a)no.such.host> wrote: > On Wed, 24 Jan 2007 09:19:22 +0100, G Frege wrote: > > On Tue, 23 Jan 2007 19:45:39 -0500, David Marcus > ><DavidMarcus(a)alumdotmit.edu> wrote: > > >>> > >>> People cannot conceive of an infinite past, [...] > >>> > >> Why not? I believe that was the usual assumption before the Big Bang > >> was discovered. > >> > > Not really... Remember? > > > "In the beginning God created the heavens and the earth. Now the earth > > was formless and empty, darkness was over the surface of the deep, and > > the Spirit of God was hovering over the waters." > > > This happened about 6000 years before Christ's birth, or so. > > It was supposed to be 4004 BC, according to Bishop Ussher. What o'clock?
From: Virgil on 24 Jan 2007 16:23 In article <1169642941.554024.145640(a)k78g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 23 Jan., 19:27, Dave Seaman <dsea...(a)no.such.host> wrote: > > On Tue, 23 Jan 2007 17:15:52 GMT, Andy Smith wrote: > > > By definition, a set is "finite" if it has the size of some natural > > number. If a set isn't finite, then it's called "infinite". > > If a colour is not red, then it is called green by set theorists. Perhaps non-red or in-red, but not green. > > > It's > > obvious that the set of all natural numbers can't be finite, since that > > would imply the existence of a largest natural number. > > It is obvious tat the set N cannot be actually infinite, because that > would imply infinity to be finished . Thus WM tries to scuttle every function having a non-finite domain on the grounds that allowing it would complete an infinity. What sort on mathematics does WM want to allow which requires every function have a finite domain? > > And they were right! > > > You can't have an actually infinite integer for exactly the same reason > > that you can't have an actually 200-cm. meter or an actually 4-sided > > triangle > > or an actually finished infinity. Or, according to WM, any sort of function at all unless it has a finite domain.
From: Virgil on 24 Jan 2007 16:24 In article <1169643065.622845.154490(a)k78g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > Summary > > 1) Every complete infinite binary tree T (containing all nodes and > edges) contains all paths. Then, unless it contains leaf nodes, it must contain endless paths. But every node that is a leaf node in one finite tree is not a leaf in a larger tree, so that can be no leaf nodes. > 2) The union tree T(oo) of all finite trees is well defined (as I have > shown elsewhere) and yields the complete infinite binary tree > containing all nodes and edges: T = T(oo). Except that "union" is not a legitimate name for the operation as WM defines it, agreed. One might note that it also contains all possible paths. > 3) The union of all finite trees includes the union of all nodes and, > with it, the union of all such subsets which are paths (because every > path is a well defined subset of the set of nodes if the structure of > the tree is well defined). Actually paths would be "subsets" of edges, as it is only sequences of nodes connected by edges that can be paths. > 4) The set of paths in T(oo) is a subset of the countable set of finite > sets of all paths in the finite trees. Trivially false. Since the "union" tree cannot contain any leaf nodes, no path can be finite. > 5) A countable union of countable sets is a countable set (according to > ZF with AC). True but irrelevant. > ==> The set of all path is countable. False, and proven false. > (==> The real numbers are > countable.) False, and proven false. > > Going on, we can say: > > 6) T(oo) = T contains only finite paths. False, and proven false. > 7) T(oo) = T contains all paths including all infinite paths. True but irrelevant. > ==> There are no infinite paths. (There are no irrational numbers.) False, and proven false. >
From: MoeBlee on 24 Jan 2007 16:24
On Jan 24, 11:48 am, Andy Smith <A...(a)phoenixsystems.co.uk> wrote: > Anyway, following Moeblee's kind advice, I am buggering off to read some > books. Just to be clear, my only advice was to read some books; I didn't also suggest that the poster not post in the meantime. MoeBlee |