Cantor Confusion
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From: Fuckwit on 4 Feb 2007 11:33 On 4 Feb 2007 08:14:30 -0800, mueckenh(a)rz.fh-augsburg.de wrote: > > What is 4 in mathematics? > In axiomatic set theory it's (usually) the set {0, 1, 2, 3}. Moreover it's usually considered a natural number, hence: 4 e N, 4 e Z, 4 e Q, 4 e R, 4 e C, since N c Z c Q c R c C. > > But 4 contains IIII --- not 0,1,2,3. > No. As you can see, 4 has exactly the elements 0,1,2,3. IIII is only an element of 4 if IIII is either 0,1,2 or 3. > > Henry VIII had 7 predecessors --- only including him they were 8 > Henries. > Was Henry VIII a natural number? >> >> ... for computable numbers some representation does exist. >> > But this representation does not necessarily enable us to determine > the trichotomy relation with numbers which are really numbers. > "All numbers are real, but some numbers are more real than others." (Animal Farm) > > I want to define *the natural numbers* [the following way] > > I > II > III > ... > Rules for the construction of these sequences of symbols: Rule 1: We may construct I. Rule 2: If n is constructed, we may construct nI. >> >> The non-existence of a pink elephant in the realm of the axiom that >> states that there is a pink elephant is not mathematics but philosophy. >> Axioms state what things exist (or do not exist) in their realm. > > No. > Yes. >>> >>> Whether it exists remains to be investigated. >>> >> Your existence is not a mathematical existence. >> > This form of existence is the only possible existence. > No. F.
From: Carsten Schultz on 4 Feb 2007 11:38 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: >> If A is infinite, induction can show that A is infinite. > > 1. Where did you read that? Reference? > 2. It reminds me of the Mückenheim-Axiom > > X is not finite -> there must be an x in X which is infinite This follows directly from the more fundamental Mückenheim axiom "every set is finite". Carsten -- Carsten Schultz (2:38, 33:47) http://carsten.codimi.de/ PGP/GPG key on the pgp.net key servers, fingerprint on my home page.
From: mueckenh on 4 Feb 2007 11:39 On 4 Feb., 14:34, Franziska Neugebauer <Franziska- Neugeba...(a)neugeb.dnsalias.net> wrote: > mueck...(a)rz.fh-augsburg.de wrote: > > On 2 Feb., 12:10, Franziska Neugebauer <Franziska- > > Neugeba...(a)neugeb.dnsalias.net> wrote: > >> mueck...(a)rz.fh-augsburg.de wrote: > >> > On 2 Feb., 02:42, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > >> [...] > >> >> However, when we construct the path: > >> >> p = {n_1} U {n_1, n_2} U {n_1, n_2, n_3}, ... > >> >> we get as path: > >> >> {n_1, n_2, n_3, n_4, ...} > >> >> the path length in this case is *not* a natural number. It is the > >> >> cardinality of N. > > >> > The pathlength is not a natural number but it is a number (by > >> > definition), namely omega. You see that there is no infinite set N > >> > without an infinte number in it. > > >> As omega is not member of N I don't see that. > > > The total pathlength is the union of all single pathlengths. > > The path-length of the infinite path is the supremum of the set of > paths-lengths of all finite paths. The latter is the supremum of the > set of natural numbers. This supremum exists. No. What does it consist of? A supremum exists in many places but not here. The simplest reason is that omega - n = omega for all n in N. > > > The total pathlength cannot be omega without omega being a pathlengt > > Omega is the path-length of the infinite path. The pathlengths is measured *and marked* by nodes. This is the advantage of the tree. While you may insist that there is an infinite umber of nodes we know that there is no node number omega. > > > (in the union). > > Non sequitur. This is merely your claim Non sequitur. This is merely your claim. The pathlengths is measured *and marked* by nodes. > > > "Infinite pathlength" means only that every finite pathlength is > > surpassed by another finite pathlength. > > This is the wrong meaning in the framwork of contemporary set theory. What do you think is the correct meaning with respect to node numbers? Regards, WM
From: David Marcus on 4 Feb 2007 12:48 Carsten Schultz wrote: > Franziska Neugebauer schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > >> If A is infinite, induction can show that A is infinite. > >=20 > > 1. Where did you read that? Reference? > > 2. It reminds me of the M=C3=BCckenheim-Axiom > > =20 > > X is not finite -> there must be an x in X which is infinite > > This follows directly from the more fundamental M=C3=BCckenheim axiom "ev= > ery > set is finite". One might naively think that the second axiom makes the first superfluous. Rather remarkable that they are both needed. -- David Marcus
From: mueckenh on 4 Feb 2007 13:01
On 4 Feb., 15:31, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > Similarly one can prove by induction: Every set of even positive > > numbers contains numbers which are larger than the cardinal number of > > the set. > > No. You make your usual mistake. No, you make your usual mistake by thinking that a potentially infinite set can have an infinite cardinal number. > Let E be the potentailly infinite set of even numbers. > Two statements > > i: For every element e of E, the set > F(e) = {2,4,6,...,e} contains cardinal numbers which are > larger > than the cardinal number of F(e) > > ii E contains numbers which are larger than the cardinal > number of E > > Statement i is true and can be shown by induction. Statement > ii is false. Of course, namely because there is no cardinal number for potentially infinite sets. > The fact that statment i is true does not mean > that statement ii is true. The fact that statement I is true means that there are no elements of the set which could increase its "number of elements" without increasing the sizes of elements in the set. A little bit of logic, but not much, is required to infer this fact. Regards, WM |