Cantor Confusion
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From: David Marcus on 1 Feb 2007 13:24 mueckenh(a)rz.fh-augsburg.de wrote: > On 30 Jan., 20:25, Virgil <vir...(a)comcast.net> wrote: > > In article <1170156979.321128.184...(a)j27g2000cwj.googlegroups.com>, > > > > I can't tell whether the union of all lengths is finite or not. > > > > As the union of all lengths is obviously equivalent to the union of all > > finite ordinals and the union of all finite ordinals is clearly not > > finite, WM can't tell much. > > The union of all lengths is not the sum of all lengths, which, > according to your standpoint might be infinite. But the union of all > lengths is a length. You may say that it is infinite or not. If it is > infinite, then it corresponds to an infinite natural number. If it is > not infinite, then the set N does not exist other than as a > potentially infinite (= finite but unbounded) set. It is amazing what you can write if you fail to define your terms. If we read it quickly, it sounds vaguely like a logical argument. Of course, it is actually nonsense. Did you have to practice for a long time to develop this technique? -- David Marcus
From: Dave Seaman on 1 Feb 2007 13:24 On Thu, 01 Feb 2007 17:58:02 GMT, Andy Smith wrote: > Dave Seaman <dseaman(a)no.such.host> writes > Anyway, OK, I am now happy that you can't have an injection from the set > of the reals to the set of the natural numbers, so finally I understand > that cardinality is a meaningful concept. So I understand successive > orders of cardinality to mean that there is no injection from any set > with greater cardinality into any set with a lower cardinality. > Two questions ( I should read a book): > 1) Is there a set with cardinality greater than N but less than R ? That question is known to be undecideable from the axioms of ZFC. Google for "Continuum Hypothesis". > 2) I am (dimly aware) that mathematicians talk about an infinite range > of cardinalities. Is that just an abstract concept, or can you point to > some set and say e.g. that has a higher cardinality than the reals? Cantor's theorem says |X| < |P(X)| for every X. In particular, the power set of the reals has a higher cardinality than the reals. -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. <http://www.mumia2000.org/>
From: Virgil on 1 Feb 2007 14:11 In article <1170330549.125961.302890(a)a34g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 30 Jan., 17:18, Franziska Neugebauer <Franziska- > Neugeba...(a)neugeb.dnsalias.net> wrote: > > > >> 2. BECAUSE it is _undefined_ until the "sum of all natural numbers" > > >> is defined. > > > > > It is defined if the set of all natural numbers s defined, > > > > Here is the definition of the set N > > 1 > 2 > 3 > ... > > Here is the definition of the sum of all elements of N > > 1 > 23 > 456 > ... Not for any one who knows what sums are. It is not even the sequence of partial sums, which would be 1 3 6 10 .... > > It is not difficult to sum all squares, cubes, etc. It is impossible to actually sum all of them, but one can work out their partial sums. To "sum" all of an endless sequence of terms one needs convergence of the partial sums, which in the instances above, one does not have.
From: Virgil on 1 Feb 2007 14:18 In article <1170331342.604220.89500(a)m58g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 30 Jan., 17:43, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > On Jan 30, 8:42 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > > You can try to utter again and agan this nonsense, but after a while > > > aI will cease to reply. Every such thing including only natural > > > numbers is covered by induction. All natural numbers are subject to > > > induction. > > > > True, you can use induction to prove something about > > any natural number. However, the question is: "Can > > you use induction to prove something about a set > > of natural numbers?" > > This question has a trivial answer: > If the set of all natural numbers is nothing than all natural numbers, > then yes. > If the set of all natural numbers is more than all natural numbers, > then no. it is more in the sense of being a entity called a set as well as being just a bunch of individuals scattered over the landscape of one's imagination. > > We can prove that every number is a number while a set of several > numbers is not a number. There is no reason to distinguish between > finite and infinite sets. Some sets of several numbers are themselves numbers according to some defnitions of "number". There is as much reason to distinguish between finite sets and infinite sets as between empty sets and non-empty sets, since in either case they are distinguishable. > Your strong belief in the inaccesible infinite uttered above is your > (and some other people's) personal opinion but has nothing to do with > mathematics. Actually, it is WM's opinions that have nothing to do with mathematics.
From: Virgil on 1 Feb 2007 14:25
In article <1170331631.022417.120260(a)h3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 30 Jan., 20:25, Virgil <vir...(a)comcast.net> wrote: > > In article <1170156979.321128.184...(a)j27g2000cwj.googlegroups.com>, > > > > I can't tell whether the union of all lengths is finite or not. > > > > As the union of all lengths is obviously equivalent to the union of all > > finite ordinals and the union of all finite ordinals is clearly not > > finite, WM can't tell much. > > The union of all lengths is not the sum of all lengths, which, > according to your standpoint might be infinite. But the union of all > lengths is a length. If a length must be finite then the union of all lengths is no more a length than the union of all naturals (as bounded initial sets of naturals) it itself a natural. > You may say that it is infinite or not. If it is > infinite, then it corresponds to an infinite natural number. A infinite number of some sort, perhaps, but not a natural. If each natural is the finite set set of all smaller naturals, the union of all of them is a set of all smaller naturals but not being finite it not itself a natural. This distinction seems to be WM's pons asinorum. |