Cantor Confusion
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From: Virgil on 20 Nov 2006 14:51 In article <1164029891.658767.153760(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > You do no longer persue your plan to prove a contradiction in modern set > > theory? > > Perhaps I will find another proof, but I think those delivered are > sufficient. Sufficient for what? Since all WM's arguments require assumptions outside of the axioms of ZF and which themselves contradict ZF, the contradictions are not within ZF, but between WM's system and the ZF system. > > > > > Doesn't this case make you wonder whether there are other things which > > > you do not yet know but which you could learn from me? > > > > > >> My opinion is > > >> > > >> A (n e omega & |{0, 1, 2, ..., n}| < |omega|) > > > > > > Seems an empty opinion. What is the symbol A refers to? > > > > A n (n e omega & |{0, 1, 2, ..., n}| < |omega|) > > That is correct. All (initial) segments of natural numbers (which > consist only of natural numbers!) have a finite number of members. Only those "initial segments" which consist of all the naturals preceding some fixed natural are finite. Subsets of the naturals which cannot be contained within any such a bounded initial segment, and they exist, are not finite. > > > > What I say is: The cardinality of the sequence (d_nn) is the cardinality > > of omega. > > And the ordinality of (d_nn) is omega too, like the ordinality of the > first column. Therefore it is clear that all segments of natural > numbers (which all are subsets of the numbers contained in the lines of > the EIT) are finite. All bounded initial segments, yes, but an unbounded segment exists.
From: stephen on 20 Nov 2006 14:54 David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Lester Zick wrote: <snip> > That's impressive. Either you are trolling or you have completely > misunderstood what people mean when they say "definitions are > abbreviations". Lester misunderstands pretty much anything anyone says. I suppose that is what happens when someone perversely uses their own personal vocabulary for too long. Stephen
From: Virgil on 20 Nov 2006 14:59 In article <1164030524.946698.35710(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > The diagonal consists of line indexes, i.e., of the line ends. > > > Therefore it is a subset of the line indexes. > > > > > > > Yes, the diagonal is the set of line indexes (a subset, but > > not a proper subset). > > Every line is a subset of the diagonal. Only if all the characters in all the lines are the same will that be necessary, but then the diagonal is like the set of all naturals, an infinite set of finite members. > And the diagonal is a subset of > the line ends. > > So you must have something more in mind when > > you claim that the fact that every element of the diagonal > > is a line index means that the diagonal cannot > > be longer than every line. What is this? > > Of course there are more conditions. They follow from the EIT. > Therefore I chose it. > 1 > 12 > 123 > ... > > Every line has only different indexes (there are not two equal indexes > in one line). > Line n has all indexes from 1 to n. > Every initial segment of line ends is a subset of a line. > Every initial segment of line ends is a subset of the diagonal. > Every initial segment of the diagonal is a line. Every initial segment is finite, but the diagonal is not. WM has not found any internal contradictions within ZF, what he finds is contradictions between his own assumptions and those of ZF.
From: imaginatorium on 20 Nov 2006 15:00 Lester Zick wrote: > On Mon, 20 Nov 2006 01:39:59 -0500, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > > >imaginatorium(a)despammed.com wrote: > >> Lester Zick wrote: > >> > On Sat, 18 Nov 2006 18:31:05 +0000 (UTC), stephen(a)nomail.com wrote: > >> > >David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >> > >> Lester Zick wrote: > >> > >>> On Thu, 16 Nov 2006 02:02:49 -0500, David Marcus > >> > >>> <DavidMarcus(a)alumdotmit.edu> wrote: > > > >> > >>> >Please give a specific example of something that you think is absurd or > >> > >>> >a contradiction. I don't know what you mean by "containment of sets and > >> > >>> >subsets". > >> > >>> > >> > >>> Well as I recollect Stephen seems to think infinite sets are proper > >> > >>> subsets of themselves. > >> > > > >> > >> Are you sure that is what Stephen thinks? > >> > > > >> > >I see Lester has resorted to lying. > >> > >> That was Stephen... I must say I've always been puzzled by accusations > >> of "lying" in this group. In particular, how can a claim to "seem to > >> recollect.." be lying? > > > >Depends on whether the person's memory is really hazy or they are being > >disingenuous. > > I seem to recollect being disingenuous on occasion. > > >> Of course, one may be annoyed by people who > >> jumble up what one says, but... > > > >Or, people who don't listen. Or, people who don't learn. > > Or people *** ***'* *****. Or people *** ***'* ***** *** ***** **** * > ****** **** **** ** * *****. Or ****** *** **** ** ********** ** ***** > *** **** ******* ***** *** ** *********** ******. Or ****** > *** *** ******* **** demonstrations of truth. Or ****** *** *** > ********** ** ********** ******* ****** *** **** ** ****** ** ****** > *** *** ***** ** *** ****** ******** ***** *** *** *** **** ** ****** > ** **** ***** ** ******. > > (Technically *** ******* ** Brian ****** ****** *** **** ** * ******* > dizinzzzzzzz yyyyyyyyyyyy zzzzz.) You spoke, Lester? Very good. Very good indeed. Actually somewhat more intellectually challenging than your usual level. Very even-handed, neigh, calm. > ~v~~ Arf arf! Brian Chandler http://imaginatorium.org
From: Virgil on 20 Nov 2006 15:08
In article <1164031126.414946.246070(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Franziska Neugebauer schrieb: > > >> Virgil wrote: > > >> > > >> > In article <1163856355.707119.306700(a)m7g2000cwm.googlegroups.com>, > > >> > mueckenh(a)rz.fh-augsburg.de wrote: > > >> > > > >> >> Virgil schrieb: > > >> [...] > > >> >> And there are lines as long as the diagonal is. > > >> > > > >> > Name one. > > > > > > The elements of the diagonal are a subset of the line ends. > > > > Though Virgil posed this question: Name one single _line_ which is as > > long as the diagonal ist. > > Name one single element of the diagonal which is not contained in a > line (which contains this and all preceding elements). > > This is what I call the one-eyedness of set theory: Its proponents see > that for every line, there is a diagonal element not contained in this > and all preceding lines. But they don't see, or at least dispel it, > that there is no element of the diagonal which is outside of any line. Every element except the first is "outside" at least one line. While it is true that there is no diagonal element that is "outside" of EVERY line, that is quite a different issue. > > The first observation leads to the theorem: The diagonal is superset of > all lines. The second observation (together with the fact that every > line is a superset of all preceding lines) leads to the theorem: There > is at least one line which is superset of the diagonal. Not in ZF. If L were such a line, then it would have to have a last element (as every line has a last element), but every line has a successor line with one more element, and that one more element is now in the diagonal but not in the line alleged to be a "superset of the diagonal". In ZF, statements with counterexamples are not theorems. |