Cantor Confusion
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From: Virgil on 9 Nov 2006 16:14 In article <1163074411.726659.53400(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > About a couple of weeks ago you presented an argument about trees, and > > as you presented your argument, as you described it, it was clearly > > reasonable to regard you as intending that as a proof in set theory (it > > would have been UN reasonable to think the contrary). But later you > > switched, so that your argument was not to be taken as in set theory. > > Why don't you try to find out how it could be presented in set theory? > Or why that cannot be done? We have already seen why it cannot be done in ZF or NBG. It contradicts the requirements of those systems. > > > > Now, when I say that I am giving a proof in set theory, and after I > > discuss that with you while having reminded you of that so many times, > > you switch not your OWN argument this time, but you switch to make the > > terms of MY argument subject to criticism for USING AN AXIOM OF SET > > THEORY! > > You misunderstood. See below. > > > > I'm really very curious what satisfaction you get from such games you > > play. > > > I hope that there are some lurkers who will learn what are the > consequences of accepting ZFC. > > > There can't be any intellectual satisfaction in such mindless games. So > > what is it get from them? > > I do not prohibit to use the axiom of infinity, but I show that the > consequences of its use are absurd. Not so. it requires specific assumptions which contradict the axioms of, say ZF, in order to produce those alleged absurdities. So the aburdities are in requiring the simultaneous assumption of conflicting statements. > It leads to such results as a > diagonal which is longer than any line of a matrix or a vase which > contains zero numbers at noon or Tristram Shandy getting ready with his > diary. None of which are contradictory in ZF alone without adding other assumptions which produce those alleged contradictions. > In the binary tree it leads the result that there are not less > edges than paths. Actually, in infinite binary trees, where no path terminates, ZF leads to there being LESS edges than paths. So WM has things backwards, as usual. > In my opinion that is enough to prove ZFC being > wrong. If you prefer to remain within this wrong system, good luck. We do not need luck nearly as much as WM needs it.
From: Virgil on 9 Nov 2006 16:24 In article <1163094691.248801.308560(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > Every part of the discussion is crystal clear except for WM's > > maunderings which seem more and more to be deliberately obfuscating. > > > > If every line is finite, say, for example, the nth line is of length > > 2*n, and the diagonal is not finite , then the diagonal MUST be longer > > that every line. > > By definition the diagonal of a matrix cannot be longer than every line > of the matrix. Not by any definition I recognize. > Considering your reaction to the vase, I guess you would even accept a > circle the diameter of which is longer than its circumference. I would accept that in WM's notion of geometry things may not be as Euclid, et al, visualized them, but I would never put my faith in any of WM's mathematics without considerable outside confirmation of each of his claims. > Please > let us stop here. I see no point in explaining to you any logical > truth. As so far WM has presented few logical truths, and none that I was not already aware of, his attempts to teach his grandmother to suck eggs is wasted. >It is useless to continue this discussion. WM dictating as if ex cathedra about things he knows little about does not constitute discussion.
From: Virgil on 9 Nov 2006 16:26 In article <1163094916.120056.271840(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > That's a wrong definition of the word "definition", at least in > > > > Mathematics. > > > > > > As you believe? > > > > As mathematicians believe. A definition, at least in the mathematical > > sense, is merely a form of abbreviation. It allows one to say more > > briefly what could be said without it but at greater length. > > Correct. An expression, a long one or a short one, is explained in > other words, more briefly or more detailled, respectively. > > > > > > > > > Would you understand: "The new number omega is the limit to which the > > > > > (natural) numbers n grow"? Or is even hat too incomprehensible to you, > > > > > because you don't know the meanings of "number", "limit", and "grow"? > > > > Which natural number(s) does WM suggest grows? > > I do not suggest anything here. This is a sentence by Cantor. If Wm does not of the same opinion, why does he post it at all?
From: Virgil on 9 Nov 2006 16:32 In article <1163095116.768859.225620(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > What I claim is that there is a line for every MEMBER of N (or omega). > > and if each line numbered n has at least n entries, there is a column > > for each member of N ( or omega). > > I do not say that the "number" of either is "equal" to omega, as I do > > not know what that means unless it means no more than what I said above > > that there is a member for each member of omega. > > But you think that there are *all* members of omega. There is in ZF a set of /all/ natural numbers, or of /all/ finite ordinals > And you think that > their number is omega. I call the set omega, but I am not at all sure what WM means by /number/. > > > > > It contradicts the square matrix which is required for Cantor's > > > diagonal argument. > > > > I do not recall anything in the Cantor proof about square matrices. > > > > The only requirement in the Cantor "diagonal" proof is that the nth > > number listed have a determinable nth decimal digit. > > > > And that does not require any sort of "square matrix". > > This condition describes a square matrix (if actual infinity is a > number). For every number there is a corresponding digit and for every > digit there is a corresponding number. And there is a numbers which > counts all the numbers and digits. WM again uses "number" where we use "set". We have a SET which contains all of the /finite/ naturals or all of the /finite/ ordinals.
From: Virgil on 9 Nov 2006 16:33
In article <1163095442.484508.128000(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > I say: The diagonal cannot be longer than every line. You say a lot of things that sensible people reject out of hand. |