
Re: Peerreviewed arguments against Cantor Diagonalization
Posted:
Oct 28, 2012 8:41 PM


On Sunday, October 28, 2012 7:26:25 PM UTC5, JRStern wrote: > On Sun, 28 Oct 2012 15:25:27 0700 (PDT), Arturo Magidin wrote: > > > > >On Saturday, October 27, 2012 4:49:32 PM UTC5, JRStern wrote: > > >> Are there any such published? > > > > > >You said elsewhere you are interested in "peerreviewed" criticisms to > > >Cantor's diagonal argument, but not from the point of view of intuitionism > > >or some other logical framework, but strictly within the context of ZFC. > > > > > >There are no such things, because the argument is a valid argument > > >within ZF. It is in fact pretty short and clear. > > > > Then there can be a book taking on the objections one at a time and > > knocking them down.
Who would waste his or her valuable time doing so, and what reputable, peerreviewing scientific publishing house would waste its resources sending it through the review process, let alone publishing it?
In the words of Augustus de Morgan:
"When a paradoxer parades capital letters and diagrams which are as good as Newton's to all who know nothing about it, some persons wonder why science does not rise and triturate the whole thing. This is why: all who are fit to read the refutation are satisfied already, and can, if they please, detect the paradoxer for themselves. Those who are not fit to do this would not know the difference between the true answer and the new capitals and diagrams on which the delighted paradoxer would declare that he had crumbled the philosophers, and not they him..."  "A Budget of Paradoxes", vol 2, page 354
The closest thing you have are books like De Morgan's (or Underwood Dudley's "The Paradoxers"); these are not "peerreviewed" in any reasonable sense of the word, and you can do a google search to see exactly what Dudley got for his efforts (he got sued by a crank; though he won the case, and was supported by the publisher, he could just as easily have been left to fend for himself by the latter)
> >Recall that given a set X, the Axiom of the Power Set states that > > >there is a set Y such that z in Y if and only if z is a subset of X. > > >We call this set the "power set of X", an denote it P(X). > > > > > >THEOREM (Cantor) Let X be any set, and let P(X) be the power set of > > >X. If f:X>P(X) is any function, then f is not onto; that is, there > > >exists B in P(X) that does not lie in the image of f. > > > > > >Proof. Let f:X>P(X) be a function. By the Axiom of Separation, > > > > > >B = {x in X  x is not an element of f(X)} > > > > > >is a subset of A, hence an element of P(X). We claim that f(y)=/=B for all y\in X. > > > > > >Indeed, let y in X. Either f(y)=/=B, or f(y)=B. If f(y)=B, > > >then y in B > y in f(y) > y not in B; since (P>not(P))>not(P) is a tautology, > > >we conclude that f(y)=/=B. So if y in X, then f(y)=/=B, proving that B is not > > >in the image of f. QED > > > > ... and not just telling us that there are other theories with the > > same consequent.
Huh?
Sorry, but that statement makes absolute no sense whatsoever to me. What exactly was the point you were attempting to make?
Given that there is a straighforward, direct proof of Cantor's Theorem that holds in ZF, why would any reputable publishing house waste *anyone's* time peerreviewing the arguments presented against? Not only is it often hard to understand what a crank is going on about, but it is also a rather thankless task. One may do it on their free time (as here), but there is absolutely no point in doing it formally. Like De Morgan says: those who know better and can understand the debunking do not need it, being able to provide it for themselves at will; those who do not know better cannot tell the difference between the crank and the debunker, in any case.
 Arturo Magidin

