On 30 Jan., 21:17, david petry <david_lawrence_pe...@yahoo.com> wrote: > On Wednesday, January 30, 2013 3:07:07 AM UTC-8, quasi wrote: > > david petry wrote: > > >Doron Zeilberger wrote the following in an opinion piece on his >website: > > >"Read Wolfgang Mueckenheim's fascinating book ! I especially > > >like the bottom of page 112 and the top of page 113, that prove, > > >once and for all, that (at least) the actual infinity is pure > > >nonsense." > > Proves once and for all? > > By that wording, Zeilberger appears to be affirming the validity > > of Mueckheim's claimed "proof" of some theorem or other which > > supposedly yields the conclusion that "infinity is pure nonsense". > > >http://www.math.rutgers.edu/~zeilberg/Opinion68.html > > Of course, in the same blog, a few months later, Zeilberger > > awkwardly tries to retract the claim of "proof", asserting > > that his earlier post was only intended as an offering of > > philosophical support. > > However, in my opinion, that's a blatant copout. > > Worse, I regard Zeilberger's attempted "clarification" as > > deceitful, as evidenced by his posted statement: > > "I have no expertise, or interest, in checking any possible > > technical claims that he [Muckemheim] may have made." > > Insufficient expertise? A straight-out lie, in my opinion. > > [...] > > I'm going to defend Zeilberger here, because I would be inclined to say exactly the same thing he has said: I have no expertise, or interest, in checking any possible technical claims that he [Muckemheim] may have made. > > Any "proof" that infinity is pure nonsense must be based on an appeal to common sense.
I have developed a new proof-technique, namely proof by ignorance, that is not based upon common sense. And I can say that I have good results among intelligent non-mathematicians as well as among mathematicians who have not a fundamental interest in set theory.
It was necessary to develop this technique since classical proofs are condemned to fail in a community of determined Cantor-believers. Compare, for instance the arguments by Löwenheim-Skolem or Banach- Tarski or even König who was, to my knowledge, the first to recognize that only a countable manifold of numbers can have finite definitions. Although Cantor did not believe in infinite definitions, modern mathematicians don't see a problem in reals that are undefinable (imagine: a number that is not a number and cannot appear, as an individual, in any branch of mathematics including any Cantor-list) or can only have infinite "definitions".
My proof shows the believer that he cannot do what he believes. The believer of uncountability believes that, by applying infinite sequences of bits, all real numbers can be distinguished. Now I construct the complete infinite Binary Tree by means of countably many paths, such that every node and every possible combination of nodes is covered by at least one path (in fact by infinitely many). But I don't tell what paths I have used. If it was possible to distinguish uncountably many paths purely by the nodes, then it could not be a problem to find further paths, because all must differ somehow from each other. But, of course, nobody can find such a path without knowing my choice. This shows that there are further pieces of information required to distinguish the paths. But as these pieces of information are necessarily finite words (otherwise they could not be communicated) their number is countable. So they cannot be used to distinguish uncountably many paths.
By this proof by ignorance the usual belief in the presence of uncountably many infinitely distinguishable paths in the Binary Tree or infinite sequences of digits is contradicted. Of course it will last some time until convinced set theorists will have realized the power of this method.