Paul
Posts:
624
Registered:
7/12/10


Re: Surprise at my failure to resolve an issue in an elementary paper by Rado
Posted:
Nov 4, 2013 10:11 AM


On Monday, November 4, 2013 2:43:37 PM UTC, David Hartley wrote: > In message <8933482d892046ce88866402fce0cea2@googlegroups.com>, Paul > > <pepstein5@gmail.com> writes > > >Agreed. The redefinition of L leads to problems soon after (3) of page > > >3. Furthermore, assuming an error in a paper should be a last resort. > > >An author is far more likely to omit steps of reasoning than to make an > > >elementary logical error. My initial post merely shows that the > > >equality I'm complaining about doesn't follow _immediately_. However, > > >just because an assertion doesn't follow immediately doesn't mean that > > >it doesn't follow. To show that it doesn't follow, we would need a > > >counterexample and I haven't seen one. > > > > > > To avoid all the tedious notation, let's define > > > > D(x,y,i) to mean x and y are subsets of B with r elements which, when > > ordered by <, differ only at the position indexed by i. > > > > The original definition of L has i in L iff > > > > for all x,y D(x,y,i) > f(x) =/= f(y) > > > > > > The problematic step in the proof assumes that if i is *not* in L then > > > > for all x,y D(x,y,i) > f(x) = f(y) > > > > > > Suppose for a counterexample, that B = N and put r = 2 and > > > > f(x) = 2^(1 + min x) if max x is even > > = 3^(1 + min x) if max x is odd > > > > Then L = {0} with the original definition but is empty with the > > suggested variation. > > > > It may be that with the specific B defined in the paper the two > > properties are equivalent but it certainly requires proof. > >
Many thanks for this interesting example. However, your acknowledgement that you haven't used the specific B defined in the paper is critically important.
I think that our understanding of the issues remains at the same place that it was when I made my last post. We understand that the equality at issue doesn't follow _immediately_. But we don't know whether it follows or not.
In order to get a counterexample to show that the equality at issue doesn't follow, we need the equality statement to be false, and we also need to use a set B which is consistent with the author's construction.
I still prefer to work with the author's definition. For reasons given earlier, I'm very sceptical of the idea that the author's definition of L is different from what he intends.
[Please please don't take the above to mean that I don't appreciate your work. I'd love to resolve this and I thank you heartily for working with me on this.]
Paul Epstein

