Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Surprise at my failure to resolve an issue in an elementary paper by Rado
Replies: 44   Last Post: Nov 10, 2013 12:23 PM

 Messages: [ Previous | Next ]
 Paul Posts: 780 Registered: 7/12/10
Re: Surprise at my failure to resolve an issue in an elementary paper

Posted: Nov 4, 2013 10:11 AM

On Monday, November 4, 2013 2:43:37 PM UTC, David Hartley wrote:
>
> <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
>
>
> >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
>
> >counter-example 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 counter-example, 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 counter-example 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

Date Subject Author
11/3/13 Paul
11/3/13 David Hartley
11/3/13 fom
11/3/13 fom
11/3/13 fom
11/4/13 fom
11/4/13 Paul
11/4/13 Paul
11/4/13 Peter Percival
11/4/13 David Hartley
11/4/13 Paul
11/4/13 David Hartley
11/4/13 Paul
11/4/13 David Hartley
11/4/13 Paul
11/5/13 Paul
11/5/13 David Hartley
11/5/13 Paul
11/5/13 David Hartley
11/5/13 Paul
11/6/13 Paul
11/6/13 Paul
11/7/13 Paul
11/7/13 David Hartley
11/7/13 Paul
11/7/13 David Hartley
11/7/13 Paul
11/7/13 David Hartley
11/7/13 David Hartley
11/7/13 Paul
11/7/13 David Hartley
11/8/13 Paul
11/8/13 David Hartley
11/7/13 Paul
11/7/13 fom
11/8/13 Paul
11/8/13 David Hartley
11/10/13 Paul
11/10/13 David Hartley
11/10/13 Paul
11/10/13 David Hartley
11/10/13 David Hartley
11/10/13 Paul
11/4/13 Paul
11/4/13 Peter Percival