Date: Nov 5, 2013 7:51 PM
Author: Paul
Subject: Re: Surprise at my failure to resolve an issue in an elementary paper<br> by Rado

On Wednesday, November 6, 2013 12:10:46 AM UTC, David Hartley wrote:
> In message <86eb8927-154a-4ad2-906f-759e79de59d9@googlegroups.com>, Paul
>
> <pepstein5@gmail.com> writes
>

> >Of course, the below is irrelevant to understanding the proof. However,
>
> >I am completely mystified by the page 1 sentences: "We now show that f
>
> >is L-canonical. We shall apply the definition of f repeatedly without
>
> >referring to this fact." It's only the second of those sentences that
>
> >confuses me. The first sentence is given for context.
>
>
>
> I can't make sense of it either. The whole section is rather odd.
>
> Firstly he hasn't actually defined L-canonical, only L-canonical on B,
>
> presumably he means here L-canonical on A. The actual theorem is trivial
>
> yet he devotes several lines to a proof. I assume Bulletin articles are
>
> supposed to be short, so it would have been much better to use that
>
> space for a more detailed proof of the important theorem. In particular
>
> explaining the step that had us both confused.
>
>
>
> The idea is lovely, the presentation is not.
>


Agreed totally with your post. Yes, I think he means L-canonical on A. Yes, if he is introducing the concept of L-canonical on a set S, he shouldn't suddenly abbreviate this to "L-canonical" without saying what he means. Yes, he's devoting lots of space to a triviality.

My mathematical background is well below Ph.D level so it's hard for me to judge what would be obvious to the intended readership. However, I can't help feeling that defining a set via a universal quantifier and then saying something that, from a pure logic standpoint, only follows immediately if the set was defined via an existential quantifier would confuse almost anyone. However, I think that we (both of us) did show considerable naivety in expecting that the definition of L was simply wrong -- that would be highly unlikely in retrospect. There's a bit of wishful thinking there. We probably hoped it was wrong since that would have let us comfortably plough on.

I actually think there is an error in the proof -- an error not just a presentational flaw. However, if so, it's very easily fixed. Referring to (b) on page 2. Let x0' = 0. And let all the x terms be larger than all the x' terms. Then we fail to find the required [X0, X1] = [X1, X2] relationship. The idea is that B(r^s) terms are much larger than the corresponding B(r^(s-1)) terms. However, the definition of B(t) seems to be wrong to make the above idea work, if we take the paper literally. B(t) should be{b_t, b_2t...} I think the construction fails if B(t) contains b0.

Perhaps I shouldn't judge prematurely because I'm not an expert and I haven't even reached the end of the paper yet, but it does seem to have been poorly edited and refereed.

As always, thanks for reading the paper and offering your thoughts.

Paul Epstein