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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

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 Paul Posts: 780 Registered: 7/12/10
Re: Surprise at my failure to resolve an issue in an elementary paper

Posted: Nov 5, 2013 7:51 PM

On Wednesday, November 6, 2013 12:10:46 AM UTC, David Hartley wrote:
>
> <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.

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