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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
by Rado

Posted: Nov 6, 2013 3:32 AM
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On Wednesday, November 6, 2013 12:51:36 AM UTC, Paul wrote:
> On Wednesday, November 6, 2013 12:10:46 AM UTC, David Hartley wrote:
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> > In message <86eb8927-154a-4ad2-906f-759e79de59d9@googlegroups.com>, Paul
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> > <pepstein5@gmail.com> writes
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> > >Of course, the below is irrelevant to understanding the proof. However,
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> > >I am completely mystified by the page 1 sentences: "We now show that f
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> > >is L-canonical. We shall apply the definition of f repeatedly without
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> > >referring to this fact." It's only the second of those sentences that
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> > >confuses me. The first sentence is given for context.
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> > I can't make sense of it either. The whole section is rather odd.
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> > Firstly he hasn't actually defined L-canonical, only L-canonical on B,
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> > presumably he means here L-canonical on A. The actual theorem is trivial
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> > yet he devotes several lines to a proof. I assume Bulletin articles are
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> > supposed to be short, so it would have been much better to use that
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> > space for a more detailed proof of the important theorem. In particular
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> > explaining the step that had us both confused.
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> > The idea is lovely, the presentation is not.
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> 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.
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> 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.
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> 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.
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> 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.
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> As always, thanks for reading the paper and offering your thoughts.
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I think there's a general principle which I've been continually flouting that operates both here and in other contexts. I wonder whether there's an established name for the principle (?). Principle: If it appears that someone else has made a mistake and there are a number of alternatives for what the mistake was, always assume the least serious mistake possible. It's a particularly good principle for a manager to apply to an employee. Avoid unnecessary accusations by making the most generous assumption possible.

Here, I don't think simply removing b0 allows the [X0, X1], [X1, X2] construction without further modification. However, I think that the way to modify it is to regard r and s as fixed and to choose all x and x' terms to be as large as possible (using colloquial language).

This approach is more consistent with my principle. It's a far less serious mistake for an author to omit the assumption that terms are sufficiently large(an experienced reader could be expected to insert that anyway just as we sometimes assume a function is continuous when the author doesn't explicitly say so) than to erroneously put an incorrect element into a set.

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

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