Paul
Posts:
474
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


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: > > > In message <86eb8927154a4ad2906f759e79de59d9@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 Lcanonical. 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 Lcanonical, only Lcanonical on B, > > > > > > presumably he means here Lcanonical 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 Lcanonical on A. Yes, if he is introducing the concept of Lcanonical on a set S, he shouldn't suddenly abbreviate this to "Lcanonical" 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^(s1)) 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. >
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

