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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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 David Hartley Posts: 463 Registered: 12/13/04
Re: Another not-completely-insignificant gap in the Rado paper
Posted: Nov 7, 2013 9:05 AM

<pepstein5@gmail.com> writes
>So far, even when I appeal to large xi terms, I don't see enough space
>between the elements to be sure of obtaining the relationships of the
>form [X0, X1] = [X1, X2] etc.
>

I've only just started to look at the rest of the proof, but here's my
first thoughts.

At each successive stage we're given a larger set to draw the elements
of the next X_i from. B(r^(s-1)) has (r-1) elements between each member
of B(r^s). That should be enough. (There could be a problem if the least
element of X1 is b_0, but that can be avoided by choosing X_1
carefully.)

>However, I think I see the issue. As written, I don't see where he
>uses the fact that B is a proper subset of B'.
>
>Therefore, perhaps the definition of B(t) is an error? Perhaps the
>element at index j in the sequence B(t) is intended to mean the term at
>index j in the C sequence where C refers to the sequence: b0, b2, b4,
>b6....

... but I think you may be right there. As specified B(t) is not a
subset of B for odd t.

>There does seem to be some small problem either with the paper, or my
>understanding of the paper, because I see no place in the paper where
>he uses the fact that he has removed the odd index elements from B'.
>
>Perhaps he redefined the b_i elements so that the i index now refers to
>their position in B rather than in B' but he definitely needs to tell
>the reader that he is doing this.

The definition of pi assumes that X_sigma0^rho0 has an even index, so it
seems he's still using indexing in B' but assuming the X_i are all
within B.

This is where he uses the fact that B =/= B'. b_(2pi+1) is not in B
allows you to change X_sigma0 to Z_sigma0 as it can't already be another
element of X_sigma0. But it has the same order-relationship as B_2pi
with all other elements of B and so does not change the
order-relationship of X_sigma0 with other X_i. In particular
(X_0,X_sigma0) = (X_0,Z_sigma0)

>I see that you need to remove the odd elements because I don't get
>enough spacing, but the construction still doesn't seem coherent
>because the [X0, X1] steps simply don't work if you follow the
>definitions literally.

So it seems it all works if you just change B(t) to {b_2t, b_4t,...}

--
David Hartley

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