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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: Another not-completely-insignificant gap in the Rado paper
Posted: Nov 7, 2013 9:38 AM
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On Thursday, November 7, 2013 2:05:55 PM UTC, David Hartley wrote:
> In message <b829deeb-b331-4460-a723-5d76a8ac54d0@googlegroups.com>, Paul
>
> <pepstein5@gmail.com> writes
>

> >So far, even when I appeal to large xi terms, I don't see enough space
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> >between the elements to be sure of obtaining the relationships of the
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> >form [X0, X1] = [X1, X2] etc.
>
> >
>
> I've only just started to look at the rest of the proof, but here's my
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> first thoughts.
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>
>
> At each successive stage we're given a larger set to draw the elements
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> of the next X_i from. B(r^(s-1)) has (r-1) elements between each member
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> of B(r^s). That should be enough. (There could be a problem if the least
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> element of X1 is b_0, but that can be avoided by choosing X_1
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> carefully.)
>
>
>
>
>

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

> >There does seem to be some small problem either with the paper, or my
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> >understanding of the paper, because I see no place in the paper where
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> >he uses the fact that he has removed the odd index elements from B'.
>
> >
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> >Perhaps he redefined the b_i elements so that the i index now refers to
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> >their position in B rather than in B' but he definitely needs to tell
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> >the reader that he is doing this.
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>
>
> The definition of pi assumes that X_sigma0^rho0 has an even index, so it
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> seems he's still using indexing in B' but assuming the X_i are all
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> within B.
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>
>
> This is where he uses the fact that B =/= B'. b_(2pi+1) is not in B
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> allows you to change X_sigma0 to Z_sigma0 as it can't already be another
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> element of X_sigma0. But it has the same order-relationship as B_2pi
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> with all other elements of B and so does not change the
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> order-relationship of X_sigma0 with other X_i. In particular
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> (X_0,X_sigma0) = (X_0,Z_sigma0)
>
>
>

> >I see that you need to remove the odd elements because I don't get
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> >enough spacing, but the construction still doesn't seem coherent
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> >because the [X0, X1] steps simply don't work if you follow the
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> >definitions literally.
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>
>
> So it seems it all works if you just change B(t) to {b_2t, b_4t,...}
>
>

David,

Thanks for your contributions. I don't see things quite the same way. I think that, as before, the author meant exactly what he said. All indexing is fine and beginning at b_0 is fine. You may need sufficient largeness of the x terms where largeness is defined in relation to the indexing of the b terms. I disagree with you that we have enough B(r^(s-1)) elements between the members of B(r ^ s). We sometimes (unless we argue further) want r instead of r-1.

The philosophy is that, whenever you find that the construction poses a problem, you appeal to the same type of argument that I (with Blass's help) found nearer the beginning of the thread.

Whenever we see a problem in the construction, we move around it by using the Blass-Epstein argument to say that, because we're working inside B', the problem (not being able to find enough B(r^(s-1) elements) is inconsistent with the assumed inequality f(x...) =/= f(x'...).

Thanks again,

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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