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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: 474
Registered: 7/12/10
Re: Another not-completely-insignificant gap in the Rado paper
Posted: Nov 7, 2013 12:36 PM
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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
>
> >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,...}
>
>


I think my previous posting is wrong. This is my problem and fix. Problem: There's a problem if all of the x' terms in the original f(x) = f(x') equality fit strictly between two of the x terms. That's the case where we don't necessarily get enough B(r^(s-1)) elements.

Fix: Simply order the x and x' terms to prevent the above happening. If all the x' terms fit between two x terms, then it is impossible for the converse to happen.

Is this correct? Or were you right about there always being enough elements.
Many thanks for your help.

Paul Epstein


Date Subject Author
11/3/13
Read Surprise at my failure to resolve an issue in an elementary paper by Rado
Paul
11/3/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper by Rado
David Hartley
11/3/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper
by Rado
fom
11/3/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper
by Rado
fom
11/3/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper
by Rado
fom
11/4/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper
by Rado
fom
11/4/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper
by Rado
Paul
11/4/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper
by Rado
Paul
11/4/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper
by Rado
Peter Percival
11/4/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper by Rado
David Hartley
11/4/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper
by Rado
Paul
11/4/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper by Rado
David Hartley
11/4/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper
by Rado
Paul
11/4/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper by Rado
David Hartley
11/4/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper
by Rado
Paul
11/5/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper
by Rado
Paul
11/5/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper by Rado
David Hartley
11/5/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper
by Rado
Paul
11/5/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper by Rado
David Hartley
11/5/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper
by Rado
Paul
11/6/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper
by Rado
Paul
11/6/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper
by Rado
Paul
11/7/13
Read Re: Another not-completely-insignificant gap in the Rado paper
Paul
11/7/13
Read Re: Another not-completely-insignificant gap in the Rado paper
David Hartley
11/7/13
Read Re: Another not-completely-insignificant gap in the Rado paper
Paul
11/7/13
Read Re: Another not-completely-insignificant gap in the Rado paper
David Hartley
11/7/13
Read Re: Another not-completely-insignificant gap in the Rado paper
Paul
11/7/13
Read Re: Another not-completely-insignificant gap in the Rado paper
David Hartley
11/7/13
Read Re: Another not-completely-insignificant gap in the Rado paper
David Hartley
11/7/13
Read Re: Another not-completely-insignificant gap in the Rado paper
Paul
11/7/13
Read Re: Another not-completely-insignificant gap in the Rado paper
David Hartley
11/8/13
Read Re: The Rado paper -- a possible further simplification
Paul
11/8/13
Read Re: The Rado paper -- a possible further simplification
David Hartley
11/7/13
Read Re: Another not-completely-insignificant gap in the Rado paper
Paul
11/7/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper
by Rado
fom
11/8/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper
by Rado
Paul
11/8/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper by Rado
David Hartley
11/10/13
Read Re: Possible major blunder in Rado's version of Canonical Ramsey
Theorem that goes far beyond omitting proof steps
Paul
11/10/13
Read Re: Possible major blunder in Rado's version of Canonical Ramsey Theorem that goes far beyond omitting proof steps
David Hartley
11/10/13
Read Re: Possible major blunder in Rado's version of Canonical Ramsey
Theorem that goes far beyond omitting proof steps
Paul
11/10/13
Read Re: Possible major blunder in Rado's version of Canonical Ramsey Theorem that goes far beyond omitting proof steps
David Hartley
11/10/13
Read Re: Possible major blunder in Rado's version of Canonical Ramsey Theorem that goes far beyond omitting proof steps
David Hartley
11/10/13
Read Re: Possible major blunder in Rado's version of Canonical Ramsey
Theorem that goes far beyond omitting proof steps
Paul
11/4/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper
by Rado
Paul
11/4/13
Read Re: Surprise at my failure to resolve an issue in an elementary paper
by Rado
Peter Percival

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