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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 3:29 PM

<pepstein5@gmail.com> writes
>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.

It is crucial that X_sigma0 and X_0 be sunsets of B. Then we know
X_sigma0^rho0 has an even index, 2pi, and that b_(2pi+1) will not be in
X_0 or X_sigma0. That's the whole reason for using B

There's nothing in the existing specification to stop you choosing X_0
and X_1 with all odd indices (unless r is even of course). That shows
immediately that using the B(t) as stated can't work.

Starting with b_0 is not fine. If the smallest member of X_1 is b_0 and
the smallest of X_0 is larger then there is no way to construct X_2.
Starting B(t) at b_2t is one way around that. Another would be to
specify that rho_0 is the smallest index at which the x_i and x'_i
differ.

You are quite right about not having enough room to do the construction.
All the elements of X_1 could lie between two successive elements of
X_0. Then all of X_2 would have to be between the corresponding elements
of X_1 and there may be only (r-1) available.

>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'...).

There is a sense in which that applies here. If X_0 and X_1 are
interleaved in that way, then we can jump straight to the last part of
the argument. The point of the X_i sequence is to get us to the position
where we can change the rho0-indexed element of X_sigma0 to the
one-higher indexed b without disturbing the relative orderings with X_0.
If X_0 and X_1 are disjoint then that can't happen anyway and we can
just put sigma0 = 1 and finish.

--
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