Paul
Posts:
764
Registered:
7/12/10


Re: Another notcompletelyinsignificant gap in the Rado paper
Posted:
Nov 7, 2013 9:38 AM


On Thursday, November 7, 2013 2:05:55 PM UTC, David Hartley wrote: > In message <b829deebb3314460a7235d76a8ac54d0@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^(s1)) has (r1) 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 orderrelationship as B_2pi > > with all other elements of B and so does not change the > > orderrelationship 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,
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^(s1)) elements between the members of B(r ^ s). We sometimes (unless we argue further) want r instead of r1.
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 BlassEpstein argument to say that, because we're working inside B', the problem (not being able to find enough B(r^(s1) elements) is inconsistent with the assumed inequality f(x...) =/= f(x'...).
Thanks again,
Paul Epstein

