Paul
Posts:
500
Registered:
7/12/10


Re: Another notcompletelyinsignificant gap in the Rado paper
Posted:
Nov 7, 2013 4:34 PM


On Thursday, November 7, 2013 8:29:09 PM UTC, David Hartley wrote: > In message <1e7387e216fc487fbe17b490ed25dd39@googlegroups.com>, Paul > > <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^(s1)) elements between the members of B(r ^ s). We sometimes > > >(unless we argue further) want r instead of r1. > > > > 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 (r1) available. > > > > >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'...). > > > > 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 rho0indexed element of X_sigma0 to the > > onehigher 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. >
Many thanks for your thoughts on this.
It would be interesting to compare and contrast with http://www.renyi.hu/~p_erdos/195001.pdf which proves the same thing. Currently, I haven't had a chance to digest either paper. I'm put off by maths where the readers have to fill in the details themselves. I enjoy it far more when I follow the author's reasoning.
That would not be a good approach if I were an academic mathematician. But I'm not so I'm taking the approach I enjoy best.
Can't wait to have attained mastery of this theorem of genius.
Paul Epstein

