2005-05-24T09:14:00.000-07:00

OK. Having only one sibling of a given rank is bad. The favorite sibling function will give a successor, and one will have the type that says "sibling but not any iteration of favorite sibling". Then there seems to be no reason (above \omega^2) that one couldn't have a model with two siblings of a single rank.

The same objection can be raised by replacing the favorite sibling function by a relation that holds between any siblings of rank \alpha and \alpha + 1 respectively.

How about making the functions one to one? I still don't see the problem here.

just a thought

2005-05-23T19:02:00.000-07:00

What if in addition to having a scott game played only on limit ordinals, we have one that's played only on eldest children? Will this allow us to avoid the L_{2\omega,\omega} sentence eliminating the Eldest + infinity case?

By the way, it came up in the last talk that a class is pseudo-elementary L_{\omega_1,\omega} iff it is \Sigma^1_1, so ordinals aren't a pseudo-elementary class.

But divisible abelian groups have \omega_1 many order types (according to Hodge's Model Theory) .

Do Weak Upward Persistance and the limit-only Scott game really give you Upward Persistance? I don't see this.

Vaught Conjecture

just a thought