[FOM] Question on Infinite Sequences of Ordinals

[Paul Larson]

The following seems relevant to the posting of Dmytro Taranovsky below.

There is a theorem of Erdos and Hajnal which shows that you can't have 
indiscernibles for sequences of ordertype omega without adding new 
omega-sequences. In particular, it shows that if M is a transitive inner 
model of ZFC closed under omega-sequences, then for any ordinal lambda 
there is a function in M from the omega-sequences from lambda to omega 
which takes all values on the omega sequences from any subset of lambda 
of ordertype lambda in V (among other things; this is a weak version). 
There is a version of the theorem on page 319 of the first edition of 
Kanamori's Higher Infinite, among other places. This version doesn't 
literally do what we want, but it can be easily modified to do so.

On the other hand, Woodin has shown that you can have infinitary
indiscernibles for the Chang model, the least transitive class model of 
ZF closed under omega-sequences (Kunen had previously shown that if 
there are uncountably many measurable cardinals in V, then this
is not a model of Choice). Woodin has shown that if there exist proper
class many Woodin limits of Woodin cardinals, then there is a sharp for 
the Chang model, a theory in a language with constants for each real, 
constants for omega_1-many L_omega_1,omega_1-indiscernibles, plus 
omega-many Skolem functions, which serves as a blueprint for the Chang 
model. Futhermore, this theory is universally Baire, which implies that 
every set of reals in the Chang model is universally Baire, and thus 
determined.




Dmytro Taranovsky wrote:

> In the constructible universe L, the indiscernibles satisfy all large
 > cardinal properties realized in L, and the theory of increasing
 > n-tuples of indiscernibles is canonical and depends only on n.
>
> I was wondering whether for some transitive models, there is an analogue
 > of the indiscernibles, but with (increasing) infinite sequences of
 > ordinals.  That is the ordinals should satisfy all large cardinal
 > properties realized in the model, and the theory should be canonical
 > and independent of the infinite sequence (of order type omega) chosen.
>
> My motivation for the question is finding out how expressive one can be
 > without invoking uncountable sets.  There should be analogues of
 > indiscernibles with countable sequences of countable ordinals, but
 > perhaps
> they are not definable in mild extensions of (H(omega_1), in) and can be
 > used to define, say, the theory of L(R).
>
> Dmytro Taranovsky