[FOM] Question on Infinite Sequences of Ordinals

[Dmytro Taranovsky]

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