[FOM] Question on Infinite Sequences of Ordinals
dmytro at MIT.EDU
Sun Apr 24 13:25:29 EDT 2005
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).
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