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I have restated the question in terms of MK set theory
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Andrew Bacon
Andrew Bacon
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The following can be stated as a sentence of second order ZFCMorse-Kelley set theory:

If L is the logic of a proper class sized Kripke frame, then L is the logic of a set sized Kripke frame.

It follows from a $\Pi^1_1$ second order reflection principle (or suitable large cardinal assumption). But I was wondering if this is something that might follow from standard results about frame validity and so be a theorem of second order ZFC alone.MK? And if not, what is the consistency strength of this principle?

The following can be stated as a sentence of second order ZFC:

If L is the logic of a proper class sized Kripke frame, then L is the logic of a set sized Kripke frame.

It follows from a $\Pi^1_1$ second order reflection principle (or suitable large cardinal assumption). But I was wondering if this is something that might follow from standard results about frame validity and so be a theorem of second order ZFC alone. And if not, what is the consistency strength of this principle?

The following can be stated as a sentence of Morse-Kelley set theory:

If L is the logic of a proper class sized Kripke frame, then L is the logic of a set sized Kripke frame.

It follows from a $\Pi^1_1$ reflection principle (or suitable large cardinal assumption). But I was wondering if this is something that might follow from standard results about frame validity and so be a theorem of MK? And if not, what is the consistency strength of this principle?

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Andrew Bacon
Andrew Bacon
  • 357
  • 1
  • 9

Logics of proper class sized Kripke frames

The following can be stated as a sentence of second order ZFC:

If L is the logic of a proper class sized Kripke frame, then L is the logic of a set sized Kripke frame.

It follows from a $\Pi^1_1$ second order reflection principle (or suitable large cardinal assumption). But I was wondering if this is something that might follow from standard results about frame validity and so be a theorem of second order ZFC alone. And if not, what is the consistency strength of this principle?