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Asaf Karagila ♦
Asaf Karagila ♦
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This is a really late answer, but the answer to your question is "No." As a dual to the Theorem 6 that Thomas Benjamin mentions above (which I believe is a result of Stavi), Janos Makowsky proved that V\v{o}penka'sVopěnka's Principle is equivalent to the statement "All logics have a compactness cardinal."

This is a really late answer, but the answer to your question is "No." As a dual to the Theorem 6 that Thomas Benjamin mentions above (which I believe is a result of Stavi), Janos Makowsky proved that V\v{o}penka's Principle is equivalent to the statement "All logics have a compactness cardinal."

This is a really late answer, but the answer to your question is "No." As a dual to the Theorem 6 that Thomas Benjamin mentions above (which I believe is a result of Stavi), Janos Makowsky proved that Vopěnka's Principle is equivalent to the statement "All logics have a compactness cardinal."

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Will Boney
Will Boney
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This is a really late answer, but the answer to your question is "No." As a dual to the Theorem 6 that Thomas Benjamin mentions above (which I believe is a result of Stavi), Janos Makowsky proved that V\v{o}penka's Principle is equivalent to the statement "All logics have a compactness cardinal."