If every theory that axiomatizes uncountable models has at least one countable model
No.12851076 ViewReplyOriginalReport
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https://plato.stanford.edu/entries/paradox-skolem/
Obviously you can carry on doing math that presupposes the existence of this larger cardinals, but ultimately, any theory you construct in FOL (eg real-closed field, algebraically-closed field, etc) will be satisfied by at least one model with a countable domain.
Obviously the issue of the ontology of mathematical objects in general is complex, but given this result, does it really make sense to so casually speak of there "existing" "higher infinities?"
Should those predisposed towards mathematical realism/Platonism still observe Occam's razor?
Obviously you can carry on doing math that presupposes the existence of this larger cardinals, but ultimately, any theory you construct in FOL (eg real-closed field, algebraically-closed field, etc) will be satisfied by at least one model with a countable domain.
Obviously the issue of the ontology of mathematical objects in general is complex, but given this result, does it really make sense to so casually speak of there "existing" "higher infinities?"
Should those predisposed towards mathematical realism/Platonism still observe Occam's razor?
