[FOM] S4 + ZFC

Michael Carroll mcarroll at pobox.com
Sun Aug 26 18:27:05 EDT 2007

Dana Scott wrote, some thirty years ago, in one of my favorite passages:
"I see that there are any number of contradictory set theories, all 
extending the Zermelo-Fraenkel axioms; but the models are all just models of 
the first-order axioms, and first-order logic is weak. ... A new idea (or 
point of view) is needed, and in the meantime all we can do is to study the 
great _variety_ of models." (Foreword to Bell's "Boolean-valued Models and 
Independence Proofs in Set Theory.")

In a recent post to FOM, Prof. Scott suggests using Boolean-valued models 
for classical ZFC to investigate modal set theory. This seems to continue 
the study of the great variety of models of a weak logic. Adding S4's modal 
axioms to ZFC is something different. It may lead nowhere, or possibly it's 
a new point of view.

In any case, Prof. Scott's taking the time to comment is a great 

Mike Carroll

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