Apparently, founding mathematics on Homotopy Type Theory instead of ZFC makes automated proof checking much simpler and more elegant. Has anybody tried reformulating Max Tegmark's Level IV Multiverse using Homotopy Type Theory instead of sets to see if the implied prior fits our anthropic observations better?
Homotopy type theory differs from ZFC in two ways. One way is that it, like ordinary type theory, is constructive and ZFC is not. The other is that it is based in homotopy theory. It is that latter property which makes it well suited for proofs in homotopy theory (and category theory). Most of the examples in slides you link to are about homotopy theory.
Tegmark is quite explicit that he has no measure and thus no prior. Switching foundations doesn't help.
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