Mark Gritter (markgritter) wrote,
Mark Gritter

Ultimate L

I'm catching up on New Scientist and read Richard Elwes' article on "Ultimate Logic". Elwes does a good job explaining what the issues are but I found his description of Woodin's new theory lacking. Standard set theory (ZFC axioms) leave the Continuum Hypothesis undecidable--- there are models which satisfy the axioms where CH is true, and models where CH is false. Coming up with a better model alone is not enough, although that is the impression that Elwes gives. You also need a new axioms (or an entirely new axiomatization) that makes your model fit and all the other weirdo models not fit.

Woodin's slides on "The Search for Ultimate L" start out pretty accessible but get obscure quickly, but I think got a better flavor for what he was trying to do. He gives as an example Projective Determinacy, which is independent of ZFC but is commonly accepted. As an additional axiom he seems to offer "there is a proper class of Woodin cardinals", along with a suggestion for a model that provides an object for every sentence in the Von Neumann hierarchy.

I don't have an opinion on the mathematics, but philosophically I am skeptical. You can escape Skolem's Paradox only by using higher-order logic or an uncountable number of axioms. And neither of these are things that we can really trust our "intuition" about. But some days I even dispute the existence of R....
Tags: mathematics
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