This is a talk for the CUNY Logic Workshop on February 15, 2013.
This talk will be based on my paper, A multiverse perspective on the axiom of constructibility.
Set-theorists often argue against the axiom of constructibility $V=L$ on the grounds that it is restrictive, that we have no reason to suppose that every set should be constructible and that it places an artificial limitation on set-theoretic possibility to suppose that every set is constructible. Penelope Maddy, in her work on naturalism in mathematics, sought to explain this perspective by means of the MAXIMIZE principle, and further to give substance to the concept of what it means for a theory to be restrictive, as a purely formal property of the theory.
In this talk, I shall criticize Maddy’s specific proposal. For example, it turns out that the fairly-interpreted-in relation on theories is not transitive, and similarly the maximizes-over and strongly-maximizes-over relations are not transitive. Further, the theory ZFC + `there is a proper class of inaccessible cardinals’ is formally restrictive on Maddy’s proposal, although this is not what she had desired.
Ultimately, I argue that the $V\neq L$ via maximize position loses its force on a multiverse conception of set theory, in light of the classical facts that models of set theory can generally be extended to (taller) models of $V=L$. In particular, every countable model of set theory is a transitive set inside a model of $V=L$. I shall conclude the talk by explaining various senses in which $V=L$ remains compatible with strength in set theory.