This will be a featured talk at the Midwest PhilMath Workshop 15, held at Notre Dame University October 18-19, 2014. W. Hugh Woodin and I will each give one-hour talks in a session on Perspectives on the foundations of set theory, followed by a one-hour discussion of our talks.

**Abstract**. I shall argue that the commonly held $V\neq L$ via maximize position, which rejects the axiom of constructibility V = L on the basis that it is restrictive, implicitly takes a stand in the pluralist debate in the philosophy of set theory by presuming an absolute background concept of ordinal. The argument appears to lose its force, in contrast, on an upwardly extensible concept of set, in light of the various facts showing that models of set theory generally have extensions to models of V = L inside larger set-theoretic universes.

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 proposal, pointing out that neither the fairly-interpreted-in relation nor the (strongly) maximizes-over relation is transitive, and furthermore, the theory ZFC + `there is a proper class of inaccessible cardinals’ is formally restrictive on Maddy’s account, contrary to what had been desired. Ultimately, I shall argue that the V≠L via maximize position loses its force on a multiverse conception of set theory with an upwardly extensible concept of set, in light of the classical facts that models of set theory can generally be extended to models of V=L. I shall conclude the talk by explaining various senses in which V=L remains compatible with strength in set theory.

This talk will be based on my paper, A multiverse perspective on the axiom of constructibility.

Will you be discussing H.J. Keisler’s old result in his paper “Forcing and the Omitting Types Theorem” (that is, “Every countable standard model of ZF has an end extension which is a model of ZFL”) and how it relates to yours?

I was not aware of this reference, but I’ll take a look at it. Barwise’s theorem strengthens this. My embedding theorem is a different result, showing that every countable model of set theory is isomorphic to a submodel of its own $L$. But that inclusion will of course not generally be an end-extension.

Also, will you be discussing what happens to $0^{\#}$ and other nonconstructible reals under your notion of “upwardly extensible concept of set”?

I discuss this in the paper, where I explain how a real though to be non-constructible can become constructible when one adds ordinals on top. We know this happens. For example, if $0^\sharp$ exists, then there is a countable transitive model $M$ inside $L$ that thinks ZFC+$0^\sharp$ exists. So $M$ has a fake $0^\sharp$, which is reveals as constructible once one goes to a high enough $L_\alpha$ that shows all of $M$ is constructible.

Interesting. By the way, what ramifications, if any, does the view that every set-theoretic universe should have extensions satisfying V=L have regarding the Whitehead Conjecture? Does it imply that every Whitehead group of cardinality $\aleph_1$ should ultimately be free?

The uncountability of the set will not be absolute under this kind of extension. When you are considering arbitrary extensions like this, every set becomes countable, and so there is no reason to expect that the group still has size $\aleph_1$ in the extension.

But will the model believe that sets of cardinality $\aleph_1$ exist?

Yes, if we are speaking of models of ZFC, then of course it will have sets that it thinks have size $\aleph_1$. The point is that in this multiverse picture, the nature of the set-theoretic background changes as one makes larger and larger extensions, and sets that seemed very large in one universe become countable and even constructible in much taller universes. But meanwhile, each of these universes can be taken as a model of ZFC, with all the familiar set-theoretic truths being true there.