Topic Archive: multiverse
This talk will give a brief overview of set-theoretic geology, the study of the collection of grounds of $V$. Forcing is naturally viewed as a method for passing from a model $V$ of set theory (the ground model) to an outer model $V[G]$ (the forcing extension). A change in perspective, however, allows us to use forcing to look inward: from a model $V$, we define an inner model $W$ of $V$ to be a ground of $V$ if $W$ is a transitive proper class satisfying ZFC and $V$ can be obtained by forcing over $W$, that is, if $V = W[G]$ for a suitable $W$-generic $G$. For a given model $V$, the collection of all of its ground models forms the context for what we call set-theoretic geology. This second-order collection, consisting of (possibly many) proper classes $W$, nonetheless admits a first-order definition – within a single universe, we have first-order access to an interesting local neighborhood of the set-theoretic multiverse. We will explore this neighborhood, pointing out various geological phenomena including bedrock models, the mantle and the outer core. This is joint work with Joel David Hamkins and Gunter Fuchs.
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 $Vneq L$ 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.
I shall give a summary account of some current issues in the philosophy of set theory, specifically, the debate on pluralism and the question of the determinateness of set-theoretical and mathematical truth. The traditional Platonist view in set theory, what I call the universe view, holds that there is an absolute background concept of set and a corresponding absolute background set-theoretic universe in which every set-theoretic assertion has a final, definitive truth value. What I would like to do is to tease apart two often-blurred aspects of this perspective, namely, to separate the claim that the set-theoretic universe has a real mathematical existence from the claim that it is unique. A competing view, which I call the multiverse view, accepts the former claim and rejects the latter, by holding that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe, and a corresponding pluralism of set-theoretic truths. After framing the dispute, I shall argue that the multiverse position explains our experience with the enormous diversity of set-theoretic possibility, a phenomenon that is one of the central set-theoretic discoveries of the past fifty years and one which challenges the universe view. In particular, I shall argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.