Rutgers Logic Seminars

Spring 2013

Rooms 423 & 705, Hill Center

Descriptive Set Theory Seminar

Hill 423

Monday Mar 25th, 3:20-4:40 pm

Simon Thomas, Rutgers

Invariant random subgroups VI

Rutgers Logic Seminar

Hill 705

Monday Mar 25th, 5:00-6:20 pm

Joel Hamkins, CUNY

Pluralism in mathematics: the multiverse view in set theory

and the question of whether every mathematical statement has

a definite truth value

Abstract: I shall describe the debate on pluralism in the philosophy of set

theory, specifically on the question of whether every mathematical and

set-theoretic assertion has a definite truth value. A traditional Platonist

view in set theory, which 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. I shall try 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, 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.