Satisfaction is not absolute
Joel David Hamkins
The City University of New York
I will discuss a number of theorems showing that the satisfaction relation of first-order logic is less absolute than might have been supposed. Two models of set theory $M_1$ and $M_2$, for example, can agree on their natural numbers $langlemathbb{N},{+},{cdot},0,1,{lt}rangle^{M_1}=langlemathbb{N},{+},{cdot},0,1,{lt}rangle^{M_2}$, yet disagree on arithmetic truth: they have a sentence $sigma$ in the language of arithmetic that $M_1$ thinks is true in the natural numbers, yet $M_2$ thinks $negsigma$ there. Two models of set theory can agree on the natural numbers $mathbb{N}$ and on the reals $mathbb{R}$, yet disagree on projective truth. Two models of set theory can have the same natural numbers and have a computable linear order in common, yet disagree about whether this order is well-ordered. Two models of set theory can have a transitive rank initial segment $V_delta$ in common, yet disagree about whether this $V_delta$ is a model of ZFC. The theorems are proved with elementary classical methods.
This is joint work with Ruizhi Yang (Fudan University, Shanghai). We argue, on the basis of these mathematical results, that the definiteness of truth in a structure, such as with arithmetic truth in the standard model of arithmetic, cannot arise solely from the definiteness of the structure itself in which that truth resides; rather, it must be seen as a separate, higher-order ontological commitment.
Professor Hamkins (Ph.D. 1994 UC Berkeley) conducts research in mathematical and philosophical logic, particularly set theory, with a focus on the mathematics and philosophy of the infinite. He has been particularly interested in the interaction of forcing and large cardinals, two central themes of contemporary set-theoretic research. He has worked in the theory of infinitary computability, introducing (with A. Lewis and J. Kidder) the theory of infinite time Turing machines, as well as in the theory of infinitary utilitarianism and, more recently, infinite chess. His work on the automorphism tower problem lies at the intersection of group theory and set theory. Recently, he has been preoccupied with various mathematical and philosophical issues surrounding the set-theoretic multiverse, engaging with the emerging debate on pluralism in the philosophy of set theory, as well as the mathematical questions to which they lead, such as in his work on the modal logic of forcing and set-theoretic geology.