Satisfaction is not absolute, Connecticut, October 2013

This will be a talk for the Logic Seminar in the Mathematics Department at the University of Connecticut in Storrs on October 25, 2013.

Abstract. The satisfaction relation $\mathcal{N}\models\varphi[\vec a]$ of first-order logic, it turns out, is less absolute than might have been supposed.  Two models of set theory, for example, can agree on their natural numbers and on what they think is the standard model of arithmetic $\langle\mathbb{N},{+},{\cdot},0,1,{\lt}\rangle$, yet disagree on their theories of arithmetic truth, the first-order truths of this structure.  Two models of set theory can have the same natural numbers and the same reals, yet disagree on projective truth.  Two models of set theory can have the same natural numbers and a computable linear order in common, yet disagree about whether it is well-ordered.  Two models of set theory can have a transitive rank initial segment $V_\delta$ in common, yet disagree about whether it is a model of ZFC.  The arguments rely mainly on elementary classical methods.

This is joint work with Ruizhi Yang (Fudan University, Shanghai), and our manuscript will be available soon, in which we prove these and several other very general facts showing that satisfaction is not absolute.  On the basis of these mathematical results, we mount a philosophical argument that a commitment to the determinateness of truth in a structure, such as the case of arithmetic truth in the standard model of arithmetic, cannot result solely from the determinateness of the structure itself in which that truth resides; rather, it must be seen as a separate, higher-order ontological commitment.

University of Connecticut Logic Seminar | Article

4 thoughts on “Satisfaction is not absolute, Connecticut, October 2013

  1. Hi, Prof. Hamkins, in August you mentioned that you would have the talk up on this site or on ArXiv. When I click on that talk in the “talks” section in this site, I am brought back to this page; when I google it, I get no results on ArXiv. Am I doing something wrong? Thanks.

    • Sorry, it’s taking longer than I expected, in part because of another project, and so it isn’t yet available. But in fact I’ve been working on it hard for the past two weeks, and I think we’ll have a draft fairly soon.

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