Satisfaction is not absolute, CUNY Logic Workshop, September 2013

This will be a talk for the CUNY Logic Workshop on September 27, 2013.

Abstract.  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 $\langle\mathbb{N},{+},{\cdot},0,1,{\lt}\rangle^{M_1}=\langle\mathbb{N},{+},{\cdot},0,1,{\lt}\rangle^{M_2}$, 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 $\neg\sigma$ 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.

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2 thoughts on “Satisfaction is not absolute, CUNY Logic Workshop, September 2013

  1. Hi Professor. How was your talk? Your claims are very interesting and I was wondering if they were met with any controversy. I was also wondering if you have a draft with some details, especially details concerning an apparent bifurcation in arithmetic.

    • Hello, Everett. The talk was a lot of fun, and I was able both to give proofs of the mathematical results and hint at the philosophical arguments that we have in mind with them. I don’t think there was any controversy at the talk, although Roman Kossak advanced the argument that one can look at the models $M_1$ and $M_2$ as not actually having different theories of arithmetic truth, since the proof does show that their theories of arithmetic truth are actually isomorphic. So he would claim of the sentence $\sigma$ whose truth value changed, that its meaning also changed. I think the paper will be ready in a few weeks, so check back here for it.

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