Satisfaction is not absolute
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arXiv
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| Format: | Preprint |
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2013
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| _version_ | 1866912515222405120 |
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| author | Hamkins, Joel David Yang, Ruizhi |
| author_facet | Hamkins, Joel David Yang, Ruizhi |
| contents | We prove that the satisfaction relation $\mathcal{N}\modelsφ[\vec a]$ of first-order logic is not absolute between models of set theory having the structure $\mathcal{N}$ and the formulas $φ$ all in common. Two models of set theory can have the same natural numbers, for example, and the same standard model of arithmetic $\langle\mathbb{N},{+},{\cdot},0,1,{\lt}\rangle$, yet disagree on their theories of arithmetic truth; two models of set theory can have the same natural numbers and the same arithmetic truths, yet disagree on their truths-about-truth, at any desired level of the iterated truth-predicate hierarchy; 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 $\langle H_{ω_2},{\in}\rangle$ or the same rank-initial segment $\langle V_δ,{\in}\rangle$, yet disagree on which assertions are true in these structures.
On the basis of these mathematical results, we argue that a philosophical commitment to the determinateness of the theory of truth for a structure cannot be seen as a consequence solely of the determinateness of the structure in which that truth resides. The determinate nature of arithmetic truth, for example, is not a consequence of the determinate nature of the arithmetic structure $\mathbb{N}=\{0,1,2,\ldots\}$ itself, but rather, we argue, is an additional higher-order commitment requiring its own analysis and justification. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1312_0670 |
| institution | arXiv |
| publishDate | 2013 |
| record_format | arxiv |
| spellingShingle | Satisfaction is not absolute Hamkins, Joel David Yang, Ruizhi Logic 03E, 03E We prove that the satisfaction relation $\mathcal{N}\modelsφ[\vec a]$ of first-order logic is not absolute between models of set theory having the structure $\mathcal{N}$ and the formulas $φ$ all in common. Two models of set theory can have the same natural numbers, for example, and the same standard model of arithmetic $\langle\mathbb{N},{+},{\cdot},0,1,{\lt}\rangle$, yet disagree on their theories of arithmetic truth; two models of set theory can have the same natural numbers and the same arithmetic truths, yet disagree on their truths-about-truth, at any desired level of the iterated truth-predicate hierarchy; 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 $\langle H_{ω_2},{\in}\rangle$ or the same rank-initial segment $\langle V_δ,{\in}\rangle$, yet disagree on which assertions are true in these structures. On the basis of these mathematical results, we argue that a philosophical commitment to the determinateness of the theory of truth for a structure cannot be seen as a consequence solely of the determinateness of the structure in which that truth resides. The determinate nature of arithmetic truth, for example, is not a consequence of the determinate nature of the arithmetic structure $\mathbb{N}=\{0,1,2,\ldots\}$ itself, but rather, we argue, is an additional higher-order commitment requiring its own analysis and justification. |
| title | Satisfaction is not absolute |
| topic | Logic 03E, 03E |
| url | https://arxiv.org/abs/1312.0670 |