Satisfaction is not absolute

• DOI: 10.48550/arxiv.1312.0670
• Publication Date: 2013-01-01
• Authors: Joel David Hamkins; Ruizhi Yang
• Keywords: Iterated function; Natural number; Predicate (mathematical logic); Rank (graph theory)

We prove that the satisfaction relation $\mathcal{N}\models\varphi[\vec a]$ of first-order logic is not absolute between models set theory having structure $\mathcal{N}$ and formulas $\varphi$ all in common. Two can have same natural numbers, for example, standard model arithmetic $\langle\mathbb{N},{+},{\cdot},0,1,{\lt}\rangle$, yet disagree on their theories truth; two numbers truths, truths-about-truth, at any desired level iterated truth-predicate hierarchy; reals, projective $\langle H_{\omega_2},{\in}\rangle$ or rank-initial segment V_\delta,{\in}\rangle$, which assertions are true these structures. On basis mathematical results, we argue a philosophical commitment to determinateness truth cannot be seen as consequence solely resides. The determinate nature truth, $\mathbb{N}=\{0,1,2,\ldots\}$ itself, but rather, argue, an additional higher-order requiring its own analysis justification.