an:05495725
Zbl 1159.03028
Oliveira, Igor Carboni; Carnielli, Walter
The Ricean objection: An analogue of Rice's theorem for first-order theories
EN
Log. J. IGPL 16, No. 6, 585-590 (2008); erratum ibid. 17, No. 6, 803-804 (2009).
00243302
2008
j
03D35
Rice's theorem; undecidable; Turing machine
\textit{H. G. Rice} [Trans. Am. Math. Soc. 74, 358--366 (1953; Zbl 0053.00301)] proved that any property defined over the set of languages accepted by a Turing machine is either trivial or undecidable. The authors prove an analogue of Rice's theorem for first-order theories expressed in the language \(\{0, s, +, \cdot\}\) (\(0\) being a constant, \(s\) a unary operation, and \(+\) and \(\cdot\) binary operations), stating that, for any nontrivial property of first-order theories, it is undecidable whether the theory axiomatized by a given finite set of axioms has that property. The short proof uses Robinson's \(Q\).
Victor V. Pambuccian (Phoenix)
Zbl 0053.00301