The Lindenbaum fixed point algebra is undecidable (Q810010)

The author studies fixed point algebras (fpa's) in the sense of \textit{C. Smoryński} [Bull. Am. Math. Soc., New Ser. 6, 317-356 (1982; Zbl 0544.03032)]. These are certain natural Boolean pairs that arise from consideration of natural mappings on the Lindenbaum algebra. \textit{R. M. Solovay} [Recursion theory, Proc. Symp. Pure Math. 42, 473-486 (1985; Zbl 0573.03030)] showed that the fpa's associated with the Lindenbaum algebra of r.e. consistent theories containing PA are all recursively isomorphic. Call this the Lindenbaum fixed point algebra. The author proves that the first-order theory of the Lindenbaum fixed point algebra is hereditarily undecidable.