Hyperdiagonalizable algebras (Q1078181)

In 1975 R. Magari introduced diagonalizable algebras (DA's) in order to study logical properties of Peano arithmetic by algebraic means. We recall that a DA is a Boolean algebra equipped with a unary operation \(\tau\) which corresponds to the formula Theor(v). By a hyperdiagonalizable algebra (HDA) we mean a DA in which, for every polynomial f, the equation \(\tau fx=\tau x\) admits a solution. This is the translation of another diagonalization property of Peano arithmetic (which is not a consequence of the diagonalization lemma): namely, for every recursive function r where exists a sentence G such that \(\vdash Theor(\overline{r\bar G})\leftrightarrow Theor(\bar G)\). In this paper we study the dual spaces of HDA's, we show that it is sufficient to consider the polynomials of the form \(fx=x\to c\) (where c is an element of the algebra), and we prove the finite model property for HDA's.