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A COMPLETENESS RESULT FOR FIXED‐POINT ALGEBRAS
Author(s) -
Montagna Franco
Publication year - 1984
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.19840303205
Subject(s) - mathematics , peano axioms , completeness (order theory) , algebra over a field , calculus (dental) , discrete mathematics , humanities , philosophy , pure mathematics , medicine , mathematical analysis , dentistry
FPAs were introduced by C. SMORY~SKI in [lo]; these structures arise from the following consideration: let T be an axiomatic theory in which every recursive function is representable; let for every two T-sentences p , q, p w T q stand for t , p t, q, and let us call a formula Fx in one free variable extensional iff, for every two T sentences p and q, p q implies F@ , F c ~ ; then every extensional formula Fx induces a mapping P from the Lindenbaum sentence algebra 8, of T into a, defined by P [ p ] T ’= [F$]-T (here denotes the equivalence class of p modulo -,). Now, let Br = { P : Fx is extensional} (of course we identify P and d iff they coincide as mappings); if we define the Boolean operations in BT in an obvious way’) the pair (a,., 23,) becomes an FPA, called the full Lindenbaum FPA of T (clearly, Condition 4 is the algebraic translation of GODEL’S Diagonalization Lemma).