Peano's Smart Children: A Provability Logical Study of Systems with Built-in Consistency

A pnovubW y 2.acgtica,2 ztudy o6 zysteni with built-in cones tevccy Alb ent V,iss etc 1 Introduction Consistency can be built into a system in various ways. The two best known constructions are Rosser's and Feferman's. Both construction's take a given formal system in the usual sense as initial data. Consider for example Peano Arithmetic (PA). A proof in the Peano System will count as a proof in the Rosser System based on PA, if there is no shorter Peano proof of the negation of its conclusion. The Feferman System can be described in various interesting ways-modulo provable equivalence in PA of the formulas defining the set of theorems. One such way is: a proof in the Peano System will count as a proof in the Feferman System based on PA, if the finite set of arithmetical Peano axioms smaller than or equal to the largest arithmetical Peano axiom used in the proof is consistent. The reasons such constructions occur in the literature are various: i) They serve as counterexamples in the study of the relation between Godel's first and second Incompleteness Theorem (see Feferman [19601). ii) They serve as didactical examples in philosophical discussions, like the debate on intensionality in Mathematics (see e.g. Auer-bach[ 1985]) and the discussion on the possible bearing of the In-completeness Theorems on the Minds & Machines problem (see e.g.