Burton Dreben

A supplement to herbrand

In [5] it was shown that to complete Herbrands argument for his Fundamental Theorem (see [6]) a weak analyzing function for certain applications of the rules of passage is needed. The following theorem describes such a function. (We use the terminology and notation of [5] except that, for each schema S and each p > 1, we shall write D(S, p) rather than Ds for the domain of order p generated by S. In addition, we shall say that an element of D(S, p) is of order k, 1 _ k _ p, if it belongs to D(S, k) but not to D(S,k -1).) THEOREM. Let S be a rectified schema having property C of order p, and let S contain in a negative occurrence, within the scopes of r (? 0) restricted quantifiers, either the schema VxW.VZ or the schema ZV.VxW. Let T be the schema obtained from S by putting Vx(WVZ) for VxW. VZ or by putting Vx(ZVW) for ZV.VxW. If N is the number of members of D(S, p), then the schema T has property C of order q P(I ? Nr)P. PROOF of the case VxW. VZ. (For ZV. VxW the argument is the same.) We begin the proof by defining a standard form for the reduction of a schema over a domain. DEFINITION I. Let S be any schema and F(S) its functional form. The strict functional form of S is the result of dropping all the (restricted)

Herbrand-Style Consistency Proofs

Publisher Summary This chapter discusses an approach to study the consistency stemming from ideas of Herbrand. This approach exploits the oldest and most naive idea in proof theory: a set of axioms is consistent if it has a model. The chapter contrasts this theory with the approach initiated by Gentzen and continued by Schutte. The Herbrand approach is best viewed as a reformulation of Hilberts evaluation method, a reformulation that frees that method from its customary dependence on the ɛ-calculus. The basic result is the fundamental theorem of Herbrand, the finitistic correlate of the Lowenheim-Skolem theorem. The chapter formulates the fundamental theorem of Herbrand and discusses how the method can be used to give a consistency proof for elementary number theory, and how the recursive satisfaction of AE-formulas and the no-counterexample interpretation follow immediately from the proof.

Annual meeting of the association for symbolic logic

PHILADELPHIA 1981 The 1981-1982 annual meeting of the Association for Symbolic Logic took place at the Franklin Plaza Hotel, Philadelphia, Pennsylvania, December 28-29, 1981, in conjunction with the annual meeting of the Eastern Division of the American Philosophical Association. A symposium on Intuitionism was held, commemorating the centennial of L. E. J. Brouwer, jointly sponsored by the Eastern Division. The speakers were Scott Weinstein, W.W. Tait, and Dana Scott. Solomon Feferman served as Chairman. Kenneth McAloon gave a survey lecture, Combinatorial versions of the incompleteness theorem: Survey and remarks. J. Michael Dunn gave a survey lecture, Topics in relevance logic. Yuri Gurevich gave an invited address, The monadic second-order logic in applications. Twenty contributed papers were presented in person and five by title. The paper of Lambek and Scott was presented by Scott; the paper of Friedman and Scedrov was presented by Skedrov. The Council met the evening of December 28 and continued at noon December 29.

Twenty-Second Annual Meeting of the Association for Symbolic Logic

S OF PAPERS 105 The above rule can be used only for steps of a main proof. Furthermore, use of identity elimination is not permitted in the subordinate proofs of the above rule except for substituting into formulas which are themselves identities or denials of identities. Without these restrictions consistency cannot be proved, and Myhill has shown that if the second restriction is omitted the Russell paradox becomes derivable. An operator can be employed which operates on a unit class to give its only member. The following three rules can also be consistently added, provided that the second and third are used only for steps of main proofs: (x)[ax = bx] (x)(ax) a