Elementary intuitionistic theories

Abstract

The present paper concerns itself primarily with the decision problem for formal elementary intuitionistic theories and the method is primarily model-theoretic. The chief tool is the Kripke model for which the reader may find sufficient background in Fitting's book Intuitionistic logic model theory and forcing (North-Holland, Amsterdam, 1969). Our notation is basically that of Fitting, the differences being to favor more standard notations in various places.The author owes a great debt to many people and would particularly like to thank S. Feferman, D. Gabbay, W. Howard, G. Kreisel, G. Mints, and R. Statman for their valuable assistance.The method of elimination of quantifiers, which has long since proven its use in classical logic, has also been applied to intuitionistic theories (i) to demonstrate decidability ([9], [15], [17]), (ii) to prove the coincidence of an intuitionistic theory with its classical extension ([9], [17]), and (iii), as we will see below, to establish relations between an intuitionistic theory and its classical extension. The most general of these results is to be obtained from the method of Lifshits' quantifier elimination for the intuitionistic theory of decidable equality.Since the details of Lifshits' proof have not been published, and since the proof yields a more general result than that stated in his abstract [15], we include the proof and several corollaries.

The eskolemization of universal quantifiers

“…Theorem 18 [23] In iMP dec every sequent S in L is equivalent to a sequent ⇒ Smoryński uses Theorem 15 and Theorem 18 to prove that iEq dec and iMP dec are decidable, but we do not see how to obtain this as an easy corollary from our Herbrand theorems for these logics. Similar theorems as the ones discussed above could be obtained for other theories.…”

“…Theorem 15 [23] In iEq dec every sequent S in L is equivalent to a sequent of the form ⇒ n i=1 A i ∧ B i , where the A i are conjunctions of atomic formulas and their negations, and the B i are propositional combinations of the formula ∃x(x = x), denoted by E 1 , and the formulas E n ∃x 1 . .…”

Encoding true second‐order arithmetic in the real‐algebraic structure of models of intuitionistic elementary analysis

“…But the full real algebraic structure turned out to be undecidable. There were results by Smoryński (in [15]) and by Cherlin (in [1]) suggesting that this might be the case. Indeed, building on Cherlin's work, in [3] we were able to interpret true first order arithmetic in the L ‐theory of Scott's model.…”