Intuitionistic Formal Systems Research Articles

Epistemic and intuitionistic formal systems

Developpement d'une methode uniforme visant a l'integration des mathematiques classiques et intuitionnistes. Application d'un theoreme de plongement de toute algebre de Heyting complete dans une algebre de Boole complete a intersections finies et a reunions arbitraires

Continuity and comprehension in intuitionistic formal systems

Within the framework of a formal system, we can ask if one can find any necessary relations between the answers to the two questions posed above. We show that, within the language of second-order arithmetic, one cannot find any such relations; even if one includes Church's thesis, which says that every constructive function is recursive. In earlier work, we have proved independence results related to question (1) in the context of the language of arithmetic. The main tool of the present paper is an extension of our earlier methods to second-order comprehension principles. It is fairly easy to prove the consistency of strong principles of set existence with the continuity of extensional functions, even in the presence of Church's thesis (see discussion in [3]). And, as mentioned, the case where one does not have strong set existence principles has been dealt with in [1]. The main problem, then, is the independence of the continuity of extensional functions from strong set existence principles. Of course, if the formal axioms considered are classically true, this independence is trivial; but we are interested in the independence in an axiomatic framework including nonclassical principles. Foremost among such principles is Church's thesis, which (in a suitable formulation) will reduce members of NN to recursive indices, and functions from NN to N to effective operations, which compute the function value recursively from an index of the argument. Thus, under Church's thesis CT, the statement all functions are continuous reduces to an arithmetical proposition about effective operations. This proposition (for the case of NN) is called KLS, after Kreisel, Lacombe, and Shoenfield, who gave a classical proof of it [5]. It also happens that this sentence KLS lies in a syntactic class for which CT is conservative (over the theories we will consider; see discussion in the text). Thus the difficult part of our problem is to prove the independence of KLS from various principles of set existence.

The unprovability in intuitionistic formal systems of the continuity of effective operations on the reals

The question of continuity of functions defined on Baire space NN or on the reals has been of great interest to constructivists. We have little hope of answering the question without a fundamental philosophical analysis of the notion of constructive function. However, one can ask whether the continuity of functions can be derived in known intuitionistic formal systems, and answer these questions by mathematical means.The constructivist does not generally wish to restrict himself to reals (or members of NN) given by a recursive law; therefore the most natural formal systems have variables for reals, sequences and functions. The axioms of these systems turn out to be compatible with the assumption that every real or sequence is given by a recursive law, and every function by an effective operation. An underivability result, therefore, is only strengthened if this assumption is taken as an axiom; and once this is done, we may as well work with (indices of) effective operations, within arithmetic.By KLS is meant the assertion that each effective operation on Baire space NN is continuous. In [B1] it is shown that this statement cannot be proved in various intuitionistic formal systems. The continuity of functions on the reals has been of even greater interest to constructivists than the continuity of functions on NN; by KLS(R) is meant the assertion that each effective operation from the real numbers R to R is continuous. It is the purpose of the present note to show that KLS(R) is also underivable in intuitionistic formal systems.

The unprovability in intuitionistic formal systems of the continuity of effective operations on the reals

The question of continuity of functions defined on Baire space NN or on the reals has been of great interest to constructivists. We have little hope of answering the question without a fundamental philosophical analysis of the notion of constructive function. However, one can ask whether the continuity of functions can be derived in known intuitionistic formal systems, and answer these questions by mathematical means.The constructivist does not generally wish to restrict himself to reals (or members of NN) given by a recursive law; therefore the most natural formal systems have variables for reals, sequences and functions. The axioms of these systems turn out to be compatible with the assumption that every real or sequence is given by a recursive law, and every function by an effective operation. An underivability result, therefore, is only strengthened if this assumption is taken as an axiom; and once this is done, we may as well work with (indices of) effective operations, within arithmetic.By KLS is meant the assertion that each effective operation on Baire space NN is continuous. In [B1] it is shown that this statement cannot be proved in various intuitionistic formal systems. The continuity of functions on the reals has been of even greater interest to constructivists than the continuity of functions on NN; by KLS(R) is meant the assertion that each effective operation from the real numbers R to R is continuous. It is the purpose of the present note to show that KLS(R) is also underivable in intuitionistic formal systems.

The nonderivability in intuitionistic formal systems of theorems on the continuity of effective operations

The primary purpose of this work is to analyze the constructive interpretations of classical theorems concerning the continuity of effective operations. These theorems are the “recursivizations” of the statements that all functions (on various spaces) are continuous. To discuss the question whether these statements are constructively valid, it would be necessary to analyze the notion of “constructively valid,” which is beyond the scope of this paper. We now proceed to describe our results. We deal with two kinds of effective operations: (i) partial operations on indices of partial functions, and (ii) total operations on indices of total functions. The two principal theorems of the subject assert that each such operation is continuous; for (i) this is due to Myhill-Shepherdson [7] and called MS here; for (ii) it is due to Kreisel, Lacombe, and Shoenfield [5] and called KLS. (Explicit formulations of these and other principles mentioned in the introduction will be given in §1 below.)Several interpretations (of, e.g., Heyting's arithmetic HA) in the literature satisfy Markov's principle MP, and for these MS and KLS are valid, since a close inspection of the classical proofs of MS and KLS shows that they are derivable in HA + MP. But we also find an interpretation in the literature under which MS and KLS are valid while MP is not (hence MS and KLS do not imply MP in systems for which this interpretation is sound). This work is in §2.

Ronald Harrop. Concerning formulas of the types A → B v C, A → (Ex)B(x) in intuitionistic formal systems. The journal of symbolic logic, vol. 25 no. 1 (for 1960, pub. 1961), pp. 27–32.

Ronald Harrop. Concerning formulas of the types A → B v C, A → (Ex)B(x) in intuitionistic formal systems. The journal of symbolic logic, vol. 25 no. 1 (for 1960, pub. 1961), pp. 27–32.

Disjunction and existence under implication in elementary intuitionistic formalisms

Let Pp, Pd, and N be the intuitionistic formal systems of prepositional calculus, predicate calculus, and elementary number theory, respectively.1 Consider the following six propositions.8(1) ├A V B only if ├A or ├B.(2) ├∋xA(x) only if ├Ã(t) for some formula Ã(x) congruent to A(x) and some term t free for x in Ã(x).