The concept of a recursively definite predicate of natural numbers was introduced by F. B. Fitch in his An extension of basic logic as follows:

Every recursive predicate is recursively definite. If R(x1, …, xn) is recursively definite so is (Ey)R(x1, …, xn−1, y) and (y)R(x1, …, xn−1, y). If R is recursively definite and S is the proper ancestral of R, then S is recursively definite, where the proper ancestral of a relation is defined as follows: if R is of even degree, say 2m, then the proper ancestral of R is the relation S such that for all x1, …, xm, y1, …, ym, S(x1, …, xm, y1, …, ym) is true if and only if there is a finite sequence of sequences (z11, …, zm1), (z12, …, Zm2), …, (z1k, …, Zmk) such that R(Z11, …, Zm1, z12, …, zm2), R(z12, …, zm2, z13, …, zm3), …, R(z1,k−1, z1k, …, Zmk) are all true, where (z11, …, zm1) is (x1, …, xm) and (z1k, …, zmk) is (y1, …, ym).

An arithmetic predicate is one which is definable in terms of the operations ‘+’ and ‘·’ of elementary arithmetic, the connectives of the classical prepositional calculus, and quantifiers.

An effective method for testing arguments consisting of singly quantified expressions has recourse to Parry's trapezoid symbolism.1 The procedure entails representing in that symbolism each premise of the argument and the negation of the conclusion. Variables are disregarded. After clearing away any negated quantifiers, the universal statements, including a denied existential conclusion, are first set down and their conjunction (KU) derived. Existential statements, including a denied universal conclusion, are then listed. The argument is valid if and only if some contradiction appears, whether as some premise empty of lines, as a KU empty of lines, or as some existential statement having no lines in common with the KU.

We systematically study conservation theorems on theories of semi-classical arithmetic, which lie in-between classical arithmetic $\mathsf {PA}$ and intuitionistic arithmetic $\mathsf {HA}$. Using a generalized negative translation, we first provide a structured proof of the fact that $\mathsf {PA}$ is $\Pi _{k+2}$-conservative over $\mathsf {HA} + {\Sigma _k}\text {-}\mathrm {LEM}$ where ${\Sigma _k}\text {-}\mathrm {LEM}$ is the axiom scheme of the law-of-excluded-middle restricted to formulas in $\Sigma _k$. In addition, we show that this conservation theorem is optimal in the sense that for any semi-classical arithmetic T, if $\mathsf {PA}$ is $\Pi _{k+2}$-conservative over T, then ${T}$ proves ${\Sigma _k}\text {-}\mathrm {LEM}$. In the same manner, we also characterize conservation theorems for other well-studied classes of formulas by fragments of classical axioms or rules. This reveals the entire structure of conservation theorems with respect to the arithmetical hierarchy of classical principles.

Chang, in [1], proves an interpolation theorem (Theorem I, remark b)) for a first-order language. The proof of Chang's theorem uses essentially nonsimple devices, like special and ω1-saturated models.

In remark e) in [1], Chang asks if there is a simpler proof of his Theorem I.

In [1], Chang proves also another interpolation theorem (Theorem II), which is not an extension of his Theorem I, but extends Craig's interpolation theorem to Lα+,ω languages with interpolant in Lα+,α where α is a strong limit cardinal of cofinality ω.

In remark k) in [1], Chang asks if there is a generalization of both Theorems I and II in [1], or at least a generalization of both Theorem I in [1] and Lopez-Escobar's interpolation theorem in [7].

Maehara and Takeuti, in [8], show that there is a completely different proof of Chang's interpolation Theorem I as a consequence of their interpolation theorems. The proofs of these theorems of Maehara and Takeuti are proof theoretical in character, involving the notion of cut-free natural deduction, and it uses devices as simple as those needed for the usual Craig's interpolation theorem. Hence this can be considered as a positive answer to Chang's question in remark e) in [1].

In attempting to reconstruct Rose's proof of Lemma 3.2 of [1], the present authors found what is apparently a different and simpler method, which moreover leads to a far stronger conclusion.

We are operating in the Heyting prepositional calculus as formulated on p. 3 of [1] or on pp. 82 and 101 of [2], and shall make use of relevant theorems on pp. 90, 113–119 of [2]. We shall use a, b, c, w, x, y, z as propositional variables.

We say that a conjunction is simple if each factor has one of the forms: (i) a, (ii) ¬a, (iii) a⊃b, (iv) a⊃(b∨c), (v) (a&b)⊃c, (vi) (a⊃b)⊃c.

In this paper a converse of Barwise's completeness theorem is proved by cut-elimination considerations applied to inductive definitions. We show that among the transitive sets T satisfying some weak closure conditions (closure under primitive-recursive set-functions is more than enough), only the unions of admissible sets satisfy Barwise's completeness theorem in the form stating that if φ ∊ T is a sentence which has a derivation (in the universe) then φ has a derivation in T. See §1 for the origin of the problem in Barwise's paper [Ba].

Stated quite briefly the proof is as follows (a step-by-step account including relevant definitions is given in the body of the paper):

Let T be a transitive prim.-rec. closed set, and let is nonempty, transitive and closed under pairs}. For each let κ(A) be the supremum of closure ordinals of first-order positive operators on subsets of A (first-order with respect to By Theorem 1 of [BGM], it is enough to prove that rank(T) in order to obtain that T is a union of admissible sets. (The rank of a set is defined by rank(x) = supy ∊ x (rank(y) + 1); since T is prim.-rec. closed, rank(T) = smallest ordinal not in T.)

Let We show how to find in T (in fact, in Lω(A)) a derivable sentence τ that has no derivation D such that rank(D) ≤ α. Thus, if τ is to have a derivation in T, rank(T) > α. α is arbitrary (< κ(A)), so rank(T) ≥ κ(A). Q.E.D.

In [2] we constructed an infinite set of natural numbers containing no subset of higher (Turing) degree. Since it is well known that there are nonrecursive sets (e.g. sets of minimal degree) containing no nonrecursive subset of lower degree, it is natural to suppose that these arguments may be combined, but this is false. We prove that every infinite set must contain a nonrecursive subset of either higher or lower degree.