This paper discusses certain problems arising within the treatment of the senses of functions in Alonzo Church's Logic of Sense and Denotation. Church understands such senses themselves to be “sense-functions,” functions from sense to sense. However, the conditions he lays out under which a sense-function is to be regarded as a sense presenting another function as denotation allow for certain undesirable results given certain unusual or “deviant” sense-functions. Certain absurdities result, e.g., an argument can be found for equating any two senses of the same type. An alternative treatment of the senses of functions is discussed, and is thought to do better justice to Frege's original theory.

This little gem is stated unbilled and proved (completely) in the last two lines of §2 of the short note Kleene [1938]. In modern notation, with all the hypotheses stated explicitly and in a strong (uniform) form, it reads as follows:

Second Recursion Theorem (SRT). Fix a set V ⊆ ℕ, and suppose that for each natural number n ϵ ℕ = {0, 1, 2, …}, φn: ℕ1+n ⇀ V is a recursive partial function of (1 + n) arguments with values in V so that the standard assumptions (a) and (b) hold with

.

(a) Every n-ary recursive partial function with values in V is for some e.

(b) For all m, n, there is a recursive function : Nm+1 → ℕ such that

.

Then, for every recursive, partial function f of (1+m+n) arguments with values in V, there is a total recursive function of m arguments such that

Proof. Fix e ϵ ℕ such that and let .

We will abuse notation and write ž; rather than ž() when m = 0, so that (1) takes the simpler form

in this case (and the proof sets ž = S(e, e)).

We present a survey of some results and problems concerning constructions which require a diagonalization of length continuum to be carried out, particularly constructions of almost disjoint families of various sorts. We emphasize the role of cardinal invariants of the continuum and their combinatorial characterizations in such constructions.

We carry out some of Galois' work in the setting of an arbitrary first-order theory T. We replace the ambient algebraically closed field by a large model M of T, replace fields by definably closed subsets of M, assume that T codes finite sets, and obtain the fundamental duality of Galois theory matching subgroups of the Galois group of L over F with intermediate extensions F ≤ K ≤ L. This exposition of a special case of [10] has the advantage of requiring almost no background beyond familiarity with fields, polynomials, first-order formulae, and automorphisms.