We define, in the spirit of Fenstad [2], a higher type computation theory, and show that countable recursion over the continuous functionals forms such a theory. We also discuss Hyland's proposal from [4] for a scheme with which to supplement S1–S9, and show that this augmented set of schemes fails to generate countable recursion. We make another proposal to which the methods of this section do not apply.

Dowker [1] raised the question of the existence of filters such that for every coloring (partition) of the underlying index set I with two colors there is a relation R on I which (i) is fat (in the sense that sets of the form {y Є I∣xRy} are in the filter) and (ii) has no bichromatic symmetric pairs (i.e., distinct indices x and y such that x R y and y R x). Additionally, he required that the filter have no anti-symmetric fat relation, for such a relation would vacuously satisfy (i) and (ii). The question of the existence of Dowker filters has been studied more recently by Rudin [3], [4], who conjectures [3] that such filters do not exist.

For ZFC the problem remains open. However, Example 2 of this paper shows that one can construct a Dowker filter provided one drops the axiom of choice in favor of the Baire Property (BP) axiom which is known to be incompatible with ZFC but relatively consistent with ZF. In fact, the filter constructed is super-Dowker in the sense that (ii) can be replaced by the requirement that all components of all symmetric pairs have the same color. But, in ZFC the existence of a super-Dowker filter implies the existence of a measurable cardinal.

Let F be a filter on an index set I. A set will be called big, small, or medium depending on whether F contains that set, its compliment, or neither, respectively. We define five cardinals associated with F:

α denotes the smallest cardinal such that there is a family of α big sets whose intersection is not big.

ν denotes the smallest cardinal such that there is a family of ν big sets whose intersection is small.

Let ZFB be ZF + “every set is the same size as a wellfounded set”. Then the following are true.

Every sentence true in every (Rieger-Bernays) permutation model of a model of ZF is a theorem of ZFB. (i.e., ZFB is the theory of Rieger-Bernays permutation models of models of ZF) ZF and ZFAFA are both extensions of ZFB conservative for stratified formulæ. The class of models of ZFB is closed under creation of Rieger-Bernays permutation models.

In [3], Shelah defined the cardinals κn(T) and , for each theory T and n < ω. κn(T) is the least cardinal κ without a sequence (pi)i<κ of complete n-types such that pi is a forking extension of pj for all i < j < κ. It is essential in computing the stability spectrum of a stable theory. On the other hand is called the number of independent partitions of T. (See Definition 1.2 below.) Unfortunately this invariant has not been investigated deeply. In the author's opinion, this unfortunate situation of is partially due to the fact that its definition is complicated in expression. In this paper, we shall give equivalents of which can be easily handled.

In §1 we shall state the definitions of κn(T) and . Some basic properties of forking will be stated in this section. We shall also show that if = ∞ then T has the independence property.

In §2 we shall give some conditions on κ, n, and T which are equivalent to the statement . (See Theorem 2.1 below.) We shall show that does not depend on n. We introduce the cardinal ı(T), which is essential in computing the number of types over a set which is independent over some set, and show that ı(T) is closely related to . (See Theorems 2.5 and 2.6 below.) The author expects the reader will discover the importance of via these theorems.

Some of our results are motivated by exercises and questions in [3, Chapter III, §7]. The author wishes to express his heartfelt thanks to the referee for a number of helpful suggestions.

The main purpose of this paper is to show how partial recursive functions and isols can be used to generalize the following three well-known theorems of combinatorial theory.

(I) For every finite projective plane Π there is a unique number n such that Π has exactly n 2 + n + 1 points and exactly n 2 + n + 1 lines.

(II) Every finite projective plane of order n can be coordinatized by a finite planar ternary ring of order n. Conversely, every finite planar ternary ring of order n coordinatizes a finite projective plane of order n.

(III) There exists a finite projective plane of order n if and only if there exist n − 1 mutually orthogonal Latin squares of order n.

The notion of a ptyx (plural, ptykes) is obtained by generalising the ordinals On and the dilators Dil introduced by Girard [1] into a typed hierarchy. The notion is due to Girard, and a detailed treatment will appear in Girard [2].

In this Introduction we will give a summary of the theory of ptykes. It will be an advantage to be familiar with the theory of dilators as introduced in Girard [1] or as presented in Girard and Normann [3].

The class On of ordinals is organised into a category ON by using strictly increasing maps f: x → y as morphisms. A dilator will be a functor F: ON → ON commuting with pullbacks and direct limits. Associated with each dilator F there is a denotation system DF, obtained as follows: For each x ∈ On and y ∈ F(x), y can be given a unique denotation (c; x0,…, xn−1;x)F, where c∈F(n) (n = {0,…, n − 1}), y = F(ϕ)(c) (where ϕ is defined by ϕ(i) = xi for i < n), and for no m < n and ψ: m → n do we have c ∈ im(ψ).

We let the trace Tr(F) be the set of pairs (c, n) occurring in a denotation.

In [1], page 85, Quine states: “… ponentials of logically true statements are logically true.” It is perhaps of some interest that this assertion is not completely precise as can be seen from counterexamples. One simple example is the following:

Let L be some language in which ‘c’ and ‘b’ are atomic sentential constants. Let ‘c’ be false and ‘b’ true. Suppose that L includes parentheses and the material conditional ‘⊃’ as logical notation, but that ‘b’, at least, counts as extra-logical.