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.

Let K be an algebraic number field and IK the ring of algebraic integers in K. *K and *IK denote enlargements of K and IK respectively. Let x Є *K – K. In this paper, we are concerned with algebraic extensions of K(x) within *K. For each x Є *K – K and each natural number d, YK(x,d) is defined to be the number of algebraic extensions of K(x) of degree d within *K. x Є *K – K is called a Hilbertian element if YK(x,d) = 0 for all d Є N, d > 1; in other words, K(x) has no algebraic extension within *K. In their paper [2], P. C. Gilmore and A. Robinson proved that the existence of a Hilbertian element is equivalent to Hilbert's irreducibility theorem. In a previous paper [9], we gave many Hilbertian elements of nonstandard integers explicitly, for example, for any nonstandard natural number ω, 2 ω P ω and 2 ω (ω 3 + 1) are Hilbertian elements in *Q, where p ω is the ωth prime number.

Since Georg Cantor discovered set theory the main problem in this area of mathematical research has been to discover what is the size of the continuum. The continuum hypothesis (CH) says that every infinite set of reals either has the same cardinality as the set of all reals or has the cardinality of the set of natural numbers, namely

In 1939 Kurt Gödel discovered the Constructible Universe and proved that CH holds in it. In the early sixties Paul Cohen proved that every universe of set theory can be extended to a bigger universe of set theory where CH fails. Moreover, given any reasonable cardinal κ, it is possible to build a model where the continuum size is κ. The new technique discovered by Cohen is called forcing and is being used successfully in other branches of mathematics (analysis, algebra, graph theory, etc.).

In the light of these two stupendous works the experts (especially the platonists) were forced to conclude that from the point of view of the classical axiomatization of set theory (called ZFC) it is impossible to give any answer to the continuum size problem: everything is possible!

In private communications Gödel suggested that the continuum size from a platonistic point of view should be ω2, the second uncountable cardinal. As this is not provable in ZFC, Gödel suggested that a new axiom should be added to ZFC to decide that the cardinality of the continuum is ω2.

We refer to the axioms in Quine's book, Mathematical logic, New York, 1940.

To prove the independence of *200, give xϵ α the truth value F in all cases and give (x)ϕ the same truth value as ϕ. Then clearly all formulas derivable from the other axioms besides *200 have the value T, whereas from *200 one can derive (∃x)(∃α)(xϵ α) which has the value F. This method of proving independence amounts to taking for a model a universe consisting of the single object Λ.

For *201 we prove a contingent independence. That is, we prove that if Quine's system is consistent, then *201 is independent. The line of argument is the following. Suppose *201 can be derived from the other axioms. Let us replace xϵ α by throughout all the axioms. Then what *201 becomes can be derived from what the other axioms become. However what *201 becomes will lead to a contradiction in Quine's system whereas the rules which the other axioms become are valid in Quine's system.

We now get down to technical details. Let us refer to the replacement of xϵ α by throughout an expression as an r replacement. Denote the result of performing an r replacement on ϕ by ϕr. Let Wα denote

Then

Note that if x and y are variables, then by D10,

We prove that any speedable computably enumerable set may be split into a disjoint pair of speedable computably enumerable sets. This solves a longstanding question of J.B. Remmel concerning the behavior of computably enumerable sets in Blum's machine independent complexity theory. We specify dynamic requirements and implement a novel way of detecting speedability—by embedding the relevant measurements into the substage structure of the tree construction. Technical difficulties in satisfying the dynamic requirements lead us to implement “local” strategies that only look down the tree. The (obvious) problems with locality are then resolved by placing an isomorphic copy of the entire priority tree below each strategy (yielding a self-similar tree). This part of the construction could be replaced by an application of the Recursion Theorem, but shows how to achieve the same effect with a more direct construction.

An r.e. degree a ≠ 0, 0′ is called strongly noncappable if it has no inf with any incomparable r.e. degree. We show the existence of a high strongly noncappable degree.

In my American Academy paper of 1937 I gave a mathematical clarification and extension of C. I. Lewis's theory of strict implication using as the base (K, T, ×, ′, ≣). In that paper the element a⊰b (read “a hook b”) was defined by (a⊰b) = (ab′=cc′). The main purpose of the present paper is to show that this definition may be replaced by the simpler definition (a⊰b) = (ab≣a), provided Postulate 15 is replaced by two simpler postulates, 15a and 15b (see below). The paper may also serve to throw further light on the essential difference between Lewis's theory of “strict” implication and the more familiar theory of “material” implication. This paper and the earlier one, when read together, are entirely self-contained, all proofs (including proofs of independence) being given in full and no reference to earlier literature being required.

We begin by reviewing briefly the rôle which logical deduction plays in the writing of any scientific book.

Suppose you sit down to write. What do you actually do? In the last analysis, your activity as an author is a process of voluntary selection, restricted by certain inescapable rules. From an indefinite number of possible propositions which may occur to your imagination, you voluntarily select certain propositions which you accept as being worthy of being written down, one after another, in your book. Every proposition which occurs to you is either accepted, or rejected, or laid aside for further consideration.

We consider the modality “ $\varphi $ is true in every $\sigma $ -centered forcing extension,” denoted $\square \varphi $ , and its dual “ $\varphi $ is true in some $\sigma $ -centered forcing extension,” denoted $\lozenge \varphi $ (where $\varphi $ is a statement in set theory), which give rise to the notion of a principle of $\sigma $ -centered forcing. We prove that if ZFC is consistent, then the modal logic of $\sigma $ -centered forcing, i.e., the ZFC-provable principles of $\sigma $ -centered forcing, is exactly $\mathsf {S4.2}$ . We also generalize this result to other related classes of forcing.

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.