A conditional Poisson process is observed, whose rate is a diffusion process of known structure. The problem is to estimate the rate from the observed point process. Recursive equations are given for the conditional moment generating function and for an unnormalized conditional probability density of the rate. By studying these equations separately in between jumps and at the jumps, series expansions are obtained for these generating functions and densities in a number of examples that arise in applications to optical modulation and communications networks.

We prove the following theorem: If S is a consistent set of axioms for set theory, among whose consequences are the axioms of pairing and extensionality, and if S′ is the extension of S to all classes of sets of S, then S′ is consistent and remains so upon addition of an axiom expressing that the universal class has a well-ordering wherein every non-initial element has an immediate predecessor. The basic idea will be use of the denumerable model of S guaranteed by the Skolem-Löwenheim theorem to obtain (i) a model of S′ in the domain of classes of natural numbers, and (ii) a correspondence between representatives (in the model of S′) of sets of S′ and natural numbers. One can then select a class of natural numbers in the model of S′ which will well-order (in the sense of the model of S′) the representatives of sets of S′ according to the numbers corresponding to them. Specifically, let the formulas of S involve only one kind of variable and the sole predicate ‘ϵ’. The formulas of S′ can be thought of as identical with those of S; but the axioms of S′ will consist of (i) those of S with quantifiers restricted to sets by clauses ‘(∃z)[y ϵ z]’, (ii) those provided by the schema ‘(∃x) ▪ y ϵ x ≡y [(∃z)[y ϵ z] ▪ A]’, where ‘A’ is any formula of S′ in which ‘x is not free, and (iii) the axiom of extensionality.

1. In this paper we construct a model for part of the system NF of [4]. Specifically, we define a relation R of natural numbers such that the R-relativiseds of all the axioms except P9 of Hailperin's finitization [2] of NF become theorems of say Zermelo set theory. We start with an informal explanation of the model. 2. Scrutiny of P1-P8 of [2] suggests that a model for these axioms might be constructed by so to speak starting with a universe that contained a “universe set” and a “cardinal 1”, and passing to its closure under the operations implicit in P1-P7, viz., the Boolean, the domain, the direct product, the converse, and the mixtures of product and inverse operations represented by P3 and P4. To obtain such closure we must find a way of representing the operations that involve ordered pairs and triples. We take as universe of the model the set of natural numbers ω; we let 0 represent the “universe set” and 1 represent “cardinal 1”. Then, in order to be able to refer in the model to the unordered pair of two sets, we determine all representatives of unordered pairs in advance by assigning them the even numbers in unique fashion (see d3 and d25); we can now define the operations that involve ordered pairs and triples, and obtain closure under them using the odd numbers. It remains to weed out, as in d26, the unnecessary sets so as to satisfy the axiom of extensionality.