General Axioms Research Articles

On the principle of dependent choices

Publisher Summary This chapter discusses the general axiom of choice (denoted by (Z)) is independent of (T), that is, cannot be proved from (T) and the usual axioms of set-theory. An independence-proof has sense only with respect to a well-defined formal system whose consistency is either proved or assumed as an hypothesis. Proof applies only to such systems of set-theory that remain self-consistent after the adjunction of the following axiom: there is a non-denumerable set of elements which are not sets.

PYLORIC STENOSIS IN NONIDENTICAL TWINS

The occurrence of malformations in twins has always incited the interest of medical observers. In the last two decades such malformations have been of special note in view of the introduction of the theory of genetic pathogenesis of some tumors. The incidence of congenital pyloric stenosis in twins has been a bulwark in the defense of this theory because it has indicated few exceptions to the general axiom that heterologous twins never suffer from identical malformations, nor does one of homologous twins ever suffer from malformation not shared by its fellow. 1 There have been many cases to suggest and to give support to the belief that there is a genetic basis for pyloric stenosis, the exact nature of which is not known. The fact that in four fifths of the cases the disease occurs in males attests to that, as does the report of Fabricius and Vogt-Moller 2 on

On the Equations of Laplace and Maxwell

An attempt is made to deduce the equations of electrostatics and of electrodynamics from general axioms instead of quantitative experiments. Among the axioms necessary to find the first are: the existence of a field, determined by differential equations without action at a distance, the principle of superposition, isotropy of space and existence of a potential. From Poisson's equation Maxwell's equations follow with four additional assumptions, the chief of which is the validity of the restricted theory of relativity.