we study different completeness definitions for two categories of lattice-valued cauchy spaces and the relations between these definitions. we also show the equivalence of a so-called completion axiom and the existence of a completion.

Nominalistic Logic (NL) is a new presentation of Paul Gilmore’s Intensional Type Theory (ITT) as a sequent calculus together with a succinct nominalization axiom (N) that permits names of predicates as individuals in certain cases. The logic has a flexible comprehension axiom, but no extensionality axiom and no infinity axiom, although axiom N is the key to the derivation of Peano’s postulates ...

Veblen has given a set of axioms for geometry.!. The first eight of these are as follows : % Axiom I. There exist at least two distinct points. Axiom II. If points A, B, C are in the order ABC, they are in the order CBA. Axiom III. If points A, B, C are in the order ABC, they are not in the order BCA. Axiom IV. If points A, B, C are in the order ABC, then A is distinct from C. Axiom V. If A and...

The reals, of course, already have an axiom system. They are a complete, linearly ordered field. This axiom system is even categorical, meaning that it completely characterizes the reals. Up to isomorphism, the reals are the only complete, linearly ordered field. Another property of axiom systems, considered to be particularly elegant ever since the birth of formal logic, is independence. In an...

The chaotic hypothesis discussed in [GC1] is tested experimentally in a simple conduction model. Besides a confirmation of the hypothesis predictions the results suggest the validity of the hypothesis in the much wider context in which, as the forcing strength grows, the attractor ceases to be an Anosov system and becomes an Axiom A attractor. A first test of the new predictions is also attempted.