In this article we discuss some elementary concepts in axiomatics, which is the study of axioms. The axiomatic system we use is that of a metric space, which will be familiar to many readers. Metric spaces were introduced by the French mathematician Maurice Fréchet in 1906 as a generalization of the real numbers and the “distance function” f (x , y) = |x − y|. The aim is to construct limits, co...

Starting from a generalization of the standard axioms for a monoid we present a stepwise development of various, mutually equivalent foundational axiom systems for category theory. Our axiom sets have been formalized in the Isabelle/HOL interactive proof assistant, and this formalization utilizes a semantically correct embedding of free logic in classical higher-order logic. The modeling and fo...

Both the Continuum Hypothesis and Martin's Axiom allow induc-tive constructions to continue in circumstances where the inductive hypothesis might otherwise fail. The search for useful related axioms procedes naturally in two directions: towards "Super Martin's Axioms ," which extend MA to broader classes of orders; and towards "Anti-Martin's Axioms" (AMA's) which are strictly weaker than CH, bu...

This research used MiChat and SayHi as materials for forensic investigations using three different tools, namely MOBILedit, Magnet Axiom, Belkasoft. These tools will show each performance in the process. We also added a rooting process an option if data cannot be extracted optimally even when these applications. The result of this study shows that cases studied with processes without root acces...

This study was designed to examine the biomechanical and physiological responses between cycling on the Axiom stationary ergometer (Axiom, Elite, Fontaniva, Italy) vs. field conditions for both uphill and level ground cycling. Nine cyclists performed cycling bouts in the laboratory on an Axiom stationary ergometer and on their personal road bikes in actual road cycling conditions in the field w...