3 0 Introduction 3 Part I 1 Formulation of the various systems 7 2 Theorems of various systems 13 On ReS and finite sets 13 Proof that HF models ZF – Infinity 16 On GJ and the class of rudimentary functions 20 Companions and the Gandy–Jensen Lemma 21 A single generating function for rud(u) 22 Other remarks on GJ 23 On fReR 25 On ReRI 26 On KP 27 3 Remarks on transitive models 27 4 Models of ReS...

New inequalities for the general case of convex functions defined on linear spaces which improve the famous Jensen’s inequality are established. Particular instances in the case of normed spaces and for complex and real n-tuples are given. Refinements of Shannon’s inequality and the positivity of Kullback-Leibler divergence are obtained.