Flow: the Axiom of Choice is independent from the Partition Principle Adonai S. Sant’Anna Ot´avio Bueno Marcio P. P. de Fran¸ca Renato Brodzinski For correspondence:
[email protected] Abstract We introduce a general theory of functions called Flow. We prove ZF, non-well founded ZF and ZFC can be immersed within Flow as a natural consequence from our framework. The existence of strongly inaccessible cardinals is entailed from our axioms. And our first important application is the introduction of a model of Zermelo-Fraenkel set theory where the Partition Principle (PP) holds but not the Axiom of Choice (AC). So, Flow allows us to answer to the oldest open problem in set theory: if PP entails AC. This is a full preprint to stimulate criticisms before we submit a final version for publication in a peer-reviewed journal. 1 Introduction It is rather difficult to determine when functions were born, since mathematics itself changes along history. Specially nowadays we find different formal concepts associated to the label function. But an educated guess could point towards Sharaf al-D¯ınal-T¯us¯ıwho, in the 12th century, not only introduced a ‘dynamical’ concept which could be interpreted as some notion of function, but also studied arXiv:2010.03664v1 [math.LO] 7 Oct 2020 how to determine the maxima of such functions [11]. All usual mathematical approaches for physical theories are based on ei- ther differential equations or systems of differential equations whose solutions (when they exist) are either functions or classes of functions (see, e.g., [26]). In pure mathematics the situation is no different.