||At the beginning mathematics enabled a systematic approach to popular games giving advantage to the perceptive player, however nowadays it is also common to reformulate conjectures and theorems in term of games, in a way that if one were able to find a winning strategy (a way of playing which always leads to victory) then the original statement would be automatically proven.
In this talk we will make a short excursion into the land of Game Theory. I will address finite and infinite games, and give some examples and applications. A game would be called determined if there is a winning strategy. It is not clear whether all games may have a winning strategy. In fact this property can be thought as a new axiom of Set Theory, and it is usually called Axiom of Determinacy. After formalizing the notion of game, winning strategy, and show some general facts, I will prove a couple of shocking set theoretical consequences of this new axiom, in particular its relation with other well-known controversial axioms: Axiom of Choice and the Continuum Hypothesis.