turkmath.org

We all know that in Mathematics, there are statements that can neither be proved nor disproved, such as the axiom of choice or the continuum hypothesis. We also sometimes hear about statements that are "true" but "not provable". But what does all this mean? Are such statements unavoidable? The notion of "model of set theory" helps clarifying these issues. Of course, we need to specify the meaning of "proving"! Starting with elementary facts about groups, we shall explain what a complete theory is and why Mathematics is hopelessly incomplete.