 Such results show that even an arithmetical theory such as PA can express statements of mathematical significance (as opposed to statements concocted for logical purposes) that require some detour through the infinite to be proved, even though they can be stated purely arithmetically. In contrast to arithmetic, the mathematical incompleteness of set theory was shown by Gödel and Cohen for important statements such as the Axiom of Choice, the continuum hypothesis, etc. It is important to emphasize here that logicians working in set theory, recursion theory and proof theory probe the mysterious role of the infinite in proving results about the finite. It could be said that set theorists are mainly concerned with understanding how the demonstrable mathematical incompletability of Zermelo-Fraenkel (with Choice, i.e. ZFC) set theory, which is a consequence of results by Gödel and Cohen, can be addressed by finding new principles that will allow us to solve some of the most pressing questions concerned with the structure of the real numbers. In other words, since ZFC cannot be taken as a sufficient basis for the mathematics of infinity much of contemporary set theory is trying to solve the problem by finding new principles, which often take the form of assuming the existence of very large cardinals (see the entry on   independence and large cardinals).   The hope is that this work will lead to settling the Continuum Hypothesis and other major problems about the projective sets (on projective sets see entry on   set theory).