What is a Set ?

A set is a tree. To specify a set we can list all the elements that belong to it and then all the elements that belong to each one of these elements and so forth (rigorously this is to be understood in terms of ordinals and assuming the axiom of choice). In Cohen's forcing technique there is the notion of a $P$-name, where $P$ is some set endowed with a preorder. A $P$-name is simply a set viewed as a tree but with each node labelled with an element of $P$. An ordinary set can be seen as a $P$-name with its tree's nodes tagged with the top element 1 of $P$. Given a $P$-name and a subset $G\subseteq P$ we can obtain a normal set by going through the tree and prunning off nodes whose labels do not belong to $G$ and then forgetting the labels and considering the resulting set - this is the interpretation or evaluation map. A particular case is in which $P$ is the set of all Borelian sets of the real interval $I$. A $P$ -name is then a probabilistically described set.