Now this statement cannot be true since this would make Epimenides a liar, leading to the falsity of his statement. The statement cannot be false either, since this would imply that Cretans are not liars, hence what Epimenides says should be true, leading to a contradiction. This follows from -Collection as follows. Given , there exists a set . So, by -Collection, there exists a set such that for all .
Circumscription is a tool to jump to certain conclusions.
The main point of circumscription is that the objects that can be shown to have a certain property by reasoning from certain facts are all the objects that satisfy . Thus one circumscribes the set of objects. The is the notation for first writing all symbols in terms of and , then passing negations in through to predicate letters, and finally replacing each occurrence of a subformula in the result by . Hence, For any infinite cardinal m , the set is defined as . The elements of are said to be hereditarily of cardinality less than m .