1.7: Russell’s Paradox
As the ideas for set theory were explored, there were attempts to define sets as broadly as possible. It was hoped that any collection of mathematical objects that could be defined by a formula would qualify as a set. This belief was known as the General Comprehension Principle (GCP). Unfortunately, the GCP gave rise to conclusions which were unacceptable for mathematics. Consider the collection defined by the following simple formula: \[V=\{x \mid x \text { is a set and } x=x\} .\] If \(V\) is considered as a set, then since \(V=V\) , \[V \in V \text {. }\] If this is not an inconsistency, it is at least unsettling. Unfortunately, it gets worse. Consider the collection \[X=\{x \mid x \notin x\} .\] Then \[X \in X \text { if and only if } X \notin X \text {. }\] This latter example is called Russell’s paradox, and showed that the GCP is false. Clearly there would have to be some control over which definitions give rise to sets. Axiomatic set theory was developed to provide rules for rigorously defining sets. We give a brief discussion in Appendix B.