Self-reference and the limits of  logic

Ancient Greece: Epimenides of Crete -- "All Cretans are liars."  If he's telling the truth, he must be lying, but if he's lying, then he's telling the truth.


18th century: Barber of Seville -- Everyone in town has barber cuts their hair, except those who cut their own hair. Who cuts the hair of the barber?


Late 19th century: A. N. Whitehead: set theory will create universal axiomatic foundation for complete and self-consistent mathematics. B. Russell: What about the set of all sets which do not contain themselves?”


Early 20th century: Vienna Circle -- “Principia Mathematica,” (e.g. Logical Type Theory) to remove self-referential paradox.


1931: Godel’s Theorem – Typographical Number Theory created to show that no system can fully represent mathematics unless it is powerful enough to do self-reference, and no system with full self-reference can escape self-contradictory statements. Therefore, there will always be theorems whose truth or falsehood cannot be ascertained in any powerful system of mathematics.