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.

** **

**18 ^{th}
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
19 ^{th} 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
20 ^{th} 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.