Gödel’s Theorems and the Epsilon Calculus

The epsilon calculus contains terms of the form ‘ xFx’ for every predicate in the language. This means that it includes what I shall call ‘empty’ terms (when there are no F s) and also what I shall call ‘indexical’ terms (when there is more than one F ). These particular terms give the epsilon calculus a very distinctive character in comparison with the standard predicate calculus. For instance, an ‘empty’ term is central to understanding how we can do things that Whitehead and Russell’s Principia Mathematica cannot do, in connection with Gödel’s theorems. And attention to ‘indexical’ terms is crucial to solving the major logical paradoxes that have been a puzzle for over a century. Most particularly they are crucial to the solution of what has been called ‘Gödel’s Paradox’, which has been claimed to show that natural language is paraconsistent.