Self-Reference Paradoxes

Liar, Russell, Berry — three self-reference tricks that power Gödel.

The idea

Self-reference occurs when a sentence, a set, or a description refers to itself, and it is the mechanism behind the classical paradoxes of logic. By itself it is harmless: "this sentence is written in English" refers to itself and is simply true.

It becomes dangerous when the property a sentence attributes to itself is undone by the attribution. Suppose a sentence asserts of itself that it is not true. Then it is true exactly when it is not, and the assumption that every sentence is either true or false collapses. The same pattern appears without sentences: a set defined to contain exactly the objects that do not contain themselves, or a number specified by a description that the specification disqualifies.

Each of these is a paradox in the strict sense: an argument from assumptions that all seemed acceptable to a contradiction, so at least one assumption must be given up. The self-reference itself is rarely the one to blame, since ordinary sentences refer to themselves without harm. What must go is the property: "true", "definable in English", and "any condition defines a set" are notions a formal system cannot use without restriction.

Ways to work on it

• Walkthrough. "This sentence is false" admits no consistent truth value.
• Practice. The set of all sets not containing themselves — and why naive set theory is broken.
• Hardest. One template behind Liar, Russell, and Berry — apply it to a fresh paradox and name each escape.

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