Gödel's Incompleteness Theorems

Gödel's Incompleteness Theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system capable of modelling basic arithmetic. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics.

Overview

The theorems, known as the First Incompleteness Theorem and the Second Incompleteness Theorem, deal with the limitations of formal systems. The first theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers. The second theorem builds upon the first, stating that such a system cannot demonstrate its own consistency.

First Incompleteness Theorem

The First Incompleteness Theorem essentially states that in any consistent formal system F within which a certain amount of arithmetic can be done, there are statements of the language of F which can neither be proved nor disproved in F. This means that there are some truths within the system that cannot be derived from the axioms.

Second Incompleteness Theorem

The Second Incompleteness Theorem is an extension of the first, stating that no consistent system can prove its own consistency. This means that the system cannot demonstrate that there are no contradictions within it.

Implications

The implications of Gödel's Incompleteness Theorems are far-reaching. They have been interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible. The theorems also imply that there can never be a mathematical "theory of everything", a complete list of axioms that can prove all mathematical truths.

See also

• Kurt Gödel
• Formal system
• Axiom
• Arithmetic
• Hilbert's program

References

• Gödel, Kurt (1931). "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I". Monatshefte für Mathematik und Physik. 38: 173–198.