undecidable proposition

logic

Written by

The Information Architects of Encyclopaedia Britannica

Last updated

July 14, 2026

Quick Summary

An undecidable proposition is a statement within a formal system that cannot be proven true or false using the system's axioms and rules of inference. This concept emerged from Kurt Gödel's incompleteness theorems in 1931, which demonstrated that any sufficiently complex formal system (like those used for arithmetic) will inevitably contain true statements that are unprovable within that system.12

The development of the concept of "mechanical procedure" or algorithm by mathematicians like Alonzo Church and Alan Turing was crucial in formalizing undecidability. It established that certain problems, such as the halting problem for Turing machines, are absolutely unsolvable, meaning no algorithm can exist to solve them for all cases. This implies that for many formal systems, there is no decision procedure to determine the truth or provability of every statement within them.234

Summary created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

4 Britannica Sources

Ask Britannica AI about this topic

Britannica doesn't have a full article on this topic, so we've curated a list of our relevant coverage.

foundations of mathematics
Recursive definitions
…formal mathematical system will contain undecidable propositions—propositions which can be neither proved nor disproved. Church and Turing, while seeking an algorithmic (mechanical) test for deciding theoremhood and thus potentially deleting nontheorems, proved independently, in 1936, that such an algorithmic method was impossible for the first-order predicate logic (see logic, history…
Read More
…formal mathematical system will contain undecidable propositions—propositions which can be neither proved nor disproved. Church and Turing, while seeking an algorithmic (mechanical) test for deciding theoremhood and thus potentially deleting nontheorems, proved independently, in 1936, that such an algorithmic method was impossible for the first-order predicate logic (see logic, history…
Read More

Turing machine
… tool that could infallibly recognize undecidable propositions—i.e., those mathematical statements that, within a given formal axiom system, cannot be shown to be either true or false. (The mathematician Kurt Gödel had demonstrated that such undecidable propositions exist in any system powerful enough to contain arithmetic.) Turing instead proved that there…
Read More
… tool that could infallibly recognize undecidable propositions—i.e., those mathematical statements that, within a given formal axiom system, cannot be shown to be either true or false. (The mathematician Kurt Gödel had demonstrated that such undecidable propositions exist in any system powerful enough to contain arithmetic.) Turing instead proved that there…
Read More