# Decidability A logic system is decidabile if there is an algorithm that can determine whether a statement always follows from the system's axioms. Hilbert's first concern was the *completeness* of mathematics: is there a way to prove all true propositions within a system of math? Hilbert's second question was about the *consistency* of mathematics: is there a way to prove that no contradictory statements arise in a system of math? Hilbert's third question was about *decidability*: is there an algorithm that can always determine whether a statement follows from the axioms? And that leaves only the third and final big question from Hilbert: is math decidable? That is, is there an algorithm that can always determine whether a statement follows from the axioms? In 1936, Alan Turing found a way to settle this question, but to do it he had to invent the modern computer. --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://mandober.gitbook.io/math-debrief/900-index/enums/enum-logic-system-properties/decidability.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.