From Sudopedia, the free Sudoku reference guide

From Sudopedia

In a domain of logical discourse, an axiom is something that cannot be derived from the other truths in the system by logical reasoning; it must be assumed to be true.

For the solver of a Sudoku, Uniqueness is an axiom--if it is to be used in deductions, it has to be assumed to be true without proof.

Since Uniqueness is an axiom, I have removed the quotes around the word axiom in the last paragraph. The quotes implied that it is called an axiom but isn't really a genuine one.

Professor Prune 12:56, 10 February 2009 (EDT)

For me, part of the problem is proving the uniqueness, therefore using it as an axiom is not valid (IMO) --Scraggy 22:56, 5 November 2009 (CET)

Uniqueness Theorem

A proof to the following theorem would certainly help in this debate:

 A proper Sudoku can always be solved using proper solving techniques.

A proper solving technique in this case means a technique that does not assume the uniqueness condition.

In other words, if you can solve every Sudoku that can be solved by a non-proper solving technique, using some other (proper) technique as well, then the uniqueness condition is valid for all solvable Sudokus. A Sudoku is not solvable if it does not have a unique solution...