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New page: A statement in formal logic is called undecidable if there is no proof or disproof of the statement in formal logic. A common misconception is that undecidable statements have no truth val...
A statement in formal logic is called undecidable if there is no proof or disproof of the statement in formal logic. A common misconception is that undecidable statements have no truth value, but this statement is not true. For example, many set theorists now believe that the [[continuum hypothesis]] (which is known to be undeciable in ZFC) is actually false.
As a more concrete example, suppose we had a "theory of colored shapes" <math>T</math> where the objects were colored shapes (red triangles, blue squares, etc), and the possible atomic sentences were of the form "shape <math>S</math> is a square." Since we do not have the language to state "shape <math>S</math> is red" then any statement of this form is undecidable in <math>T</math>, though it could be true or false in some larger theory <math>T'</math>.
As a more concrete example, suppose we had a "theory of colored shapes" <math>T</math> where the objects were colored shapes (red triangles, blue squares, etc), and the possible atomic sentences were of the form "shape <math>S</math> is a square." Since we do not have the language to state "shape <math>S</math> is red" then any statement of this form is undecidable in <math>T</math>, though it could be true or false in some larger theory <math>T'</math>.