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Added section on undecidability and faith, as discussed on the talk page.
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 [[atom]]ic 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 [[atom]]ic 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>.
==Undecidability and Faith==
The existence of undecidable statements proves the illogic of atheistic attempts to demand proof for the existence of God. Because we know that there exist true statements that we can never logically prove, there are necessarily things that must be believed on the basis of faith, rather than logic. The atheistic mantra that "the burden of proof lies with the believer" ignores this classic result in mathematical logic, and exposes their ignorance. It is notable that [[Kurt Godel]] who demonstrated the existence of undecidability in mathematical logic was a devoutly religious man.
==Famous Undecidable Statements==
==Famous Undecidable Statements==