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An axiomatic theory (such as a geometry) is said to be complete if each valid statement in the theory is capable of being proven true or false.

A theory is decidable iff there is an algorithm which can determine whether or not any sentence r is a member of the theory.

The paradox "This statement is false," stated in the fourth century BC. It is a sharper version of the Epimenides paradox, "All Cretans are liars...One of their own poets has ...

A disjunction that is true if only one, but not both, of its arguments are true, and is false if neither or both are true, which is equivalent to the XOR connective. By ...

If at least one solution can be determined for a given problem, a solution to that problem is said to exist. Frequently, mathematicians seek to prove the existence of ...

A disjunction that remains true if either or both of its arguments are true. This is equivalent to the OR connective. By contrast, the exclusive disjunction is true if only ...

A formal theory is said to be incomplete if it contains fewer theorems than would be possible while still retaining consistency.

A set of maximum degree to which all other degrees of recursively enumerable sets can be many-one reduced. If set A is many-one complete, then it is one-one complete, and ...

The operation of interchanging true and false in a logical statement. The negation of A is often called "NOT-A," and can be denoted !A, or with the negation sign ¬, so not-A ...

If A=>!B and B=>!A (i.e., (A=>!B) ^ (B=>!A), where !A denotes NOT, => denotes implies, and ^ denotes AND), then A and B are said to be inequivalent, a relationship which is ...