In Mathematics
An effective theory is a formal theory whose set of axioms is recursively enumerable, that is, it is theoretically possible to write a computer program that, if allowed to run forever, would output the axioms of the theory one at a time and not output anything else.
Godel's first incompleteness theorem demonstrates that such a theory cannot at the same time be complete, consistent, and include elementary arithmetic. See also Proof sketch for Gödel's first incompleteness theorem#Hypotheses of the theory.
Read more about this topic: Effective Theory
Famous quotes containing the word mathematics:
“The three main medieval points of view regarding universals are designated by historians as realism, conceptualism, and nominalism. Essentially these same three doctrines reappear in twentieth-century surveys of the philosophy of mathematics under the new names logicism, intuitionism, and formalism.”
—Willard Van Orman Quine (b. 1908)
“Mathematics alone make us feel the limits of our intelligence. For we can always suppose in the case of an experiment that it is inexplicable because we don’t happen to have all the data. In mathematics we have all the data ... and yet we don’t understand. We always come back to the contemplation of our human wretchedness. What force is in relation to our will, the impenetrable opacity of mathematics is in relation to our intelligence.”
—Simone Weil (1909–1943)