Foundations of Mathematics

Axiomatic System

What are theorems, corollaries, and lemmas?

In mathematics and logic, a theorem is a statement demonstrated to be true by accepted mathematical operations and arguments. In general, a theorem is usually based on some general principle that makes it part of a larger theory; it differs from an axiom in that a proof is required for its acceptance. Some of the more well-known theorems are named after their discoverers, such as the Pythagorean theorem (involving right triangles) and Fermat’s last theorem. It is interesting to note that American Richard Feynman (1918–1988), one of the most brilliant physicists of the 20th century, stated that any theorem, no matter how difficult to prove in the first place, is viewed as “trivial” by mathematicians once it has been proved. Thus, according to Feynman, there are only two types of mathematical objects: trivial ones and those that have not yet been proved.

A corollary is a theorem that has been proved in only a few steps from an established theorem, or one that follows as a direct consequence of another theorem or axiom. And finally, a lemma is a theorem proved as a preliminary or intermediate step in the proof of another, more basic theorem; or a brief theorem used to prove a larger theorem.



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