What is the difference between lemma, axiom, definition, corollary, etc?

Question

In learning basic analysis I am encountering a lot of language that seems somewhat tough to define:

Axiom, proposition, definition, lemma, theorem, law, corollary

Are there clear separations in definitions between all these things or is there a lot of overlap? Or is it sometimes a judgment call? How do most people use these words?

Answer 1

There is often a difference, although usage might depend on author / language /... I will be as concrete as possible, at the expense of being a bit sloppy:

• Axiom: a fundamental logical statement that you assume to be true in order to build a theory. Nothing grows out of nothing: even to construct logic or mathematics you need to start from some assumptions that you just accept as reasonable.
• Definition: one cannot do mathematics using just logical symbols: it is just too cumbersome. Often one introduces simplifications, notations, names to talk about things that come up frequently. It is an agreement about calling something in a certain way.
• Lemma: a true statement that can be proved (proceeding from other true statements or from the axioms) and that is immediately (or almost immediately) used to prove something more important (a theorem / proposition).
• Theorem: an important and/or difficult to prove true mathematical statement.
• Proposition: a true mathematical statement that is not as important / difficult as a theorem. Let's say, an ordinary true mathematical statement.
• Corollary: a true mathematical statement that follows quite directly as a consequence of a theorem or proposition (e.g. as a special case).
• Law: not very much used in pure mathematics, it is more common e.g. in physics to refer to a true fact about nature.

Note that sometimes tradition gets in the way: you can accept Zorn's Lemma as an axiom, and similarly there are lemmas that have become theorems in theor own right. There is not always a sharp distinction!

Answer 2

Some are quite different than others:

Axiom: This is what you are taking to be the ground truth. For instance, Peano axioms axiomatize natural numbers, and you can use Dedekind cuts to axiomatize reals (You can read Dedekind's cut and axioms for more info)

You can use this somewhat interchangably with the word "definition". (You can say this is how you "define" natural or real numbers), but generally a definition goes as:

"Defintion: We say a natural number is BOO if it satisfies the following axioms: 1) If you add one to any number which is BOO, the resulting number is not BOO. 2) If you add one to any number which is not BOO, the resulting number is BOO. 3) 1 is not BOO.

Now, let's make a proposition. (Generally, this just means a statement that can be either true or false. It is something that they propose; this is I think mostly used if they are not going to prove it for a while.)

Proposition: "Every BOO number is even."

Theorem: This is essentially a mathematical truth; anyone claiming one of these better give you a proof of it. Since we will prove the above proposition, let's rewrite it as:

Theorem: "Every BOO number is even."

Now, to help us prove it, we are going to prove two mini-theorems, more commonly referred to as lemmas.

Lemma 1: "2 is BOO"

Proof: 1 is not BOO by axiom 3, so by axiom 2, 1+1 = 2 is BOO.

Lemma 2: "If n is BOO, n+2 is BOO"

Proof: If n is BOO, by axiom 1, n+1 is not in BOO, but then by axiom 2, (n+1)+1 = n+2 is in BOO.

Now, the proof of the theorem would be to show that every number of the form 2 + 2 + 2 + ... + 2 would be divisible by 2. You can use the fact that multiplication by 1/2 distributes over summation, but that would require perhaps a lot more definitions than what you want to use. You can also use something called induction, which is a proof method. It depends on how you want to define even-ness.

Now, let's write a corollary from our theorem:

Corollary: 2 is even. (We know this follows from our theorem, since we proved earlier that every BOO number is even and we know that 2 is BOO.)

Law: It refers to big observations, you can write a "law of BOOs as: Every number is either BOO or BOO + 1." Laws are not that common, but we have a few big ones: Law of large numbers, and De Morgan's laws. You can imagine them to be very big or useful theorems. (There are a lot of big and useful theorems like central limit theorem that are not called laws).

Hope that answers it with a toy example!