Lemma/Proposition/Theorem, which one should we pick?

Solution 1:

There seem to be two issues here. One is why certain well-known results are called Lemmas, such as Zorn's, Yoneda's, Nakayama's, and so on. I don't know the answer to this; presumably it is a mixture of what was written in some original source and the results of the transmission of that original source through the mathematical tradition. (As one interesting example of how labels can be changed in the course of transmission, there is a result in the theory of automorphic forms and Galois representations, very well known to experts, universally referred to as "Ribet's Lemma"; however, in the original paper it is labelled as a proposition!)

The second issue is how contemporary writers label the results in their papers. My experience is that typically the major results of the paper are called theorems, the lesser results are called propositions (these are typically ingredients in the proofs of the theorems which are also stand-alone statements that may be of independent interest), and the small technical results are called lemmas. This probably varies quite a bit from writer to writer (and perhaps also from field to field?).

Solution 2:

I don't know if there are any hard and fast rules, but here is a rough start for others to nitpick:

• A Theorem is a major result that you care about (e.g. "the goal of this paper is to prove the following theorem").
• A Lemma is a useful result that needs to be invoked repeatedly to prove some Theorem or other. Note that sometimes Lemmas can become much more useful than the Theorems they were originally written down to prove.
• A Proposition is a technical result that does not need to be invoked as often as a Lemma.

Solution 3:

While generally the terms are used as suggested by Qiaochu, there are some authors who are bothered by these nebulous subjective terms. For example, Kaplansky wrote in the preface of his classic textbook Commutative Rings

In the style of Landau, or Hardy and Wright, I have presented the material as an unbroken series of theorems. I prefer this to the n-place decimal system favored by some authors, and I have also grown tired of seeing a barrage of lemmas, propositions, corollaries, and scholia (whatever they are). I admit that this way the lowliest lemma gets elevated to the same eminence as the most awesome theorem. Also, the number of theorems becomes impressive, so impressive that I felt the need to add an index of theorems.