Lemma vs. Theorem

Question

I've been using Spivak's book for a while now and I'd like to know what is the formal difference between a Theorem and a Lemma in mathematics, since he uses the names in his book. I'd like to know a little about the ethymology but mainly about why we choose Lemma for some findings, and Theorem for others (not personally, but mathematically, i.e. why should one classify a finding as lemma and not as theorem). It seems that Lemmas are rather minor findings that serve as a keystone to proving a Theorem, by that is as far as I can go.

NOTE: This question doesn't address my concern, so please avoid citing it as a duplicate.

Answer 1

First off there is no "formal difference" between a theorem and a lemma. Formally, if you view mathematics from the perspective of set theory (ZFC), you must conclude that anything commonly called a "lemma" in the literature is by definition "a theorem of ZFC," i.e. a finite sequence of true formulas of ZFC which flow logically from one formula to the next ending on a formula representing the statement of the theorem.

So, lemmas are invoked with literary freedom that it be understood that they really are theorems, but somehow "little ones". But why bother?

A lemma comes typically in two forms: (i) a useful trick or (ii) a technical step in a proof. Let me demonstrate some examples.

A useful trick in real analysis is called "Fatou's Lemma," which helps us interchange limit operations and integrals. Very roughly, it states that

"if $\displaystyle\lim_{n \rightarrow \infty} f_n(x) \rightarrow f(x)$ for all $x$, then

$$\int \lim f_n(x) dx = \int f(x) dx \leq \lim \displaystyle\int f_n(x) dx ,"$$

which, it turns out, becomes "half of the work" in proving a lot of very useful and frequently used inequalities like the Monotone Convergence Theorem and Lebesgue's Dominated Convergence Theorem. On its own, Fatou's Lemma is not so remarkable, and it quickly becomes a minor routine step in very major and fundamental theorems in real analysis -- this is why it is itself a lemma, not a theorem.

Another good example of a theorem of the (i) type is "Zorn's lemma". Zorn's lemma is a technical statement about partially ordered sets but it is invoked frequently in proofs studying ideals in ring theory (I'm sure it has many more uses but I'm unfamiliar with them).

The strange thing about Zorn's lemma is that it is logically equivalent to the Axiom of Choice, i.e. from Zorn's lemma you can prove the Axiom of Choice and from the Axiom of Choice you can prove Zorn's lemma. In other words, if you studied the axioms of set theory but instead of assuming the axiom of choice you assumed Zorn's Lemma as an axiom (let's call this Zorn's Axiom for now), then you could eventually deduce the Axiom of Choice (perhaps Lemma of Choice?) as a consequence of Zorn's Axiom. So Zorn's lemma is a lemma ONLY BECAUSE we assume the Axiom of Choice rather than Zorn's lemma as an axiom of standard set theory: it is a lemma only because of how we choose to organize mathematics.

A type (ii) lemma is something highly technical that, if proven in the middle of the theorem you really are trying to prove, you may have difficulty getting back on track since it takes too long. This happens ALL THE TIME in mathematics. Here is an example I came across recently from the proof of Dirichlet's theorem on arithmetic progressions in Tom Apostol's "Introduction to Analytic Number Theory":

Theorem (Dirichlet's Theorem): If $h$ and $k$ are relatively prime integers, then there are infinitely many primes in the arithmetic progression $\{hn+k \colon n = 1,2,3,\ldots\}$.

To prove this theorem, he proves a number of lemmas, such as

Lemma 7.4: If $x > 1$ we have

$$\displaystyle\sum_{p \leq x; p \equiv h (mod k)} \frac{\log p}{p} = \frac{1}{\phi(k)} \log x + \frac{1}{\phi(k)} \displaystyle\sum_{r=2}^{\phi(k)} \overline{\chi_r(h)} \displaystyle\sum_{p \leq x} \frac{\chi_r(p)\log p}{p} + \mathscr{O}(1),$$

and

Lemma 7.5 For $x > 1$ and $\chi \neq \chi_1$, we have

$$\displaystyle\sum_{p \leq x} \frac{\chi(p)\log p}{p} = -L_{\chi}'(1) \displaystyle\sum_{n \leq x} \frac{\mu(n)\chi(n)}{n} + \mathscr{O}(1),$$

and so forth. He has, in total, about 5 or 6 such lemmas which are steps in the proof of the theorem stated above. The reason these things, while complicated and substantial (far more than Fatou's lemma!), are called lemmas, is that if you began proving Dirichlet's Theorem and proved these in the middle of that proof, you would easily get lost.

So really, what a lemma is to you is whatever you want it to be. It is a word that exists in our vocabulary that is part of the proper name of a concept like Zorn's lemma or it can be simply a word to promote a more readable exposition.

Answer 2

@Peter asked about the etymology. Both words are Greek, of course. “Lemma” comes from the verb “lambano”, which means I take, and my handy little Greek dictionary gives the meaning of lemma to be "income, gain, gratification, profit”! How this came to have a mathematical meaning I have no idea of. “Theorem” give less cause for wonder: it comes from “theoro”, which means “I look at, view, behold, observe”, and the derivative noun theorema has the dictionary meaning “sight, spectacle”. I wish it had meant “observation”, but this seems not so, at least in the classic period.

Answer 3

Lemmas are results that help prove theorems. Usually, they provide the intermediate steps on the way to proving a theorem. They are typically not the main result you aim to prove. Nonetheless, some lemmas are so useful that they become among the most important tools in an area, like the lemma Bruno mentioned, or Urysohn's Lemma, or Schwarz's Lemma, or Nakayama's Lemma, or...

Answer 4

While generally the term is used as suggested in the other answer(s), it's worth mention that some esteemed authors reject 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.

Answer 5

There is no mystery regarding the use of these terms: an author of a piece of mathematics will label auxiliary results that are accumulated in the service of proving a major result lemmas, and will label the major results as propositions or (for the most major results) theorems. (Sometimes people will not use the intermediate term proposition; it depends on the author.)

Exactly how this is decided is a matter of authorial judgement.

There is a separate issue, which is the naming of certain well-known traditional results, such as Zorn's lemma, Nakayama's lemma, Yoneda's lemma, Fatou's lemma, Gauss's lemma, and so on. Those names are passed down by tradition, and you don't get to change them, whatever your view is on the importance of the results. As to how they originated, one would have to investigate the literature.

Answer 6

I guess it will do no harm if I copy here relevant part from Wikipedia's Theorem#Terminology. I think it answers this question to some extent.

However, it is possible that some points here might be obsolete - the reference given for this at Wikipedia is Wentworth, G.; Smith, D.E. (1913). "Art. 46, 47". Plane Geometry. Ginn & Co. But I found the description given below fine.

This is taken from the current revision of the article.

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A number of different terms for mathematical statements exist, these terms indicate the role statements play in a particular subject. The distinction between different terms is sometimes rather arbitrary and the usage of some terms has evolved over time.

• An axiom or postulate is a statement that is accepted without proof and regarded as fundamental to a subject. Historically these have been regarded as "self evident", but more recently they are considered assumptions that characterize the subject of study. In classical geometry, axioms are general statements while postulates are statements about geometrical objects. A definition is also accepted without proof since it simply gives the meaning of a word or phrase in terms of known concepts.
• A proposition is a generic term for a theorem of no particular importance. This term sometimes connotes a statement with a simple proof, while the term theorem is usually reserved for the most important results or those with long or difficult proofs. In classical geometry, a proposition may be a construction that satisfies given requirements; for example, Proposition 1 in Book I of Euclid's elements is the construction of an equilateral triangle.
• A lemma is a "helping theorem", a proposition with little applicability except that it forms part of the proof of a larger theorem. In some cases, as the relative importance of different theorems becomes more clear, what was once considered a lemma is now considered a theorem, though the word "lemma" remains in the name. Examples include Gauss's lemma and Zorn's lemma.
• A corollary is a proposition that follows with little or no proof from one other theorem or definition.
• A converse of a theorem is a statement formed by interchanging what is given in a theorem and what is to be proved. For example, the isosceles triangle theorem states that if two sides of a triangle are equal then two angles are equal. In the converse, the given (that two sides are equal) and what is to be proved (that two angles are equal) are swapped, so the converse is the statement that if two angles of a triangle are equal then two sides are equal. In this example, the converse can be proven as another theorem, but this is often not the case. For example, the converse to the theorem that two right angles are equal angles is the statement that two equal angles must be right angles, and this is clearly not always the case.