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In Mathematics, there are a lot of references to the following words/phrases.

  1. Axioms
  2. Theorems
  3. Corollaries
  4. Claims
  5. Lemmas
  6. Definitions

I often use them interchangeably (which is definitely wrong on my part), but I am still not sure on the usage they have; or how to differentiate one from the others.

Are there some duplicate terms in the set? Are there any more terms? What differences can be observed in their usages?

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No two words in your list are equivalent; all of them have their own precise meaning in mathematics:

Axiom (or postulate)

An axiom 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.

Theorem

A theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms.

The derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other interpretations, depending on the meanings of the derivation rules.

The proof of a mathematical theorem is a logical argument demonstrating that the conclusions are a necessary consequence of the hypotheses, in the sense that if the hypotheses are true then the conclusions must also be true, without any further assumptions.

The concept of a theorem is therefore fundamentally deductive (in contrast to the notion of a scientific theory, which is empirical).

Corollary

A corollary is a proposition that follows with little or no proof from one other theorem or definition.

Lemma

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.

Conjecture (also: Hypothesis, Claim)

A conjecture is a as-yet unproven proposition that appears correct, for example:

A conjecture becomes a theorem when a formal proof for it becomes established, or until a counter-example or anti-proof determines that it is not true - a good example of this being Fermat's conjecture (often called Fermat's last Theorem for historical reasons), which has now been proven true.

Definition

A definition is used to unambiguously define a word for ease of use later (for instance, the definition of a "prime number" being "An irreducible element in the field of Integers").

  • Lemma and corollary still sound the same to me. They are propositions, with the only difference that lemma has wider applications than corollary? – hjpotter92 Mar 25 '13 at 9:35
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    @DreamEater: A lemma and a corollary are both theorems, but apart from are not similar. A lemma is a "helping theorem". It may be complex to prove and prove a result that is important for other areas of the subject. A corollary is a theorem that follows directly from the proof of an entirely different theorem. – Matt Mar 25 '13 at 9:56

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