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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?

closed as not a real question by snailboat, Dude, kiamlaluno, hjpotter92, user114 Mar 25 '13 at 12:52

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6

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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