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Definition: Theorem, Lemma, Proposition, Corollary, Postulate, Statement, Fact, Observation, Expression, Fact, Property, Conjecture and Principle

Most of the time a mathematical statement is classified with one the words listed above.

However, I can't seem to find definitions of them all online, so I will request your aid in describe/define them.

Also, when is a mathematical statement a theorem versus a lemma ? I've read that a theorem is important while a lemma is not so important and used to prove a theorem. However a theorem is sometimes used to prove some other theorem. This implies that some theorems are also lemmas ?

Is it subjective with respect to the author, which statements become a theorem, lemma, etc. ?

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    $\begingroup$ I think you mean "subjective", in which case the answer is yes. $\endgroup$
    – user64687
    Jan 20, 2014 at 15:10
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    $\begingroup$ Don't forget postulate, statement, fact, observation, expression, corollary, property, etc... $\endgroup$
    – DanielV
    Jan 20, 2014 at 15:11
  • $\begingroup$ It can actually be subjective to the continent as well. Europe has different conventions than the US for the use of lemma (or it might have been corollary) iirc. $\endgroup$
    – DanielV
    Jan 20, 2014 at 15:13
  • $\begingroup$ A conjecture will usually not have a proof, at least initially $\endgroup$
    – Henry
    Jan 20, 2014 at 15:15
  • $\begingroup$ Thanks DanielV, I've added your post. $\endgroup$
    – Shuzheng
    Jan 20, 2014 at 15:34

2 Answers 2

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I have taken this excerpt out from How to think like a Mathematician

  • Definition: an explanation of the mathematical meaning of a word.
  • Theorem: a very important true statement that is provable in terms of definitions and axioms.
  • Proposition: a statement of fact that is true and interesting in a given context.
  • Lemma: a true statement used in proving other true statements.
  • Corollary: a true statement that is a simple deduction from a theorem or proposition.
  • Proof: the explanation of why a statement is true.
  • Conjecture: a statement believed to be true, but for which we have no proof.
  • Axiom: a basic assumption about a mathematical situation (model) which requires no proof.

I think it does a great job of describing what those words mean in a sentence. Later in the chapter, he goes onto describe how we have some conjectures which have been called "Theorems" even though they weren't proven. For example, Fermat's Last Theorem was referred to as a Theorem even though it hadn't been proven. If you haven't read the book then I highly recommend it if you are a undergraduate in your first two years of math.

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  • $\begingroup$ The definition of proposition you posed is not correct. From The Free On-line Dictionary of Computing (18 March 2015) [foldoc]: proposition <logic> A statement in {propositional logic} which may be either true or false. Each proposition is typically represented by a letter in a {formula} such as "p => q", meaning proposition p implies proposition q. $\endgroup$
    – crow
    Mar 12, 2017 at 4:21
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    $\begingroup$ @crow The goal of this answer is to explain the difference between a Theorem and a Lemma in terms of hierarchy. For example, when should one use "Theorem" and when should they use "Lemma." $\endgroup$
    – Jeel Shah
    Mar 12, 2017 at 15:20
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Theorem vs. Lemma is totally subjective, but typically lemmas are used as components in the proof of a theorem. Propositions are perhaps even weaker, but again, totally subjective.

A conjecture is a statement which requires proof, should be proven, and is not proven. A principle is perhaps the same as a conjecture, but perhaps a statement which is asserted but taken as true even without proof, like an axiom.

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  • $\begingroup$ I always thought the lemmas were lesser than propositions in terms of intellectual merit... $\endgroup$
    – Joe Tait
    Feb 10, 2014 at 14:46
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    $\begingroup$ A lemma is a theorem which is "not the main point", typically depended upon by a theorem which is the main point, where the point is whatever the author is trying to prove $\endgroup$
    – crow
    Mar 12, 2017 at 4:22

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