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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 '14 at 15:10
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    $\begingroup$ Don't forget postulate, statement, fact, observation, expression, corollary, property, etc... $\endgroup$ – DanielV Jan 20 '14 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 '14 at 15:13
  • $\begingroup$ A conjecture will usually not have a proof, at least initially $\endgroup$ – Henry Jan 20 '14 at 15:15
  • $\begingroup$ Thanks DanielV, I've added your post. $\endgroup$ – Shuzheng Jan 20 '14 at 15:34
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I have taken this excerpt out from How to think like a Mathematician

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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 '17 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 '17 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 '14 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 '17 at 4:22

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