Definition: Theorem, Lemma, Proposition, Conjecture and Principle etc.
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 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.
$\endgroup$ 2 $\begingroup$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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