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What is the distinction between a proposition and a theorem? How do people decide which of the 2 to use in, say, textbooks? Somehow I think proposition sounds less serious... Thanks.

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At the mathematical level, propositions and theorems are the same: true statements that have a hypothesis and a conclusion (and a proof). At the exposition level, theorems are the results you want to stress.

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I would add that within one work theorems generally tend to be a bit deeper than propositions. – Brian M. Scott Sep 28 '11 at 10:36
I would add that it depends very much on the taste of the author, there are authors that call every lemma a "Theorem", and authors that even call deep theorems "Lemma" because they are used to prove deeper ones – Julian Kuelshammer Sep 28 '11 at 10:39

Suppose we have a formal system with a language $\mathcal{L}$ and a set of axioms such that the set $T$ of sentences (that is, well-formed formulae with no free variables) $\varphi \in \mathcal{L}$ contains those axioms and is closed under logical consequence. $T$ is thus known as a theory, and every sentence of $T$ is a theorem of $T$.

The distinction between 'theorem', 'proposition' and 'lemma' is thus largely sociological or expository: it provides a means for a mathematician to indicate what is new or important, and what results are merely stepping stones to bigger results. The distinction between them is imprecise: what is a theorem is one context may be merely a lemma in another.

For example, Zorn's Lemma is often discussed or proved without using it as a step in some further proof. A first course in set theory will often include the proofs that Zorn's Lemma is derivable from the Axiom of Choice, and vice versa. This is a nice illustration of the point above, since in one direction we derive something nominally known as a lemma—not a theorem—and in the other we derive something usually known as an axiom from a lemma we do not prove, but merely assume. Note that it is a trivial consequence of the definition above that axioms are also theorems.

The question of when to use one or the other is thus down to two major factors: how their use figures in the presentation of a result, or series of results, and precedence: how they have been used in the past. Note that pedagogical considerations are likely to carry more weight: a textbook in mathematical logic might present the Compactness Theorem as a theorem, but the Completeness Theorem as merely a corollary.

The best way to understand this in practice is to read a lot of textbooks and make note of how the best presentations of the results choose to draw these distinctions. In time it will become obvious in your own work what should be a lemma, what should be a theorem, and so on. The particular nuances given to these terms by different authors are not standard, and may vary both with the field of inquiry and the individual author.

It may help to think of proofs in narrative terms: individual results should be labelled in whatever way best structures the story, so that your readers have the best possible chance to understand your result in its proper context. Of course, this advice applies far more generally, and as with other matters of exposition, practice in both reading and writing proofs is key.

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By definition, a proposition is "A statement or assertion that expresses a judgment or opinion.", a theorem is "A general proposition not self-evident but proved by a chain of reasoning; a truth established by means of accepted truths."

So as I see the main difference is that a proposition is more evident. It is used as something supporting. A theorem, on the contrary, has a more important place in the certain theory, it is something more fundamental.

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There is no requirement that something labelled proposition be more evident than something labelled theorem; the latter is simply considered more substantial and probably more important, at least in the given context. – Brian M. Scott Sep 28 '11 at 10:34
Downvote. Dictionaries do not define mathematical terms. – Samuel Sep 28 '11 at 11:01

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