What is the difference between a proposition and a theorem? How do people decide which of the two to use in, e.g., textbooks?
Somehow I think "proposition" sounds less serious.
What is the difference between a proposition and a theorem? How do people decide which of the two to use in, e.g., textbooks?
Somehow I think "proposition" sounds less serious.
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.
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.
Actually, according to the very good book Mathematics for Computer Science
(by Eric Lehman and F Thomson Leighton), propositions and theorems are not the same (see page 7 and page 8).
Proposition is some statement (think of it as some verbally told/claimed expression), which (important point ->) can be either (1)True or (2)False.
Theorem is a Proposition which has passed the mathematical verification process and is proved to be True.
Note, that verification can be achieved in some different ways/strategies. E.g. induction, predicates, or etc.
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.