What is the usual distinction between Proposition and Theorem?
a subjective sense of importance
Technically,
[math]5+4=9[/math]
and
[math]5+4=10[/math]
are both propositions, the former is also a theorem of, for example, Peano arithmetic.
Colloquially, unprovable (w.r.t. a certain theory at hand) propositions like [math]5+4=10[/math] are not even mentioned in math books and the word is just used for less important theorems.
>>7684805
Not OP, but when would you choose when to use proposition, lemma, and corollary?
>>7684958
You realise that propositions in maths textbooks are provable right?
>>7684958
OP here. This is the philosophical/logical distinction. What I was after was the distinction in general mathematical texts.
>>7684805
A proposition is just something that may or may not be true. A theorem is something that you have proved to be true.
>>7685064
A lemma is a theorem that is of little interest on its own but is used as a stepping stone in proving a more important theorem. A corollary is a theorem that you can straightforwardly prove based on a larger theorem and which is a particularly important special case of the larger theorem.
>>7685072
Not that anon, but yes. They are also often proved. All theorems are propositions, since propositions are well formulated claims, but 0=1 is well formulated AND false (and thus not provable).
Back to the topic, propositions and theorems differ in a way that propositions are often stuff like "A group homomorphism sends identity elements to identity elements and inverses to inverses", while theorems are stuff like "All sets generate a free abelian group". The author asks themself "Do I want to refer to this result later or will I just tacitly use it?", and the choice decides whether they call the result theorem or proposition.
>>7685078
And I answered this too.
>>7685467
The author can be of either gender.
>>7685951
>ALL are proved
Justification for this proposition is left as an exercise
>>7685944
The author considers whether to refer to this result later or to just tacitly use it and calls the result either theorem or proposition.
No need to submit to the linguistic nightmare.
>>7686242
I see no nightmare.
>>7684805
I want to put my penis in the vagina of this blue alien and proceed to have sexual intercourse with her, if you catch my drift.
>lemma
>proposition
>corollary
>conjecture
>hypothesis
>theorem
math people will defend this
>>7686424
I don't follow. What do you mean?
>>7685951
A theorem is provable by definition.
And if they are proven in the text is another question.
>>7686449
Lemma is a general result to be used to prove certain claims by showing the hypothesis leads to the satisfaction of the lemma's conditions.
Proposition already decoded here.
A result following with ease from a previous result.
A sophisticated guess that things are in general as they have been shown to be in all the cases considered before the conjecture is published.
The frame of reference for a proof to be carried out in.
A provable claim, it is up to each author whether they consider these provable claims theorems or propositions.
