I will write the German names of the following English terms in brackets as it is important that I use the correct terms. This is a question about terms and I'm not completely sure if I've translated everything correctly.
I'm currently writing digital lecture notes (in German). The makes use of the following "theorem-like" mathematical statements:
- Remark ("Bemerkung" in German)
- Proposition ("Proposition" in German)
- Theorem ("Satz" in German)
He proved all of those statements. As it feels strange to me to prove remarks, I called them corollary ("Korollar" in German). Today, he told me that a corollary precedes a theorem. So I should rename them. When I told him that it feels strange to me to call them remarks and prove them, he suggested calling them "consequence" ("Folgerung" in German). But this feels somehow arbitrary to me.
I know that many authors have a lot of different naming schemes. My question is: Is there an author/institution that has a consistant naming scheme and writes why he/she chose this scheme?
This is what I think is commonly used:
- Propositions are not as important as Theorems
- Corollaries and Lemmata (is the plural correct?) are used to prove theorems
- Theorems are the most important statements
Although Wikipedia has a good list to define and distinguish the terms (see Theorem) all of the cites go to "Wentworth, G.; Smith, D.E. (1913). "Art. 46, 47". Plane Geometry. Ginn & Co." which I can't access.
Is there a source that explains or defines terms like Theorem, Proposition, Lemma and Corollary?
(The source does not necessarily need to be an online resource. I can access quite a lot with university: http://www.bibliothek.kit.edu/cms/index.php)