I'm a starting amateur mathematician. I'm studying Gödel's incompleteness theorem and have a couple of rookie questions that I can't seem to sort out. 1) In the text I'm reading it talks repeatedly about systems containing "addition and multiplication". Since multiplication can be derived from addition, why is it relevant to mention multiplication? 2) If the G sentence is added to the set of axioms, there will be always another sentence G' that is undecidable, and if this one is added then there will be a G'', G''', etc. If there appears to be a systematic way of building this sentences, why it is not valid to include axioms to build all this kind of sentences? Thanks in advanced!! The subject is thrilling and I'm looking forward to read and learn much more.