Is Firstly Grammatically Correct?

Is Firstly Grammatically Correct? Even though they are both adverbs, ‘first‘ and ‘firstly’ are hardly interchangeable in all situations: we never say “I firstly noticed it yesterday.” One might say “firstly, what are you doing in my home?” or “firstly, I hope you have insurance”—but if you want to avoid criticism, ‘first’ is the best

Is Every Totally Ordered Set Well-ordered?

Is Every Totally Ordered Set Well-ordered? By definition, any well-ordered set is totally ordered. However, the converse is not true – the set of integers which is totally ordered, is not well-ordered under the standard ordering (since itself and some its subsets do not have least elements). Although, any finite totally ordered set is well-ordered.