User blog comment:Deedlit11/Ordinal Notations VI: Up to a weakly compact cardinal/@comment-5150073-20140515153351/@comment-1605058-20140523101027

Depending on a context, "first" may not mean the same as "smallest". For example, first ordinal written by this-and-this ITTM and smallest ordinal written in that way might not be equal, because it follows from the context that in the first case we mean "first" as in "appearing on the earliest stage".

However, when there is no context, or it doesn't follow directly from context what we mean, it's generally considered that "first" means "smallest", if such exists.

So if we are given a function f and no context about its origin, use etc. it's natural to assume first fixed point is the smallest fixed point.

NB. this definition works well only in well-ordered sets. If we have a function sin(1/x) then it doesn't have smallest positive fixed point. In that case, talking about either first nor smallest fixed point is meaningless, unless we give more detail on what we mean by "first".