A mathematician uses different kinds of math alphabets:
typeset with \mathbb{} (you need to add the amsfonts or amssymb package)
\mathbb{ABCDEFGH}
typeset with \mathfrak{} (you need to add the amsfonts or amssymb package)
\mathfrak{ABCDEFGHabcdefgh1234567}
typeset with \mathcal{}
\mathcal{ABCDEFGH}
typeset with (you need to add mathrsfs package)
\mathscr{ABCDEFGH}
Notice that there are only capital letters in blackboard, calligraphic and RSFR alphabets.
From time to time it is also necessary to use some letters from the
\alpha \beta \gamma \delta \epsilon \zeta \eta