In page 13 of Mathematical Logic, Tourlakis defines the set of all well-formed-formulae as:
... the smallest set of strings, WFF, that satisfies: 1) All Boolean variables are in WFF, and so are the symbols ⊤ and ⊥. 2) If A and B are strings in WFF, then so are the strings (¬A), (A∧B), (A∨B), (A→B), (A≡B).
But as I understand it, the set of Boolean variables is infinite.
So, how can a set be a smallest one when the cardinality is infinite?
In this question somebody asked about it and the answer was
The "smallest set" condition there is crucial: if we omit it, there are lots of sets satisfying the definition
and I would go even further as to say, indeed there are infinite sets that satisfy the condition of being sets of Well Formed formulas, it's not unique because I can always find an element (Boolean variable) that belongs to another set of WFF.