Best Notation for Introducing Finite Numbers of Elements

Question

What is the best way to quantify over a finite number of elements of a set? For instance, suppose that one wished to quantify over $n$ real numbers $a_1,\ldots,a_n$ in order to state a property of their sum. In such a case, I often see authors quantify over the upper bound of the sum. For instance: $$ \phi_1 \equiv \forall n \in \mathbb{N}~.~\sum_{i = 1}^n \ln a_i = \ln \left ( \prod_{i = 1}^n a_i \right ) $$ If an author wishes to be explicit, he or she will sometimes quantify over the $a_i$ using ellipses: $$ \phi_2 \equiv \forall n \in \mathbb{N}\forall a_1,\ldots,a_n \in \mathbb{R}~.~\sum_{i = 1}^n \ln a_i = \ln \left ( \prod_{i = 1}^n a_i \right ) $$ Sometimes, I see authors use index notation with a suitable data type. For instance, if $A^*_n$ denotes the set of multisets of size $n$ of elements of $A$, then one could introduce elements more precisely. Otherwise, one could sum or multiply over the elements of the multiset in the following way: $$ \phi_3 \equiv \forall n \in \mathbb{N}\forall \alpha \in \mathbb{R}^*_n~.~\sum_{a \in \alpha} \ln a = \ln \left ( \prod_{a \in \alpha} a \right ) $$ Of course, a multiset is usually defined in terms of a family of elements. If we are given a function indexing the elements of a set, then we can write: $$ \phi_4 \equiv \forall n \in \mathbb{N}\forall f : n + 1 \rightarrow \mathbb{R}~.~\sum_{i = 1}^n \ln f(i) = \ln \left ( \prod_{i = 1}^n f(i) \right ) $$ If none of the $\phi_i$ contain ideal notations for introducing a finite number of elements, what is the most ideal notation? For brevity, I would usually opt for the first, but I feel that it's incomplete. I hope this isn't a completely trivial question.

Answer 1

This is nothing to fret about. There are many different ways to say the same thing.

Use what you consider to be the clearest, and always be considerate of your intended audience. For advanced audiences, you are not going to need to be 100% rigorous about indexing conventions. Even for weaker audiences you might want to avoid overloading them with notation.

You will definitely need exact notation if the validity of the result depends on the clarity, but otherwise you can use your discretion to pick the "best" notation for the application.