# Best Notation for Introducing Finite Numbers of Elements

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.

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I was just going to throw in that $\mathbb{R}^\ast$ is probably more recognized as "nonzero reals" than "positive reals", so you might want to change that when you write in the future. – rschwieb Aug 14 '12 at 16:55