Is there a symbol that denotes a finite set?

Question

Let's say I want to say there exists a finite set $S$.

If I write $\exists S$ it is not obvious from the notation that $S$ is finite.

How can I show that $S$ is finite in notation form?

Answer 1

How can I show that S is finite in notation form?

The answer depends strongly on the context.

In contexts where the elements of $S$ are important, the notation $S=\{x_1,...,x_n\}$ is convenient. Consider, for example, the following sentences where $X$ and $Y$ are given vector spaces:

(1) There exists $S\subset X$ such that $\operatorname{span}(S)=Y$.

(2) There exists $S=\{x_1,...,x_n\}\subset X$ such that $\operatorname{span}(S)=Y$.

Sentence (1) says that $Y$ is a vector subspace of $X$.

Sentence (2) says that $Y$ is a finite-dimensional vector subspace of $X$. Also, it introduces an upper bound for the dimension of $Y$ (which is $n$) and a particular form for any element $y$ of $Y$, namely, $$y=\alpha_1x_1+\cdots +\alpha_nx_n.$$

So, in the context of linear algebra, the notation $S=\{x_1,\ldots ,x_n\}$ says that $S$ is finite in very useful way.

To get more accurate answers, give more details on your context.