How to define the set of all finite words $\mathbb{W}$ from an alphabet $\Sigma$ using set union and Cartesian product? I wish to define the set of all words $\mathbb{W}$ from the alphabet $\Sigma$.
For instance if $\Sigma=\{0,1\}$, then the words are $\mathbb{W}:=\{ \varnothing, 0,1,00,...\}$
Can I write
$$
\mathbb{W}:=\bigcup_{i=0}^? \Sigma^i
$$
In this case I get all words as tuples.
$(0,1,0)$ would be a word and I get shorten the notation to $(0,1,0)=010$ which is a binary word.
Problem is I do not know how to capture the notion that the words are finite, whilst still grabbing all of them? What do I replace the quotation mark with?

$$
\mathbb{W}:=\bigcup_{i=0}^\infty \Sigma^i
$$
In this case, it seems I get an uncountable set.

$$
\mathbb{W}:=\bigcup_{i=0}^n \Sigma^i
$$
In this case, it seems I am missing all words greater than n.

$$
\mathbb{W}:=\bigcup_{i=0}^{<\infty} \Sigma^i
$$
Is less than infinity acceptable notation for a sum greater than any natural number but not infinite?
 A: If we consider a binary alphabet $\Sigma=\{0,1\}$ we can represent the set of all $2^i$ binary words of length $i>0$ as
\begin{align*}
\Sigma^i:=\{(x_1,x_2,\ldots,x_i):x_j\in \Sigma, 1\leq j\leq i\}
\end{align*}
We often use a special symbol $\varepsilon:=\sum^0$ to denote the empty word of length $0$.


*

*The set $\mathbb{W}$ of all binary words having finite length can be written as
\begin{align*}
\color{blue}{\mathbb{W}=\bigcup_{i\geq 0}\Sigma^i=\bigcup_{i=0}^{\infty}\Sigma^i=\varepsilon\cup\Sigma^{1}\cup\Sigma^{2}\cup\cdots}\tag{1}\\
\end{align*}
Each representation in (1) is admissible and describes the set of all binary words with finite length. Here we have a countable union of sets $\Sigma^i$ each with finite size $\left|\Sigma^i\right|=2^i$ which is a countable set.

The upper bound $\infty$ in $\bigcup_{i=0}^{\infty}$ means that we do not have any finite upper bound. Nevertheless the index $i\in\mathbb{N}$ for all indices $i$ under consideration.
Hint: We have a similar situation when considering an infinite sum
\begin{align*}
\sum_{i\geq 0}a_i=\sum_{i=0}^\infty a_i=a_0+a_1+a_2+\cdots
\end{align*}
which is a sum of countably many summands $a_i, i\in\mathbb{N}$.
