Product of Uncountably Infinite Number of 1s

Question

Just like the title says: What is the product of an uncountable number of 1s? Intuitively the answer is 1, but how does one go about defining such a product in general?

Answer 1

As 5xum said in the comments, it depends on how you define the product of an uncountable set of real numbers. Suppose that $\Lambda$ is an uncountable index set, and $a_\lambda\in\Bbb R$ for each $\lambda\in\Lambda$. Perhaps the most natural definition of $\prod_{\lambda\in\Lambda}a_\lambda$ is the following.

Let $\mathscr{F}$ be the set of all finite subsets of $I$; $\mathscr{F}$ is directed by $\subseteq$. For each $F\in\mathscr{F}$ let $a_F=\prod_{\lambda\in F}a_\lambda$; then $\nu=\langle a_F:F\in\mathscr{F}\rangle$ is a net of real numbers, and we say that $\nu$ converges to $a\in\Bbb R$ if and only if for each $\epsilon>0$ there is an $F_\epsilon\in\mathscr{F}$ such that $|a_F-a|<\epsilon$ whenever $F_\epsilon\subseteq F\in\mathscr{F}$. If $\nu$ does converge to some $a\in\Bbb R$, we define $\prod_{\lambda\in\Lambda}a_\lambda=a$; otherwise, the product is undefined. (It’s not hard to see that $\nu$ can converge to at most one real number.)

If we apply this definition to the specific example in your question, in which $a_\lambda=1$ for all $\lambda\in\Lambda$, we find that $a_F=1$ for each $F\in\mathscr{F}$, so clearly $\nu$ converges to $1$, and the product is therefore $1$ (with this definition of product).