When I was an elementary school student, I encountered some puzzles like "What's product of numbers of hair of each people in the world?", which answer was zero, because there's people with no hair(bald), and 0*any number = 0. There was similar question like "What's the product of every number in a telephone?".
After know that '0' makes the whole answer 0, I made three questions and asked first two questions to my friends (to induce the last question):
"What's the product of every numbers?" (or $\prod_{x\in \mathbb{R}} x = ?$)
"What's the sum of every numbers?" (or $\sum_{x\in \mathbb{R}} x = ?$)
They answered '0' in the first question easily, and they also answered '0' in the second question, though they spent several seconds to think (that $x+(-x) = 0$).
After they answered the second question, I asked the last question:
"The answer of the first question was zero, because 'every numbers' includes zero. Then, what's the product of every numbers, except zero? (or $\prod_{x\in \mathbb{R}^{*}} x = ?$)"
My answer was -1, because except 1 and -1, every number $x$ has it's inverse, $\frac{1}{x}$, and the product of two numbers is 1. Therefore, the answer = 1*-1*1*1*.... = -1.
It was quite interesting, because some answered 'infinite?', some answered '1', and some answered '-1', which was the correct answer I thought, before one of my friend says that
"Because the set of real numbers are uncountable, your question doesn't have an answer."
I don't remember whether my friend said "doesn't have an answer", "'s answer is infinite", or other things...
Anyway, the question is, what is the answer of $\prod_{x\in \mathbb{R}^{*}} x$? is it -1? or undecidable?
If it's not -1 (undecidable or infinite or others), then what's the answer of above two questions? or $\prod_{x\in \mathbb{Q}^{*}} x$?
If it's 1, then what's the answer in hyperreal number?