Is$\ \infty \times 0$ undefined in the extended real numbers?

Question

And if it is, why? Is it a kind of postulate related to the fact that infinitely many points make a line?

Answer 1

In the context of calculus is undefined because the usual problems (lim product = product lim fails): $$\lim_{n\to\infty}n\frac1n=1,\qquad\lim_{n\to\infty}n^2\frac1n=\infty,\qquad\lim_{n\to\infty}n\frac1{n^2}=0.$$ But in Measure Theory $$\infty\times 0=0$$ and in Set Theory $$\kappa\cdot 0 = 0$$ for any cardinal $\kappa$.

See the discussion in http://mathforum.org/kb/message.jspa?messageID=6750333