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

Vincenzo Oliva
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  • You can just choose for defining it. $\infty\times 0:=0$ is a convenient option and is often practicized. – drhab Oct 01 '14 at 10:40

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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

Martín-Blas Pérez Pinilla
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    It should be noted that, in the context of Measure Theory, $\infty\times 0=0$ is a (very useful) convention, whereas in the context of Set Theory, $\kappa\cdot 0 = 0$ is a theorem ($\kappa\cdot 0$ is the cardinal of the Cartesian product of a set of cardinal $\kappa$ and the empty set). – Taladris Oct 01 '14 at 10:47