Usually in statistical physics, when your system has a large number N of particles, you take the continuum limit -- you let [itex]N\rightarrow\infty[/itex], and convert sums to integrals (with an appropriate normalization factor). My understanding is that as a finite number tends to infinity, the infinity is still countable. What confuses me is the jump to uncountable infinity that allows us to use the continuum. Is there a rigorous math way to explain why this is ok, or is it just a physicist's way of saying countable infinity [itex]\approx[/itex] uncountable infinity? (To me, this seems absolutely unjustifiable, even in an approximation. The concepts are fundamentally different.) Thanks!