4 ms·

I'm pretty sure it's not just "sufficiently large x" but rather as x approaches infinity. The properties as you approach infinity are completely different, and

• gus_massa 11y ago
  For the Banach–Tarski paradox it's not enough to have an x that is a \HUGE number of elements, you need an actual infinite. For this and most (all?) of the unintuitive results that use the Axiom of Choice, it is even not enough to have a countable number of elements, like the integer numbers or the rational numbers. You need a bigger infinite like the number of elements in the real number. In this case the number of points in a sphere of real numbers in R^3. (More info about the different types of infinites: http://en.wikipedia.org/wiki/Cardinal_number http://en.wikipedia.org/wiki/Cardinal_number )
• tbrownaw 11y ago
  The "odd" or "counterintuitive" properties as you approach infinity, are a perfectly reasonable consequence of the general agreement to throw away information and treat infinity as all the same. But they're not all the same, they're simply large enough that any non-infinite numbers are below epsilon. Much like how infinitesimals are all equal to zero (below epsilon for any non-infinitesimal), but if they were all the same calculus wouldn't work.