## Saturday, November 17, 2012

### Why infinitesimals are too small to help with infinite lotteries: Part III

In two preceding parts (I and II), I argued that assigning the same infinitesimal probability to every outcome of a lottery with countably many tickets assigns too small a probability to those outcomes, no matter which infinitesimal was chosen.

Here I want to note that if we're dealing with hyperreal infinitesimals, then one can't get out of those arguments by assigning a different infinitesimal probability to each outcome. In fact, my second argument worked whether or not the same infinitesimal is assigned. The first did need the same infinitesimal to be assigned, but one can generalize. Suppose I assign infinitesimal probability un to the nth ticket. Now it turns out that given any countable set of hyperreal infinitesimals, there is an infinitesimal bigger than them all. So, suppose that u is an infinitesimal bigger than all the un. Since u would be too small for the probability of the tickets, a fortiori, the un will be too small, too.