## Friday, January 11, 2013

### Hyperintegers, cardinality and probability

Suppose F is an ordered field extending the reals. Then a subset Z of F closed under addition, subtraction and multiplication can be called a set of hyperintegers provided that for any x in F there is a unique n in Z such that nx<n+1. Every real closed field has a set of hyperintegers. Given a set of hyperintegers, we can call the non-negative ones "hypernaturals". Introduce the notation [x] for the set of all the hypernaturals smaller than x, for x in F. If x is itself a hypernatural, then [x] = {0,...,x−1}.

Here's an amusing and perhaps surprising little fact:

• If F has hyperintegers Z, and M is an infinite element in F, then [M] has at least the cardinality of the continuum, and in particular is uncountable.
(An infinite element is bigger in absolute value than every real.)

If F is a field of hyperreals, this follows from the fact that [M] is an internal set and not finite.

The proof is very simple. For x in F, let floor(x) be the unique n in Z such that nx<n+1. Let A be the set of numbers of the form floor(xM) for x a real number in [0,1). Observe that A is a subset of [M]. Moreover, A has the same cardinality as the set of real numbers in [0,1), since the function f(x)=floor(xM) from the real numbers in [0,1) onto A is one-to-one. To see that it's one-to-one, observe that if f(x)=f(y) for real x and y, then |xMyM|<1. Unless x=y, then M<1/|xy|, and M is finite. So x=y. So A has the cardinality of the continuum as the set of real numbers in [0,1) does. Thus, [M] has at least the cardinality of the continuum, since A is a subset of [M].

This helps improve on and slightly generalize the argument here that infinitesimals are too small to model outcomes of infinite fair countable lotteries. Suppose we say that the probability of getting an outcome n in a lottery with possible outcomes 0,1,2,... is an infinitesimal u in an ordered field F extending the reals that has hyperintegers. But u is of the right size (if it's the reciprocal of a hyperinteger) or just slightly too small (otherwise) for being the probability of getting outcome n in a fair lottery on the set [1/u]. But the set [1/u] is much much bigger than the set {0,1,2,...}, since [1/u] is uncountable while {0,1,2,...} is countable.