Why are floating point infinities, unlike NaNs, equal?

Why doesn't infinity comparison follow the logic applied to NaNs? This code prints out false three times:

double a = Double.NaN;
double b = Double.NaN;
System.out.println(a == b); // false
System.out.println(a < b); //  false
System.out.println(a > b); //  false

However, if I change Double.NaN to Double.POSITIVE_INFINITY, I get true for equality, but false for the greater-than and less-than comparisons:

double a = Double.POSITIVE_INFINITY;
double b = Double.POSITIVE_INFINITY;
System.out.println(a == b); // true
System.out.println(a < b); //  false
System.out.println(a > b); //  false

This seems dangerous. Assuming that infinite values result from overflows, I imagine it's more likely that two variables that ended up as infinities wouldn't actually be equal in perfect arithmetic.