  How many odd numbers are there? How many even numbers? From Galileo to
Cantor, the suggestion was that there are the same number of odd, even and
natural numbers, because all three sets can be mapped in one-one fashion to
each other. This jars with our intuition.
  Cardinality fails to discriminate between sets that are intuitively different
in size. The class of \emph{surreal numbers} $\boldsymbol{\mathsf{No}}$ is the
largest possible ordered field. In this work we define a function, the
\emph{magnum}, mapping the power set of the natural numbers to a subclass of
the surreals, the {surnatural numbers} $\boldsymbol{\mathsf{Nn}}$ (the
non-negative omnific integers). Set magnums are found to be consistent with our
intuition about relative set sizes. The magnum of a proper subset of a set is
strictly less than the magnum of the set itself, in harmony with Euclid's axiom
``the whole is greater than the part''.
  Two approaches are taken to specify magnums. First, they are determined by
following the genetic assignment of magnums in the way the surreal numbers
themselves are defined. Second, the counting sequence, which is defined for
every set of natural numbers, is extended using a Transfer Axiom, to evaluate
it on the surnatural numbers. The two methods are shown to be consistent.
Several theorems are proved, establishing that the family of sets with magnums
form what we call a magnum space. Finally, these theorems are employed to
evaluate the magnums of a number of subsets of $\mathbb{N}$.
