We could just throw away these short hands and just [use] integers, positive integers, negative integers. This depends on who you consider "we" to be. If w

We could just throw away these short hands and just [use] integers, positive integers, negative integers. This depends on who you consider "we" to be. If we're talking about computer programmers, in most languages integers are all you get: natural numbers aren't available (although the Haskell community want them [1]). On the other hand, from the perspective of mathematics, the natural numbers are very different to the integers. For one thing, they're wellordered whereas the integers are not. Obviously a set-theoretic definition of the natural numbers (as the finite ordinals) will start at 0 since it's just identified with ø. Set-theoretic definitions of the integers are typically in terms of equivalence classes of pairs of natural numbers (or arbitrary elements from these equivalence classes if you want to work in ACA0) [2]. [1] http://hackage.haskell.org/trac/haskell-prime/wiki/Natural http://hackage.haskell.org/trac/haskell-prime/wiki/Natural [2] See SoSOA p.10 for details. http://www.math.psu.edu/simpson/sosoa/ http://www.math.psu.edu/simpson/sosoa/