There are many different rigorous definitions of infinitesimals. Many don't require creating* any new sets, though many do require intuitionist logic[2]. My fav

There are many different rigorous definitions of infinitesimals. Many don't require creating* any new sets, though many do require intuitionist logic[2]. My favorite are nilpotent infinitesimals which give rise to synthetic differential geometry[3] and dual numbers[4]. Compared to hyperreal infinitesimals, nilpotent infinitesimals are much easier to construct. Here[5] is a gentle introduction to nilpotent infinitesimals and intuitionist logic, and here[6] is a very good book on synthetic differential geometry. * I didn't say construct, because that means something very specific in mathematics[1] and non-standard analysis is not traditionally constructive. [1] http://en.wikipedia.org/wiki/Constructivism_(mathematics) http://en.wikipedia.org/wiki/Constructivism_(mathematics) [2] http://en.wikipedia.org/wiki/Intuitionistic_logic http://en.wikipedia.org/wiki/Intuitionistic_logic [3] http://en.wikipedia.org/wiki/Synthetic_differential_geometry http://en.wikipedia.org/wiki/Synthetic_differential_geometry [4] http://en.wikipedia.org/wiki/Dual_number http://en.wikipedia.org/wiki/Dual_number [5] http://math.andrej.com/2008/08/13/intuitionistic-mathematics-for-physics/ http://math.andrej.com/2008/08/13/intuitionistic-mathematics... [6] http://home.imf.au.dk/kock/sdg99.pdf http://home.imf.au.dk/kock/sdg99.pdf