Using the well-orderedness property of the non-negative integers, plus the usual axioms of R as an ordered field and Z as an ordered ring, one can show that the the Archimedean property ("For all real x there exists an integer n with n > x") implies the existence of the ceiling function ("For all real x there exists a smallest integer n with n greater than or equal to x"). But does one need well-orderedness (or equivalently the axiom of induction) to prove this? That is, from the assumption that the ceiling function exists, can one derive the axiom of induction? It seems to me that the proposition that the ceiling function exists has an intermediate character: it cannot be derived from the ordered field and ordered ring axioms, but neither is it strong enough to imply induction. Are there a couple of nice non-standard models of (R,Z,0,1,+,x,>) that make this completely patent? Jim