Abstract:
We observe that the field of complex and p–adic numbers, the ring of n × n
matrices, the Euclidean spaces Rn (n > 1), and topological manifolds are models of
the notion of infinitesimal. These models do not satisfy the axioms of total order
and/or contain zero divisors. As a first consequence the notion of infinitesimal is
logically independent of the notions of zero divisor and total order. As second consequence
the notion of ultrafilter is not required for the definition of infinitesimal.
We define the notion of infinitesimal using the cofinite filter as in C. Schmieden
and D. Laugwitz [7]. We prove a translation theorem between expressions using
the Є– δ formalism and expressions using infinitesimals. The language employed
is many sorted. The language contains in addition to the basic carrier set (sort),
a function symbol intended to interpret the notion of size, absolute value, norm
or distance from 0, a unary predicate symbol to be interpreted the range of this
distance function, a binary symbol to be interpreted as addition in the range of
the metric, a constant symbol to be interpreted as zero as element of the range of
metric only, a symbol to denote the total order of the range of the metric function,
does not contain symbols to be interpreted as addition or multiplication or their
inverses.