Even though OpenMath has been around for more than 10 years, there is still confusion
about the ``semantics of OpenMath''. As the upcoming MathML 3 recommendation will
semantically base Content MathML on OpenMath Objects, this question becomes more
pressing.

One source of confusions about OpenMath semantics is that it is given on two levels: a
very weak algebraic semantics for expression trees, which is extended by considering
mathematical properties in content dictionaries that interpret the meaning of (constant)
symbols. While this two-leveled way to interpret objects is well-understood in logic, it
has not been spelt out rigorously for OpenMath.

In this paper we look at the semantics of OpenMath from a foundational point of view and
reconcile this ``semantics'' with the foundations of mathematics established in the
early 20th century; the traditional way of assigning meaning to mathematical objects.