D. Hilbert, in his famous program, formulated many open mathematical
problems which were stimulating for the development of mathematics and a
fruitful source of very deep and fundamental ideas. During the whole
20th century, mathematicians and specialists in other fields have been
solving problems which can be traced back to Hilbert's program, and
today there are many basic results stimulated by this program. It is
sure that even at the beginning of the third millennium, mathematicians
will still have much to do. One of his most interesting ideas, lying
between mathematics and physics, is his sixth problem: To find a few
physical axioms which, similar to the axioms of geometry, can describe a
theory for a class of physical events that is as large as possible. We
try to present some ideas inspired by Hilbert's sixth problem and give
some partial results which may contribute to its solution. In the
Thirties the situation in both physics and mathematics was very
interesting. A.N. Kolmogorov published his fundamental work
Grundbegriffe der Wahrschein- lichkeitsrechnung in which he, for the
first time, axiomatized modern probability theory. From the mathematical
point of view, in Kolmogorov's model, the set L of ex- perimentally
verifiable events forms a Boolean a-algebra and, by the Loomis-Sikorski
theorem, roughly speaking can be represented by a a-algebra S of subsets
of some non-void set n.