  This work has been prompted by the surprising lack of mathematical coherence
in the common usage of some of the fundamental entities in the theory of
probability, with an inherent risk of contradiction. While disentangling the
intricacies, we realized that the same issue has been raised many times, with
only partial solutions, notably by Boole, Hilbert, De Finetti and Renyi, among
others. In particular, a restoration of foundational coherence in the usage of
probability theory appears to be a missing piece in the solution of Hilbert VI
problem.
  Here we solve the problem by a new formalization of probability theory based
on a minimal collection of axioms with additional context dependent conditions,
whose overall consistency is then semantically verified. In Elementary
Probability, i.e. probabilities involving boolean combinations of finitely many
events, our theory leads to algebraization and, using Tarski Seidenberg
reduction, to a proof of decidability of all problems. Inconsistency in
Elementary Probability, on the other hand, is equivalent to, suitably
redefined, arbitrage or Dutch Book. In the continuous case this leads to
nonstandard analysis.
