I wonder if there are some other attitudes to the probability than those related to Kolmogorov? Did somebody try to do those things differently?
There are many variations to the axioms of probability. The main distinctions are
Finite additivity instead of countable additivity. This was favored by early statisticians like Bernoulli.
Bayesian axioms of probability defined using conditional probabilities. Kolmogorov's axioms are a special case of this.