The classical definition of probability is a probability measure, but it is a definition rather than a theorem, meaning it does not require a proof. The discussion highlights that while Wikipedia provides examples using cardinality to illustrate this definition, such examples do not prove the definition itself. Kolmogorov's axioms offer a formal framework for probability, allowing for the derivation of additional postulates and the connection to Bayes' theorem, which represents a different probability measure. The conversation also touches on the historical context of Bayesian and classical approaches, noting their differing interpretations and measures of probability. Ultimately, understanding these frameworks helps clarify the nature of probability in mathematical terms.