When reading a course on probability, I learned that they are based on Kolmogorov's axioms. I would like to know if these axioms can be deduced from ZFC axioms, or if they are added to ZFC axioms in order to work on probability?
2026-04-01 05:54:00.1775022840
Can Kolmogorov axioms be deduced from ZFC axioms?
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They cannot be derived because they are about a different theory. Simimlarly, the Peano axioms cannot be derived from ZFC - but we can verify (i.e., prove that the according statements are theorems of ZFC) that the set $\omega$ that is guaranteed to exist essentially by the Axiom of Infinity in ZFC is a model of PA acioms (with $0$ represented by $\emptyset$ and $S$ represented by $x\mapsto x\cup\{x\}$).
Maybe a better example is that the axioms of group theory cannot be derived from ZFC, but there are many different models in ZFC (i.e., groups that are described as certain sets with certain operations). Likewise there are many models of probabilty spaces in ZFC.