# Can Kolmogorov axioms be deduced from ZFC axioms?

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?

• The ZFC axioms are about sets in general, and Kolmogorov's axioms are about particular sets called probability spaces. From ZFC, you can deduce that certain sets satisfy Kolmogorov's axioms (and are therefore probability spaces), and also that other sets don't satisfy those axioms. – user49640 Jun 14 '17 at 12:22
• You mean that Kolmogorov's axioms are only relevant to indicate a specific kind of set from set theory (ZFC) ? It sounds to me like vector space's axioms. Am I right ? – toto Jun 14 '17 at 12:25
• Yes, it's a similar principle to the vector space axioms (although technically different). – user49640 Jun 14 '17 at 12:25

## 1 Answer

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.

• Okay, so, Kolmogorov axioms is only a way to say : "If a set X satisfies Kolmogorov axioms (sounds like properties instead of axioms actually). Then, theorems about probability can be applied to this set X.". It's like axioms of Group theory of vector space axioms, am I right ? – toto Jun 14 '17 at 12:31