Kolmogoroff's Axioms of Probability and Completness

Question

In Kolmogoroff Classic Foundations of the Theory of Probability, right at the beginning he gives the (now well-known axioms)

Let $E$ be a collection of elements $\xi,\eta,\zeta,\ldots$ which we shall call elementary events, and $\mathcal F$ a set of subsets of $E$; the elements of the set $\mathcal F$ will be called random events.

i) $\mathcal F$ is a field of sets.

ii) $\mathcal F$ contains the set $E$.

iii) To each set $A$ in $\mathcal F$ is assigned a non-negative real number $P(A)$. This number $P(A)$ is called the probability of the event $A$.

iv) $P(E)$ equals $1$.

v) If $A$ and $B$ have no element in commom, then $$ P(A + B) = P(A) + P(B) $$

A system of sets $\mathcal F$, together with a definite assignment of numbers $P(A)$, satisfying Axioms i)-v), is called a field of probability. Our system of Axioms i)-v) is consistent. This is proved by the following example. Let $E$ consist of the single element $\xi$ and let $\mathcal F$ consist of $E$ and the null set $0$. $P(E)$ is then set equal to $1$ and $P(0)$ equals $0$. Our system of axioms is not, however, complete, for in various problems in the theory of probability different fields of probability have to be examined.

By a field the means a collections of sets closed under union, intersection and difference, and he denotes union, intersection and difference by $+, \cdot$ and $-$. Now my question is related to the bold part, that before is totally clear to me, but what he means by these axioms are not complete? As written on Wikipedia and axiom system is complete if for every statement, either itself or its negation is derivable, and which statements are not derivable here? Or does he means something different when he talks about completeness here?

Answer 1

The key-point is Kolmogorov's reference [page 1] to David Hilbert, The Foundations of Geometry (1899) where it is stated an

Axiom of Completeness (Vollstandigkeit).

In modern term, this axiom is intended to ensure the categoricity of the system [see here : Ch.5.2 Categoricity].

Categoricity for a theory $T$ means, roughly speaking, that all models of $T$ are isomorphic.

I think that Kolmogorov here is alluding to the possibility that his axioms admit different non-isomorphic models.

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For the history of the elucidation of the difference and relationship between completeness and categoricity, see :

• Steve Awodey & Erich Reck, Completeness and Categoricity. Part I : Nineteenth-century Axiomatics to Twentieth-century Metalogic (2002), page 10 :

[about Hilbert's remarks about the Axiom of Completeness] First, note that Hilbert is again explicit that his axioms allow for different interpretations or models. Thus, a "Cartesian" geometric space just based on the set of rational numbers and certain algebraic numbers fulfills all his axioms for Euclidean geometry besides the Axiom of Line Completeness.

Second, what that axiom adds is to insure that any system of objects satisfying all of the axioms is essentially the same as - in Hilbert’s words, "is none other than" - ordinary Cartesian space, as based on the set of real numbers. [...] In fact, what this last axiom does, against the background of the others, is to make Hilbert’s whole system of axioms categorical.

See also page 11 :

it appears that what is meant by "completeness" in Hilbert’s works from this period [around 1900] might be something else instead. In his article "Uber den Zahlbegriff" published in 1900 and obviously written not long after Grundlagen, he comments again about the case of geometry. [...] According to the [...] this passage, the axioms of geometry are supposed to allow for proofs of "all geometrical propositions" [...]. This opens up the possibility that what Hilbert really means by "completeness" [at this time], is what we have called logical completeness: the (informal) provability of all truths of geometry from his axioms.

Overall it seems fair to say, however, that Hilbert is just not entirely clear on the notion of "completeness".