What's a model in axiomatic theory?

Question

In the first answer here:

Kolmogorov's probability axioms

It is wondered whether there is only one model of the axioms (up to an isomorphism).

Could somebody explain this concept? What's a model, and what is precisely meant about there being more 'models' to Kolmogorov's axioms? Do they mean, perhaps, that Kolmogorov's axioms could also be used in the "model of measure theory" since they make perfectly good sense in that setting as well?

Answer 1

A model for the "theory of a something" is just a something. A model for the "theory of a probability space" that is for Kolmogorov's axioms is simply one particular probability space, like the one describing the experiment of rolling a single fair die once. Every probability space is obviously a model for the "theory of a measure space" since said theory is just more general.

That all models are isomorphic would be highly undesirable (because the experiment earlier is nothing like for example fairly picking a number in $[0,1]$ at random), that is your notion of isomorphism would be very boring, and I don't think anything like that was implied in the link.

Answer 2

A model of Kolmogorov's axioms is called a probability space and may be denoted as a $3$-tupel $(X, \mathcal F, \mathbb P)$, where $X$ is a nonempty set, $\mathcal F \subseteq \mathcal P(X)$ is a $\sigma$-Algebra and $\mathbb P \colon \mathcal F \to [0,1]$ is a probability measure. (Note however that this isn't a model in the usual sense of model theory.)

Given two probability spaces $(X, \mathcal F, \mathbb P)$ and $(Y, \mathcal G, \mathbb Q)$ we say that they're isomorphic iff there is a bijection $f \colon X \to Y$ such that $f " A \in \mathcal Q$ and $f^{-1} " B \in \mathcal F$ as well as $\mathbb P(A) = \mathbb Q(f"A)$ and $\mathbb Q(B) = \mathbb P(f^{-1}"B)$ for all $A \in \mathcal F$ and all $B \in \mathcal G$.

There are many non-isomorphic probability spaces. For example, for each finite set $X$ with we may let $\mathcal F = \mathcal P(X)$ and define $\mathbb P \colon \mathcal F \to [0,1]$ by $\mathbb P(A) = \frac{\#A}{\#X}$.

Answer 3

What's a model

Models in model theory are sets which interpret a first-order language (the usual logical symbols $=$, $\land$, $\lor$, $\neg$, $\forall$, $\exists$, and additional function, relation, and constant symbols.)

only one model of the axioms (up to an isomorphism)

In model theory, the term for this is categorical. In first-order logic one isn't able to control the size of models of a theory (this is the Loewenheim-Skolem theorem), so we usually say that a theory $T$ is $\kappa$-categorical if all of its models of size $\kappa$ (for $\kappa$ an infinite cardinal) are isomorphic. A fundamental result is Morley's 1965 categoricity theorem (this inspired Shelah, for example, to embark on his programme on classification theory): if a theory in a countable language is categorical in some uncountable $\kappa$, it's categorical in all uncountable $\kappa$.

Theories are just collections of sentences; a complete theory is one whose models all satisfy the same sentences. For example, the theory of algebraically closed fields in characteristic zero is complete. It's also $\aleph_1$-categorical, but not countably categorical, because there are countable algebraically closed extensions of $\mathbb{Q}$ of different transcendence degree.

Now, while probability spaces are outside the purview of classical first-order logic, you might be interested in recent work on continuous model theory (see section 16 of this paper for probability spaces in particular.)