Where did the notion of 'vaughtian pair' come from?

Question

I can see the notion of vaughtian pair is crutial in Morley's categoricity theorem but I don't know exact intuition behind this notion. Is this concept corresponds to some algebra notions like pregeomtry?

Answer 1

I can't say I'm 100% confident in my answer, though it may give you an idea.

Morley and Vaught wrote in the journal for Mathematica Scandinavica (Vol.11, No.1 1963 pp37-57) and proved Vaught's two cardinal theorem. This was proved in 6.2 of the journal article.

For two cardinals $\lambda>\mu>\omega$, a two cardinal $(\lambda, \mu)$-model of a theory $T$ is a model $\mathcal{M}\vDash T$ of size $\lambda$ with a definable subset $\phi(\mathcal{M})$ of size $\mu$. The two cardinal theorem states that if $T$ has a $(\lambda, \mu)$-model, then it has a $(\omega_1, \omega)$-model.

In 6.2 of the article, I believe they make use of these Vaughtian pairs, though they do not refer to these with such a name. Certainly other books such as Model Theory: An Introduction by D. Marker, or A Course In Model Theory by K. Tent and M.Ziegler proves the two cardinal theorem using Vaught pairs. It might be in the journal that the concept of Vaughtian pair is created, though I recommend you read the article to verify the claim.

Of course, the notion of Vaughtian pair does correlate with pregeometry highly. We know that if a complete (also countable?) theory $T$ has no Vaught Pairs, then any definable minimal set of a model is also strongly minimal. It was observed (for example by Baldwin and Lachland) that in strongly minimal sets, one could define a closure operator which defines a pregeometry on the set. However, it is not guaranteed that $T$ would contain a definable minimal set, although it is guaranteed when $T$ is $\omega$-stable.

These two hypotheses are key in the Baldwin-Lachlan characterisation of uncountably categorical theories: $T$ is uncountably categorical for any given uncountable cardinal $\kappa$ if and only if $T$ is $\omega$-stable and has no Vaught pairs. For the 'if' direction of this theorem, by what was discussed above, we can find a strongly minimal set in some model, and in this set there will be a notion of dimension well defined. One can also prove that because $T$ is $\omega$-stable, there is a prime model of $T$ prime over this strongly minimal set. After some more work (arguably a lot more work than outlined here) using the idea of dimension of the strongly minimal sets, one can show that any two uncountable models of the same size of $T$ must be isomorphic.