Choice needed for bases over a fixed field — Problem 6.1

6. Chieu-Minh Tran

It is about Morley’s categoricity in power theorem. According to [1], there is the following consequence.

Theorem 6.1.

For a countable theory $T$, if it is $\kappa$ categorial for some $\kappa>\aleph_{0}$, then it is $\lambda$ categorial for all uncountable $\lambda$.

And [2] replaced $\kappa$ in the above theorem by recursion theoretic complexity (arithmetic degree). Then, the sequel goes to categoricity without Choice. The first research about this topic date back to [3] shows that we still have the theorem holds if we assume ZF + there is some uncountable well ordered set of reals.

Theorem 6.2.

[ZF $+$ there is an uncountable well ordered set of reals].

The following conditions on a countable (first order) $T$ are equivalent:

$T$ is categorical in some cardinal $\aleph_{\alpha}>\aleph_{0}$,in $\mathbf{V}$, of course

$T$ is categorical in every cardinal $\aleph_{\beta}>\aleph_{0}$, in $\mathbf{V}$, of course

$T$ is, in $\mathbf{L}[T]$, totally transcendental (i.e. $\aleph_{0}$-stable) with no two cardinal models (i.e., for no model $M$ of $T$ and formula $\varphi(x,y)\in\mathbf{L}(\tau_{T})$ and $\bar{a}\in{}^{\ell g(\bar{y})}M$ do we have $\aleph_{0}\leq|\varphi(M,\bar{a})|<\|M\|$ and $\|M\|$ is a cardinal, i.e. the set of elements of $M$ is well-orderable hence its power is a cardinal)

If $\mathbf{V}^{\prime}\subseteq\mathbf{V}$ is a transitive class extending $\mathbf{L}$, $T\in\mathbf{V}^{\prime}$ and $\mathbf{V}^{\prime}$ satisfies ZFC, then the conditions in (3) hold

For some $\mathbf{V}^{\prime}$ clause (4) holds.

However, for non-well-ordered cardinals, the situation deteriorates quickly. The sequel is the following. For such a theory $T$, we want to show it is very close to a stongly minimal $T^{\prime}$. It is like a vector space over fixed field or algebraically closed field. And since there is no well order, it is hard to take a basis and match them to one another. So there is the question.

Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today.