Does every finitely axiomatizable theory have irredundant bases of every finite cardinality? Let $T$ be a finitely axiomatizable first-order theory, and let $n$ be an arbitrary positive integer. An axiom set $A$ for $T$ is defined to be irredundant iff $A$ has no redundant axioms. My question is, does $T$ always have an irredundant axiom set whose cardinality is $n$?
 A: Here's a partial result: the answer is positive if $T$ has no finite models.
Let $\theta$ be a single sentence axiomatizing $T$ and fix $n>1$. Consider the following set of axioms:

*

*"If there are more than $n-1$ elements in the universe, then $\theta$."


*"There is not exactly $1$ element in the universe."


*"There are not exactly $2$ elements in the universe."


*...


*"There are not exactly $n-1$ elements in the universe."
This is an independent axiomatization of $T$ consisting of $n$ sentences.
We can push this basic idea to a much wider class of theories. However, note that some restriction is necessary: the empty theory has no irredundant axiomatizations of any cardinality $>0$. More generally, if there are only $k$ structures (up to elementary equivalence) which don't satisfy $T$, then $T$ can't have an irredundant axiomatization of size $>k$ (think about the role of sentences in an axiomatization of $T$ as ruling out non-models).
A: Here is another partial result. Assume that $T$ is not axiomatisable by the empty set, and that your language contains infinitely many unary predicate symbols $Q_0,Q_1,\ldots$
Then the answer is yes:
Let $A$ be some finite nonempty axiomatisation of $T$. Then $\{\bigwedge A\}$ is an irredundant axiom set for $T$ of size $1$. On the other hand, whenever you have some irredundant axiom set $A\cup\{\theta\}$ for $T$ of size $n>0$, then $A\cup\{(\exists x Q_i(x))\lor\theta,(\lnot\exists x Q_i(x))\lor\theta\}$ is an irredundant axiom set for $T$ of size $n+1$, where $Q_i$ is some predicate not occuring in $A\cup\{\theta\}$. In this way you get irredundant sets of arbitrary positive size.
