Question: Does "first-order logic" (i.e. any elementary first-order theory, see below) permit infinite conjunctions? (Of only countable cardinality or of arbitrary infinite cardinality?)
It seems like any infinite axiom schema is implicitly an infinite conjunction (cf. this related question), and there are numerous examples of elementary first-order theories with infinite axiom schemas. So probably the answer is yes, but if so this is rarely stated explicitly, so I want to confirm. Many sources actively give the impression that only finite conjunctions are allowed.
Background:
Following Knight and Keisler's introduction to infinitary logic, $\omega$ denotes the first infinite ordinal (countable cardinality), and $\omega_1$ denotes the first uncountable ordinal, for a given vocabulary $L$ and infinite cardinals $\mu \le \kappa$, $L_{\kappa \mu}$ is the infinitary logic with:
- no greater than $\kappa$ variables,
- conjunctions and disjunctions over sets of formulas of size strictly less than $\kappa$, and
- existential and universal quantifiers over sets of variables of size strictly less than $\mu$.
Based on these definitions, Keisler and Knight claim that $L_{\omega \omega}$ is ordinary elementary first order logic with (countably many variables,) finite conjunctions, finite disjunctions, and finite quantifiers.
They define an elementary first-order theory (which is basically what I mean by "first-order logic" in the question title) as any set (of unspecified cardinality?) of sentences in $L_{\omega \omega}$ closed under logical consequence.
Infinite disjunctions definitely are NOT allowed in first-order logic, see either [this related question] or [this other related question]. It is also probably not a coincidence that all of the examples of $L_{\omega_1 \omega}$ theories given in section 1.1 of Knight and Keisler's introduction to infinitary logic (i.e. "simplest possible" theories expressible in $L_{\omega_1 \omega}$ and not $L_{\omega \omega}$) use countable disjunctions but do not use infinite conjunctions.
Another related question is about how quantifiers in the elementary first-order object theory correspond to infinitary conjunctions/disjunctions in the metatheory for infinite models. However I am asking about infinitary conjunctions/disjunctions in the object theory, not the metatheory.