Standard first-order logic (finitary logic) allows only finite conjunctions, disjunctions, and quantifications; infinitary logics generalize this by allowing infinite conjunctions, disjunctions, or quantifications. Understanding the limitations of first-order logic motivates studying infinitary logics and their applications in set theory and model theory.
Your prerequisite on logical consequence and entailment introduced the machinery for determining when one set of sentences forces another to be true. That analysis implicitly assumed that every formula is a finite string — you can write it down, count its symbols, and verify its syntactic form in finitely many steps. Finitary logic is simply first-order logic with this finiteness constraint made explicit: every well-formed formula is a finite expression, and every proof is a finite sequence of steps. This constraint is not arbitrary — it is what makes logic mechanically checkable and gives it the completeness and compactness properties you may have encountered.
But finiteness has a cost: some natural mathematical properties cannot be expressed in finitary first-order logic. The canonical example is connectivity of a graph: you cannot write a single first-order sentence that is true in a structure exactly when the underlying graph is connected, because connectivity requires saying "for every pair of vertices, there exists a path of some finite length between them" — and that path can be arbitrarily long, requiring an infinite disjunction over all possible lengths. Similarly, you cannot express "this structure is finite" or "this relation is well-founded" in finitary FOL. These expressibility gaps motivate extending the language.
Infinitary logics relax the finiteness constraint in controlled ways. The most studied is L_{ω₁,ω} (pronounced "L omega-one-omega"), which allows countably infinite conjunctions and disjunctions but still only finite strings of quantifiers. In this logic, you can write the sentence φ_0 ∧ φ_1 ∧ φ_2 ∧ ... — an infinite conjunction — as a single formula. This is enough to characterize many structures up to isomorphism that FOL cannot pin down: the structure of the natural numbers with successor is categorical in L_{ω₁,ω} (there is, up to isomorphism, exactly one model), whereas in finitary FOL it has non-standard models by the compactness theorem.
The tradeoff is severe: the beautiful theorems of finitary FOL — completeness, compactness, Löwenheim-Skolem — fail in infinitary logics. Compactness says a set of FOL sentences has a model if every finite subset does; this fails for L_{ω₁,ω} because infinite conjunctions can enforce constraints that no finite fragment captures. Completeness (every semantic consequence is provable) also fails: there is no effective proof system that captures all L_{ω₁,ω} validities. You gain expressive power and lose proof-theoretic tractability — the central tradeoff in the design of any formal logic.
Infinitary logics appear naturally in set theory and descriptive set theory, where infinite combinations arise organically. They also provide the right language for certain model-theoretic constructions, particularly when characterizing specific structures up to isomorphism. For practical logical reasoning and automated deduction, finitary first-order logic remains the workhorse — but recognizing where its expressiveness ends is essential for understanding what formal logic can and cannot do.