Infinitary Logics

Infinitary logics are logics which allow for conjunction and disjunction symbols to occur an infinite amount of times in a sentences, and allow for quantification over an infinite amount of variables.

Infinitary logics also allow for infinitely-long proofs, that it, proofs with an infinite amount of steps (recall: a proof is a sequence logically-valid statements of a logic).

i.e., Consider ${\mathcal{L}}_{\kappa ,\lambda }$ for $\kappa ,\lambda >{\aleph }_0$, ${\mathcal{L}}_{\kappa ,\lambda }$ permits formulae of length $<\kappa$ and quantification over $\lambda$-many variables. Thus, $\kappa$ is the least upper bound of the length of ${\mathcal{L}}_{\kappa ,\lambda }-$sentences. Proofs are of length $<\kappa$.

First-order logic is ${\mathcal{L}}_{\omega ,\omega }$

Infinitary logic are missing The Compactness Theorem and by extension the The Löwenheim-Skolem Theorem. Although some may have fragments of compactness (i.e., $\kappa -$compactness) and of the the Löwenheim-Skolem Theorem (see The Hanf Number and The Löwenheim Number)

Some infinitary logics are also not complete (Soundness & Completeness).

${\mathcal{L}}_{\infty ,\omega }$ (furthermore, ${\mathcal{L}}_{\infty ,\kappa }$ for all infinite cardinal $\kappa$, and ${\mathcal{L}}_{\infty ,\infty }$) can characterize any model up to isomorphism.

• Since infinitary logics can have infinitely-long formulae (and has a non-recursive syntax), Gödel encoding and diagonalization cannot be expressed. Thus, a provability or truth-predicate cannot be defined (however this may be done in a fragment of the logic but not the full logic).

References

Model Theoretic Logics [Infinitary Logic](Infinitary logic - Wikipedia)