Editing Infinitary logic (section)

==Concepts expressible in infinitary logic==
In the language of [[set theory]] the following statement expresses [[Axiom of regularity|foundation]]:

:<math>\forall_{\gamma < \omega}{V_{\gamma}:} \neg \land_{\gamma < \omega}{V_{\gamma +} \in V_{\gamma}}.\,</math>

Unlike the axiom of foundation, this statement admits no non-standard interpretations. The concept of [[well-foundedness]] can only be expressed in a logic that allows infinitely many quantifiers in an individual statement. As a consequence many theories, including [[Peano arithmetic]], which cannot be properly axiomatised in finitary logic, can be in a suitable infinitary logic. Other examples include the theories of [[non-archimedean field]]s and [[torsion-free group]]s.{{cn|date=May 2025}} These three theories can be defined without the use of infinite quantification; only infinite junctions<ref>{{cite journal |last=Bennett |first=David W. |date=1980 |title=Junctions |journal=[[Notre Dame Journal of Formal Logic]] |volume=21 |issue=1 |pages=111–118 |doi=10.1305/ndjfl/1093882943 |doi-access=free}}</ref> are needed.

Truth predicates for countable languages are definable in <math>\mathcal L_{\omega_1,\omega}</math>.<ref>{{cite web |url=https://logic.amu.edu.pl/images/9/95/Pogonowski10vi2010.pdf |title=Inexpressible longing for the intended model |last=Pogonowski |first=Jerzy |date=10 June 2010 |website=Zakład Logiki Stosowanej |publisher=[[Adam Mickiewicz University in Poznań|Uniwersytet im. Adama Mickiewicza w Poznaniu]] |page=4 |access-date=1 March 2024}}</ref>