Editing Infinitary logic (section)

{{Short description|Logic that allows infinitely long proofs}}
An '''infinitary logic''' is a [[Formal logical system|logic]] that allows infinitely long [[statement (logic)|statements]] and/or infinitely long [[Mathematical proof|proofs]].<ref>{{cite book |last=Moore |first=Gregory H. |date=1997 |chapter=The prehistory of infinitary logic: 1885–1955 |editor-last1=Dalla Chiara |editor-first1=Maria Luisa |editor-link1=Maria Luisa Dalla Chiara |editor-last2=Doets |editor-first2=Kees |editor-last3=Mundici |editor-first3=Daniele |editor-last4=van Benthem |editor-first4=Johan |editor-link4=Johan van Benthem (logician) |title=Structures and Norms in Science |publisher=Springer-Science+Business Media |pages=105–123 |doi=10.1007/978-94-017-0538-7_7 |isbn=978-94-017-0538-7}}</ref> The concept was introduced by Zermelo in the 1930s.<ref>{{cite journal |last=Kanamori |first=Akihiro |author-link=Akihiro Kanamori |date=2004 |title=Zermelo and set theory |url=https://math.bu.edu/people/aki/10.pdf |journal=The Bulletin of Symbolic Logic |volume=10 |issue=4 |pages=487–553 |doi=10.2178/bsl/1102083759 |access-date=22 August 2023}}</ref>

Some infinitary logics may have different properties from those of standard [[first-order logic]]. In particular, infinitary logics may fail to be [[Compactness (logic)|compact]] or [[Completeness (logic)|complete]]. Notions of compactness and completeness that are equivalent in [[finitary logic]] sometimes are not so in infinitary logics. Therefore for infinitary logics, notions of strong compactness and strong completeness are defined. This article addresses [[Hilbert system|Hilbert-type]] infinitary logics, as these have been extensively studied and constitute the most straightforward extensions of finitary logic. These are not, however, the only infinitary logics that have been formulated or studied.

Considering whether a certain infinitary logic named [[Ω-logic]] is complete promises to throw light on the [[continuum hypothesis]].<ref>{{cite book |last=Woodin |first=W. Hugh |author-link=W. Hugh Woodin |date=2011 |chapter=The Continuum Hypothesis, the generic-multiverse of sets, and the Ω Conjecture |chapter-url=https://dokumen.tips/documents/the-continuum-hypothesis-the-generic-multiverse-of-logic-continuum-hypothesis.html |editor-last1=Kennedy |editor-first1=Juliette |editor-link1=Juliette Kennedy |editor-last2=Kossak |editor-first2=Roman |title=Set Theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies |publisher=Cambridge University Press |pages=13–42 |doi=10.1017/CBO9780511910616.003 |isbn=978-0-511-91061-6 |access-date=1 March 2024 |archive-date=1 March 2024 |archive-url=https://web.archive.org/web/20240301200503/https://dokumen.tips/documents/the-continuum-hypothesis-the-generic-multiverse-of-logic-continuum-hypothesis.html |url-status=dead }}</ref>