Infinitary logic

An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs.[1] The concept was introduced by Zermelo in the 1930s.[2]

Some infinitary logics may have different properties from those of standard first-order logic. In particular, infinitary logics may fail to be compact or 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-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.[3]

  1. ^ Moore, Gregory H. (1997). "The prehistory of infinitary logic: 1885–1955". In Dalla Chiara, Maria Luisa; Doets, Kees; Mundici, Daniele; van Benthem, Johan (eds.). Structures and Norms in Science. Springer-Science+Business Media. pp. 105–123. doi:10.1007/978-94-017-0538-7_7. ISBN 978-94-017-0538-7.
  2. ^ Kanamori, Akihiro (2004). "Zermelo and set theory" (PDF). The Bulletin of Symbolic Logic. 10 (4): 487–553. doi:10.2178/bsl/1102083759. Retrieved 22 August 2023.
  3. ^ Woodin, W. Hugh (2011). "The Continuum Hypothesis, the generic-multiverse of sets, and the Ω Conjecture". In Kennedy, Juliette; Kossak, Roman (eds.). Set Theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies. Cambridge University Press. pp. 13–42. doi:10.1017/CBO9780511910616.003. ISBN 978-0-511-91061-6. Archived from the original on 1 March 2024. Retrieved 1 March 2024.