Infinitary Logics

  • Jon Barwise
Part of the Synthese Library book series (SYLI, volume 149)


We begin with two examples. The sentence “No one has more than a finite number of ancestors” might be written symbolically as
$$forall x[{A_0}(x)\nu {A_1}(x)\nu {A_2}(x)\nu \ldots etc.]$$
which would be read: for every person x (∀x) either x has no ancestors (A0(x)) or (v) he has one ancestor (A 1(x)), or he has two ancestors etc. Even if we interpret ‘person’ to mean past, present or future person, most of us would consider this sentence as being true. On the other hand, if one were to try to put an a priori upper bound n on the number of possible ancestors a person might have, and assert
$$forall x[{A_0}(x)\nu {A_1}(x)\nu {A_2}(x)\nu \ldots \nu {A_n}(x)]$$
then we could no longer agree, for one could always conceive of man some show surviving for another n=1 generations. Thus what we have is an infinitary sentence which is not logically equivalent to any of its finite approximations.


Test Sentence Negation Normal Form Provable Formula Abstract Model Theory Regular Formula 
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Added in Proof

  1. 1.
    Wilfred Seig has drawn my attention to a little known anti-formalist paper of Zermelo which studies the propositional part of Lω, ω: ‘Grundlagen einer allgemeinen Theorie der mathematischen Satzsysteme’, Fundamenta Mathematica 25 (1935), 136–146.Google Scholar
  2. 2.
    A technical version of this paper will appear as ‘The Role of the Omitting Types Theorem in Infinitory Logic’, in Arch. f. math. Logik u. Grundl. Forschung.Google Scholar

Copyright information

© D. Reidel Publishing Company 1981

Authors and Affiliations

  • Jon Barwise
    • 1
    • 2
  1. 1.University of WisconsinMadisonUSA
  2. 2.Stanford UniversityUSA

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