infinitary logic is a logic that allows infinitely long statements or infinitely long proofs, for example, by allowing conjunctions, disjunctions and quantifier sequences, to be of infinite length. The price to pay for adopting many of these more expressive logics is the failure of completeness or of compactness. References
An infinitary logic was introduced for the first time in
Über Stufen der Quantifikation und die Logik des Unendlichen, Jahresbericht der Deutschen Mathematiker-Vereinigung 41/42 (1931), pp. 568–570. ( gdz)
Other early works include Novikov and
D. A. Bochvar,
Über einen Aussagenkalkül mit abzählbaren logischen Summen und Produkten, Mathematičeskii Sbornik 17 (1940), pp 65–100.
Infinitary logic has been recently studied extensively by
Saharon Shelah and in categorical logic by Mihaly Makkai.
Rami Grossberg, Classification theory for abstract elementary classes. Logic and Algebra, ed. Yi Zhang, Contemporary Mathematics, Vol 302, AMS, (2002), pp. 165–204, pdf
Rami Grossberg, Saharon Shelah,
On the number of non isomorphic models of an infinitary theory which has the order property, Part A, Journal of Symbolic Logic, 51, (1986) 302–322, jstor pdf
Revised on November 22, 2016 09:36:55