nLab infinitary logic

Idea

An 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.

Related entries

• model theory

• first-order logic

• accessible category

• geometric theory

• Skolem paradox

References

An infinitary logic was introduced for the first time in

• Ernst Zermelo, 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 work of 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