This paper is a step toward showing what is achievable using non-classical metatheory—particularly, a substructural paraconsistent framework. What standard results, or analogues thereof, from the classical metatheory of first order logic can be obtained? We reconstruct some of the originals proofs for Completeness, Löwenheim-Skolem and Compactness theorems in the context of a substructural logic with the naive comprehension schema. The main result is that paraconsistent metatheory can ‘re-capture’ versions of standard theorems, given suitable restrictions and background assumptions; but the shift (...) to non-classical logic may recast the meanings of these apparently ‘absolute’ theorems. (shrink)

In this paper an attempt is made to present Skolem's argument, for the relativity of some set-theoretical notions as a sensible one. Skolem's critique of set theory is seen as part of a larger argument to the effect that no conclusive evidence has been given for the existence of uncountable sets. Some replies to Skolem are discussed and are shown not to affect Skolem's position, since they all presuppose the existence of uncountable sets. The paper ends with an assessment of (...) the assumptions on which Skolem's argument rests from a present-day perspective. (shrink)

In this paper I discuss Ernst Zermelo’s ideas concerning the possibility of developing a system of infinitary logic that, in his opinion, should be suitable for mathematical inferences. The presentation of Zermelo’s ideas is accompanied with some remarks concerning the development of infinitary logic. I also stress the fact that the second axiomatization of set theory provided by Zermelo in 1930 involved the use of extremal axioms of a very specific sort.1.