W duchu Tarskiego: o alternatywach teorio-dowodowej metalogiki

the standard metalogical set-ups seem to be all based on the idea of consequence (or proof). However, metalogic can also be effectively constructed using some non-standard primmitive ideas. In this paper an outline is given to four metalogical frameworks, alternative to the standard set-ups. They are based, respectively, on the idea of consistency; on an omission (or separation) operator; on an extension operator (called in the paper a Lindenbaum operator); and on the idea of maximality. All these metalogics, including the standard consequence-based one, are one-to-one and proof-preserving translatable, and, in this sense, are seen pairwise equivalent and „externally” consistent.