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CARNAP ON EXTREMAL AXIOMS, “COMPLETENESS OF THE MODELS,” AND CATEGORICITY

Abstract This paper provides a historically sensitive discussion of Carnap’s theory of extremal axioms developed first in the late 1920s. The main focus is set on the unpublished documents of the projected second part of his manuscript Untersuchungen zur allgemeinen Axiomatik (RC 081-01-01 to 081-01-33). Carnap’s theory will be assessed with respect to two interpretive issues. The first concerns his mathematical sources, that is, the mathematical axioms on which his extremal axioms were based. The second concerns Carnap’s understanding of the relationship between the “completeness of the models” and other metatheoretic notions investigated by him at the time, most notably that of categoricity. The paper surveys Carnap’s different attempts to explicate the extremal properties of a theory and puts his results in context with related metamathematical research at the time.