Der Satz von Hahn-Banach per Disjunktionselimination

Abstract

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The Jacobson Radical of a Propositional Theory

“…4 As compared with Gentzen’s and Lorenzen’s approaches, Scott’s entailment relation follow the contexts-as-sets paradigm, which has caused reservations [78, 79]. The relevance of the notion of entailment relation to point-free topology and constructive algebra has been pointed out in [14], and has been used very widely, e.g., in [20–22, 24, 25, 29, 32, 80, 89, 93, 100, 114, 115]. Consequence and entailment have further caught interest from various other angles [6, 35, 41, 52–54, 83, 99, 104, 117].…”

A universal algorithm for Krull's theorem

“…Incidentally, dynamical methods have been applied to Krull's theorem for prime ideals, the main focus of our work, only very recently [52], which however has immediately prompted a systematic presentation of these methods: a dynamical counterpart [50] of the syntactical conservation criterion [40,41] the semantics of which include the universal Krull-Lindenbaum theorem [39], the latter being crucial for the present paper as indicated above. With the axioms-as-rules paradigm [24,25,43], this conservation criterion further subsumes [42,45,60] and a wealth of cases from the literature, e.g. [4,10,19,23,26].…”

Towards formal Baer criteria

“…In this paper we attempt to put Baer's criterion, which allows to detect injective objects in categories of modules, on constructive grounds by couching it in point-free terms. The strategy we adopt has been informed by the localic version of the Hahn-Banach theorem, due to which the latter be interpreted as a conservation result of formal theories [9,10,25,11,33].…”