407 research outputs found
Consequences of a Goedel's misjudgment
The fundamental aim of the paper is to correct an harmful way to interpret a
Goedel's erroneous remark at the Congress of Koenigsberg in 1930. Despite the
Goedel's fault is rather venial, its misreading has produced and continues to
produce dangerous fruits, as to apply the incompleteness Theorems to the full
second-order Arithmetic and to deduce the semantic incompleteness of its
language by these same Theorems. The first three paragraphs are introductory
and serve to define the languages inherently semantic and its properties, to
discuss the consequences of the expression order used in a language and some
question about the semantic completeness: in particular is highlighted the fact
that a non-formal theory may be semantically complete despite using a language
semantically incomplete. Finally, an alternative interpretation of the Goedel's
unfortunate comment is proposed. KEYWORDS: semantic completeness, syntactic
incompleteness, categoricity, arithmetic, second-order languages, paradoxesComment: English version, 19 pages. Fixed and improved terminolog
Tameness in generalized metric structures
We broaden the framework of metric abstract elementary classes (mAECs) in
several essential ways, chiefly by allowing the metric to take values in a
well-behaved quantale. As a proof of concept we show that the result of Boney
and Zambrano on (metric) tameness under a large cardinal assumption holds in
this more general context. We briefly consider a further generalization to
partial metric spaces, and hint at connections to classes of fuzzy structures,
and structures on sheaves
Carnap's problem for intuitionistic propositional logic
We show that intuitionistic propositional logic is \emph{Carnap categorical}:
the only interpretation of the connectives consistent with the intuitionistic
consequence relation is the standard interpretation. This holds relative to the
most well-known semantics with respect to which intuitionistic logic is sound
and complete; among them Kripke semantics, Beth semantics, Dragalin semantics,
and topological semantics. It also holds for algebraic semantics, although
categoricity in that case is different in kind from categoricity relative to
possible worlds style semantics.Comment: Keywords: intuitionistic logic, Carnap's problem, nuclear semantics,
algebraic semantics, logical constants, consequence relations, categoricity.
Versions: 3rd version has minor additions, and correction of an error in 2nd
version (not in 1st version
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