An Open Formalism against Incompleteness

Gödel's Incompleteness Theorem and the Philosophy of Open Systems

In recent years a number of criticisms have been raised against the formal systems of mathematical logic. The latter, qualified as closed systems, have been contrasted with systems of a new kind, called open systems, whose main feature is that they are always subject to unanticipated outcomes in their operation and can receive new information from outside at any time [cf. Hewitt 1991]. While Gö...

Rotation operator formalism for open-shell omplexes

Biconsequence Relations: A Four-Valued Formalism of Reasoning with Inconsistency and Incompleteness

We suggest a general formalism of four-valued reasoning, called biconsequence relations, intended to serve as a logical framework for reasoning with incomplete and inconsistent data. The formalism is based on a four-valued semantics suggested by Belnap. As for the classical sequent calculus, any fourvalued connective can be defined in biconsequence relations using suitable introduction and elim...

AN INCOMPLETENESS THEOREM FOR ân-MODELS

Let n be a positive integer. By a ân -model we mean an ù-model which is elementary with respect to Σn formulas. We prove the following ân -model version of Gödel’s Second Incompleteness Theorem. For any recursively axiomatized theory S in the language of second order arithmetic, if there exists a ân-model of S, then there exists a ân-model of S + “there is no countable ân-model of S”. We also p...

An implication of Gödel’s incompleteness theorem

A proof of Gödel’s incompleteness theorem is given. With this new proof a transfinite extension of Gödel’s theorem is considered. It is shown that if one assumes the set theory ZFC on the meta level as well as on the object level, a contradiction arises. The cause is shown to be the implicit identification of the meta level and the object level hidden behind the Gödel numbering. An implication ...