Keywords
incompleteness
Gödel’s theorem
axiomatic theory
logic
Gödel’s theorem
axiomatic theory
logic
How to Cite
Gödel’s Incompleteness Theorem and the Relation of Logical Consequence. (2010). Teorie vědy Theory of Science, 32(1), 59-95. https://doi.org/10.46938/tv.2010.65
Abstract
In his proof of the first incompleteness theorem, Kurt Gödel provided a method of showing the truth of specific arithmetical statements on the condition that all the axioms of a certain formal theory of arithmetic are true. Furthermore, the statement whose truth is shown in this way cannot be proved in the theory in question. Thus it may seem that the relation of logical consequence is wider than the relation of derivability by a pre-defined set of rules. The aim of this paper is to explore under which assumptions the Gödelian statement can rightly be considered a logical consequence of the axioms of the theory in question. It is argued that this is the case only when the all the theorems of the theory in question are understood as statements of the same kind (and true in the same sense) as statements of arithmetic and statements about provability in the theory, and only if the language of the theory contains logical expressions allowing to include certain predicates of meta-language in the language of the theory.Since 2019, TEORIE VĚDY / THEORY OF SCIENCE journal provides open access to its content under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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