Information Transfer and Thermodynamic Point of View on Goedel Proof
Bohdan Hejna · InTech eBooks · 2018
Formula of an arithmetic theory based on Peano Arithmetics (including it) is a chain of symbols of its super-language (in which the theory is formulated).Such a chain is in convenience both with the syntax of the super-language and with the inferential rules of the theory (Modus Ponens, Generalization).Syntactic rules constructing formulas of the theory are not its inferential rules.Although the super-language syntax is defined recursively-by the recursive writing of mathematical-logical claims-only those recursively written super-language's chains which formulate mathematical-logical claims about finite sets of individual of the theory, computable totally (thus recursive) and always true are the formulas of the theory.Formulas of the theory are not those claims which are true as for the individual of the theory, but not inferable within the theory (Great Fermat's Theorem).They are provable but within another theory (with both Peano and further axioms).Also the chains expressing methodological claims, even being written recursively (Goedel Undecidable Formula) are not parts of the theory.The same applies to their negations.We show that the Goedel substitution function is not the total one and thus is not recursive.It is not defined for the Goedel Undecidable Formula's construction.For this case, the structure of which is visible clearly, we are adding the zero value.This correction is based on information, thermodynamic and computing considerations, simplifies the Goedel original proof, and is valid for the consistent arithmetic theories directly.