Part of the book: Thermodynamics
The formulations of the undecidability of the Halting Problem assume that the computing process, being observed, the description of which is given on the input of the ’observing’ Turing Machine, is, at any given moment, the exact copy of the computing process running in the observing machine itself (the Cantor diagonal argument). In this way an infinite cycle is created shielding what is to be possibly discovered - the possible infinite cycle in the observed computing process. By this type of our consideration and in the thermodynamic sense the equilibrium status of a certain thermodynamic system is described or, even created. This is a thermodynamic image of the Cantor diagonal method used for seeking a possible infinite cycle and which, as such, has the property of the Perpetuum Mobile - the structure of which is recognizable and therefore we can avoid it. Thus we can show that it is possible to recognize the infinite cycle as a certain original equilibrium, but with a ’step-aside’ or a time delay in evaluating the trace of the observed computing process.
Part of the book: Recent Advances in Thermo and Fluid Dynamics
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.
Part of the book: Ontology in Information Science
We will demonstrate that the I. and the II. Caratheodory theorems and their common formulation as the II. Law of Thermodynamics are physically analogous with the real sense of the Gödel’s wording of his I. and II. incompleteness theorems. By using physical terms of the adiabatic changes the Caratheodory theorems express the properties of the Peano Arithmetic inferential process (and even properties of any deductive and recursively axiomatic inference generally); as such, they set the physical and then logical limits of any real inference (of the sound, not paradoxical thinking), which can run only on a physical/thermodynamic basis having been compared with, or translated into the formulations of the Gödel’s proof, they represent the first historical and clear statement of gnoseological limitations of the deductive and recursively axiomatic inference and sound thinking generally. We show that semantically understood and with the language of logic and meta-arithmetics, the full meaning of the Gödel proof expresses the universal validity of the II. law of thermodynamics and that the Peano arithmetics is not self-referential and is consistent.1
Part of the book: Ontological Analyses in Science, Technology and Informatics