Comparison of effective health ages between the sophisticated model for researchers and simplified model for patients using GH-Method: math-physical medicine (No. 323)

This is the fourth article the author has written regarding the subject of effective health age (“Health Age”) related to the medical branch of geriatrics. Originally, he used his metabolism indexes data which were collected and processed via a sophisticated software for researchers. Later, he developed a simplified APP on the iPhone for other patients. This specific article discusses the differences of health input data and output results based on metabolism indexes and estimated health ages between these two different software versions. A comparison study between the difference of estimated health ages by using two different computer software versions was completed. The finding indicates that the complex metabolism model of his chronic software version would gain an extra 1.4% of accuracy on estimating his health age when compared to the simplified APP version. The author is not a fortune teller who uses a crystal ball to predict his or other people’s future life expectancy. Rather, he is a scientist who applies solid and sophisticated scientific techniques, such as math-physical medicine with biomedical evidence, to develop a simple arithmetical formula which can serve as a useful tool for the general population to maintain their health and achieve their desired longevity.

Comparison of effective health ages between the sophisticated model for researchers and simplified model for patients using GH-Method: math-physical medicine (No. 323)

This is the fourth article the author has written regarding the subject of effective health age (“Health Age”) related to the medical branch of geriatrics. Originally, he used his metabolism indexes data which were collected and processed via a sophisticated software for researchers. Later, he developed a simplified APP on the iPhone for other patients. This specific article discusses the differences of health input data and output results based on metabolism indexes and estimated health ages between these two different software versions. A comparison study between the difference of estimated health ages by using two different computer software versions was completed. The finding indicates that the complex metabolism model of his chronic software version would gain an extra 1.4% of accuracy on estimating his health age when compared to the simplified APP version. The author is not a fortune teller who uses a crystal ball to predict his or other people’s future life expectancy. Rather, he is a scientist who applies solid and sophisticated scientific techniques, such as math-physical medicine with biomedical evidence, to develop a simple arithmetical formula which can serve as a useful tool for the general population to maintain their health and achieve their desired longevity.

On Some Properties of Positive and Strongly Positive Arithmetical Sets

The notions of positive and strongly positive arithmetical sets are defined as in [1]-[4] (see, for example, [2], p. 33). It is proved (Theorem 1) that any arithmetical set is positive if and only if it can be defined by an arithmetical formula containing only logical operations ∃, &,∨ and the elementary subformulas having the forms 𝑥𝑥=0 or 𝑥𝑥=𝑦𝑦+1, where 𝑥𝑥and 𝑦𝑦 are variables.Corollary:the logical description of the class of positive sets is obtained from the logical description of the class of strongly positive sets replacing the list of operations &,∨ by the list ∃, &,∨. It is proved (Theorem 2) that for any one-dimensional recursively enumerable set 𝑀𝑀 there exists 6- dimensional strongly positive set 𝐻𝐻 such that 𝑥𝑥 ∈𝑀𝑀 holds if and only if (1, 2𝑥𝑥, 0, 0, 1, 0)∈𝐻𝐻+, where 𝐻𝐻+ is the transitive closure of 𝐻𝐻.

On the degrees of unsolvability of modal predicate logics of provability

The modal predicate logic of provability identifies the “□” of modal logic with the “Bew” of proof theory, so that, where “Bew” is a formula representing, in the usual way, provability in a consistent, recursively axiomatized theory Γ extending Peano arithmetic (PA), an interpretation of a language for the modal predicate calculus is a map * which associates with each modal formula an arithmetical formula with the same free variables which commutes with the Boolean connectives and the quantifiers and which sets (□ϕ)* equal to Bew(⌈ϕ*⌉). Where Δ is an extension of PA (all the theories we discuss will be extensions of PA), MPL(Δ) will be the set of modal formulas ϕ such that, for every interpretation *, ϕ* is a theorem of Δ. Most of what is currently known about the modal predicate logic of provability consists in demonstrations that MPL(Δ) must be computationally highly complex. Thus Vardanyan [11] shows that, provided that Δ is 1-consistent and recursively axiomatizable, MPL(Δ) will be complete , and Boolos and McGee [5] show that MPL({true arithmetical sentences}) is complete in {true arithmetical sentences}. All of these results take as their starting point Artemov's demonstration in [1] that {true arithmetical sentences} is 1-reducible to MPL({true arithmetical sentences}).The aim here is to consolidate these results by providing a general theorem which yields all the other results as special cases. These results provide a striking contrast with the situation in modal sentential logic (MSL); according to fundamental results of Solovay [8], provided Γ does not entail any falsehoods, MSL({true arithmetical sentences}) and MSL(PA) (which is the same as MSL(Γ)) are both decidable.

Three universal representations of recursively enumerable sets

In his celebrated paper of 1931 [7], Kurt Gödel proved the existence of sentences undecidable in the axiomatized theory of numbers. Gödel's proof is constructive and such a sentence may actually be written out. Of course, if we follow Gödel's original procedure the formula will be of enormous length.Forty-five years have passed since the appearance of Gödel's pioneering work. During this time enormous progress has been made in mathematical logic and recursive function theory. Many different mathematical problems have been proved recursively unsolvable. Theoretically each such result is capable of producing an explicit undecidable number theoretic predicate. We have only to carry out a suitable arithmetization. Until recently, however, techniques were not available for carrying out these arithmetizations with sufficient efficiency.In this article we construct an explicit undecidable arithmetical formula, F(x, n), in prenex normal form. The formula is explicit in the sense that it is written out in its entirety with no abbreviations. The formula is undecidable in the recursive sense that there exists no algorithm to decide, for given values of n, whether or not F(n, n) is true or false. Moreover F(n, n) is undecidable in the formal (axiomatic) sense of Gödel [7]. Given any of the usual axiomatic theories to which Gödel's Incompleteness Theorem applies, there exists a value of n such that F(n, n) is unprovable and irrefutable. Thus Gödel's Incompleteness Theorem can be “focused” into the formula F(n, n). Thus some substitution instance of F(n, n) is undecidable in Peano arithmetic, ZF set theory, etc.