Infinite-valued _ukasiewicz logic and Farey sequences

Abstract

The paper explores the relations between MacNaughton's criterion for infinite-valued _ukasiewicz logic, prime numbers and Farey sequences. The author gives a definition of prime numbers in terms of infinite-valued _ukasiewicz logic. According to MacNaughton's criterion, the set of functions expressed in infinite-valued _ukasiewicz logic coincides with the set of certain continuous piecewise linear functions. The paper shows that natural number $n$ is prime only if infinite-valued _ukasiewicz logic contains functions that the restriction to a proper finitely valued _ukasiewicz logic coincide with functions $N_{1/n}(x)$. While every such function has piecewise linear counterparts, linear parameters for which may be obtained in certain Farey sequences. Therefore, it is possible to find all regarded linear functions in the point with coordinates $(\frac{i}{n},\frac{1}{n})$. All such functions have equations $f(x)=b+kx$ with integer parameters $b$ and $k$, and $\frac{1}{n}=b+k\frac{i}{n}$, so it makes it possible to find the required parameters in certain Farey sequences. DOI: 10.21146/2074-1472-2018-24-2-123-128