Related papers: Preliminary investigations on induction over real …
In introductory books about natural numbers, a common kind of assertion - often left as exercise to the reader - is that certain forms of induction on $\mathbb{N}$ (regular/ordinary, complete/strong) are equivalent one to each other and to…
We introduce real induction, a proof technique analogous to mathematical induction but applicable to statements indexed by an interval on the real line. More generally we give an inductive principle applicable in any Dedekind complete…
Heritability is a central concept in the long-standing debate about nature versus nurture in biological and social sciences. However, existing notions of heritability are based on strong assumptions and do not use explicit causal models. We…
The purpose of this paper is to give an easy to understand with step-by-step explanation to allow interested people to fully appreciate the power of natural deduction for first-order logic. Natural deduction as a proof system can be used to…
When are all positions of a game numbers? We show that two properties are necessary and sufficient. These properties are consequences of that, in a number, it is not an advantage to be the first player. One of these properties implies the…
We advance a famous principle - causality principle - but under a new view. This principle is a principium automatically leading to most fundamental laws of the nature. It is the inner origin of variation, rules evolutionary processes of…
Natural numbers satisfying an unusual property are mentioned by the author in [5], in which their infinitude is also proved. In this paper, we start with an arbitrary natural number which is not a multiple of 10 and non-palindromic, form…
This article solves the Hume's problem of induction using a probabilistic approach. From the probabilistic perspective, the core task of induction is to estimate the probability of an event and judge the accuracy of the estimation.…
An extension to the factorisation principle as suggested by Fermat is presented.We start from a symmetry of natural numbers and obtain the factorisation principle therefrom.Later it is extended further to test the primality of any natural…
The theory of natural selection has two forms. Deductive theory describes how populations change over time. One starts with an initial population and some rules for change. From those assumptions, one calculates the future state of the…
While the prime numbers have been subject to mathematical inquiry since the ancient Greeks, the accumulated effort of understanding these numbers has - as Marcus du Sautoy recently phrased it - 'not revealed the origins of what makes the…
We show that induction of covariant representations for C*-dynamical systems is natural in the sense that it gives a natural transformation between certain crossed-product functors. This involves setting up suitable categories of…
Prime numbers are fascinating by the way they appear in the set of natural numbers. Despite several results enlighting us about their repartition, the set of prime numbers is often informally qualified as misterious. In the present paper,…
We prove some theorems which give sufficient conditions for the existence of prime numbers among the terms of a sequence which has pairwise relatively prime terms.
Among three natural numbers there is always one which is larger than or equal to the Nim sum of the remaining two numbers. This amazing fact has many applications.
In this paper we continue our research on the concept of liken. This notion has been defined as a sequence of non-negative real numbers, tending to infinity and closed with respect to addition in $\mathbb{R}$. The most important examples of…
Mathematical induction is a fundamental tool in computer science and mathematics. Henkin initiated the study of formalization of mathematical induction restricted to the setting when the base case B is set to singleton set containing 0 and…
Let $g \geq 2$ be an integer. A natural number is said to be a base-$g$ Niven number if it is divisible by the sum of its base-$g$ digits. Assuming Hooley's Riemann Hypothesis, we prove that the set of base-$g$ Niven numbers is an additive…
Every natural number greater than two may be written as the sum of a prime and a square-free number. We establish several generalisations of this, by placing divisibility conditions on the square-free number.
In set theory without the axiom of regularity, we consider a game in which two players choose in turn an element of a given set, an element of this element, etc.; a player wins if its adversary cannot make any next move. Sets that are…