Na\"ive Infinitesimal Analysis: Its Construction and Its Properties

This paper aims to build a new understanding of the nonstandard mathematical analysis. The main contribution of this paper is the construction of a new set of numbers, $\mathbb{R}^{\mathbb{Z}_< }$, which includes infinities and infinitesimals. The construction of this new set is done na\"ively in the sense that it does not require any heavy mathematical machinery, and so it will be much less problematic in a long term. Despite its na\"ivety character, the set $\mathbb{R}^{\mathbb{Z}_< }$ is still a robust and rewarding set to work in. We further develop some analysis and topological properties of it, where not only we recover most of the basic theories that we have classically, but we also introduce some new enthralling notions in them. The computability issue of this set is also explored. The works presented here can be seen as a contribution to bridge constructive analysis and nonstandard analysis, which has been extensively (and intensively) discussed in the past few years.