The Representable Continuum Set

Dedicated to dmvo:

\documentclass{article}
\title{The Representable Continuum Set}
\author{Anton Salikhmetov}

\begin{document}
\maketitle

\begin{abstract}
This paper introduces a new mathematical structure representing
all rational numbers’ sequences that can be defined to provide a
countable replacement for the usual continuum set of real
numbers, the question whether it is appropriate, say, for physics
being still open.
\end{abstract}

The Representable Continuum Set consists of all combinators that
are computable for all Church numerals and for each Church
numeral return a lambda expression of the list form
${\lambda x. x b [m] [n]}$, where $m$ and $n$ are natural
numbers, and $b$ is a boolean value representing sign, which
corresponds to a rational number ${\pm m / n}$.
If $S_i$ is the corresponding rational number for ${(M [i])}$,
an element $M$ from this set is considered as
${\lim_{n \rightarrow \infty} \sum_{i = 0}^n S_i}$.
\end{document}