An alternative to back-and-forth

TL;DR

This paper demonstrates how the Rasiowa-Sikorski Lemma can simplify proofs traditionally using back-and-forth techniques, providing clearer arguments and extending applications to Graph Theory.

Contribution

It introduces a novel approach using the Rasiowa-Sikorski Lemma to simplify and clarify proofs, including original proofs in Graph Theory.

Findings

Simplified proofs of known results using Rasiowa-Sikorski Lemma

Clearer presentation of back-and-forth arguments

Extension of techniques to Graph Theory

Abstract

We exhibit how the Rasiowa-Sikorski Lemma simplifies, in a sense, proofs of results that make use of the technique known as back-and-forth, often resulting in not very illustrative arguments. The first two sections seek to show one simple and one complicated proofs of known results, in the hopes that the reader appreciates how the arguments end up, in our view, considerably clearer than those found in classic literature. The final section shows how the same techniques can be adapted to areas commonly considered distant to Set Theory, in this instance, Graph Theory. Sections one and two are based on my bachelor thesis, under the direction of Dr. Roberto Pichardo Mendoza, whom I deeply thank for his advice and revision of this work. All results mentioned in this paper are well known, however, as far as we know, the proofs in the last two sections are original.