Formal logic perspectives on non-standard analysis and its applications to the Paris-Harrington Theorem

This review presents the mathematical background necessary for understanding the original proof of the Paris-Harrington Theorem, the first natural non-provable result in Peano's arithmetic. The approach here chosen is based on the use of the non-standard analysis interpretation of combinatorics principles in Ramsey's theory. The vast majority of the work is therefore devoted to developing the necessary ideas from set theory, filters, hypernatural analysis, abstract logic, and Ramsey's theory. We also provide demonstrations of several of the results used in our method, including Los's Theorem and the Finite and Infinite Ramsey's Theorem, as well as conceptual justifications for Zorn's Lemma and Gödel's Incompleteness Theorems. No previous expertise of the fields here discussed is necessary and introductory knowledge of set theory and logic should be enough for this work.