Abstract
This paper suggests motivations and goals of the program known as Reverse Mathematics, providing some illustrative examples of the many techniques and problems involved in working within subsystems of second order arithmetic, namely, in particular, RCA0, WKL0, ACA0, ATR0. Some examples from Combinatorics, — the Free Set Theorem and Ramsey’s Theorem — show how some theorems of ordinary mathematics may not fit in one of the subsystems mentioned aboves. As application of Reverse Mathematics to the History of Mathematics, we comment on König’s duality theorem and Cantor’s proof that every countable closed set is a set of uniqueness. Also, more technically, we present some results related to Lebesgue spaces (every open covering has a Lebesgue number) and Atsuji spaces (every continuous function defined on them is uniformly continuous); we show that the known proof of “every Atsuji space is Lebesgue” needs ACA0, and we conjecture that the statement is actually equivalent to ACA0. Finally, we discuss some limitations for Reverse Mathematics that may lead to projects of research in this field of mathematics.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
