An important open problem in Reverse Mathematics is the reduction of the first-order strength of the base theory from IΣ1IΣ1 to IΔ0+expIΔ0+exp. The system ERNA, a version of Nonstandard Analysis based on the system IΔ0+expIΔ0+exp, provides a partial solution to this problem. Indeed, weak Königʼs lemma and many of its equivalent formulations from Reverse Mathematics can be ‘pushed down’ into ERNA, while preserving the equivalences, but at the price of replacing equality with ‘≈’, i.e. infinitesimal proximity . The logical principle (...) corresponding to weak Königʼs lemma is the universal transfer principle from Nonstandard Analysis. Here, we consider the intermediate and mean value theorem and their uniform generalizations. We show that ERNAʼs Reverse Mathematics mirrors the situation in classical Reverse Mathematics. This further supports our claim from Sanders [19] that the Reverse Mathematics of ERNA plus universal transfer is a copy up to infinitesimals of that of WKL0. We discuss some of the philosophical implications of our results. (shrink)

The program of Reverse Mathematics (Simpson 2009) has provided us with the insight that most theorems of ordinary mathematics are either equivalent to one of a select few logical principles, or provable in a weak base theory. In this paper, we study the properties of the Dirac delta function (Dirac 1927; Schwartz 1951) in two settings of Reverse Mathematics. In particular, we consider the Dirac Delta Theorem, which formalizes the well-known property \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} (...) \begin{document}$${\int_\mathbb{R}f(x)\delta(x)\,dx=f(0)}$$\end{document} of the Dirac delta function. We show that the Dirac Delta Theorem is equivalent to weak König’s Lemma (see Yu and Simpson in Arch Math Log 30(3):171–180, 1990) in classical Reverse Mathematics. This further validates the status of WWKL0 as one of the ‘Big’ systems of Reverse Mathematics. In the context of ERNA’s Reverse Mathematics (Sanders in J Symb Log 76(2):637–664, 2011), we show that the Dirac Delta Theorem is equivalent to the Universal Transfer Principle. Since the Universal Transfer Principle corresponds to WKL, it seems that, in ERNA’s Reverse Mathematics, the principles corresponding to WKL and WWKL coincide. Hence, ERNA’s Reverse Mathematics is actually coarser than classical Reverse Mathematics, although the base theory has lower first-order strength. (shrink)

Kohlenbach's proof mining program deals with the extraction of effective information from typically ineffective proofs. Proof mining has its roots in Kreisel's pioneering work on the so-called unwinding of proofs. The proof mining of classical mathematics is rather restricted in scope due to the existence of sentences without computational content which are provable from the law of excluded middle and which involve only two quantifier alternations. By contrast, we show that the proof mining of classical Nonstandard Analysis has a very (...) large scope. In particular, we will observe that this scope includes any theorem of pure Nonstandard Analysis, where 'pure' means that only nonstandard definitions are used. In this note, we survey results in analysis, computability theory, and Reverse Mathematics. (shrink)