Reverse Mathematics (RM) is a program in the foundations of mathematics where the aim is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. Generally, the minimal axioms are equivalent to the theorem at hand, assuming a weak logical system called the base theory. Moreover, many theorems are either provable in the base theory or equivalent to one of four logical systems, together called the Big Five. For instance, the Weierstrass approximation theorem, i.e., that a continuous function can be approximated uniformly by a sequence of polynomials, has been classified in RM as being equivalent to weak König’s lemma, the second Big Five system. In this paper, we study approximation theorems for discontinuous functions via Bernstein polynomials from the literature. We obtain many equivalences between the latter and weak König’s lemma. We also show that slight variations of these approximation theorems fall far outside of the Big Five but fit in the recently developed RM of new ‘big’ systems, namely the uncountability of ${\mathbb R}$, the enumeration principle for countable sets, the pigeon-hole principle for measure, and the Baire category theorem.

Continuous functions on the unit interval are relatively tame from the logical and computational point of view. A similar behaviour is exhibited by continuous functions on compact metric spaces equipped with a countable dense subset. It is then a natural question what happens if we omit the latter ‘extra data’, i.e., work with ‘unrepresented’ compact metric spaces. In this paper, we study basic third-order statements about continuous functions on such unrepresented compact metric spaces in Kohlenbach’s higher-order Reverse Mathematics. We establish that some (very specific) statements are classified in the (second-order) Big Five of Reverse Mathematics, while most variations/generalisations are not provable from the latter, and much stronger systems. Thus, continuous functions on unrepresented metric spaces are ‘wild’, though ‘more tame’ than (slightly) discontinuous functions on the reals.

The smooth development of large parts of mathematics hinges on the idea that some sets are ‘small’ or ‘negligible’ and can therefore be ignored for a given purpose. The perhaps most famous smallness notion, namely ‘measure zero’, originated with Lebesgue, while a second smallness notion, namely ‘meagre’ or ‘first category’, originated with Baire around the same time. The associated Baire category theorem is a central result governing the properties of meagre (and related) sets, while the same holds for Tao’s pigeonhole principle for measure spaces and measure zero sets. In this paper, we study these theorems in Kohlenbach’s higher-order Reverse Mathematics, identifying a considerable number of equivalent and robust theorems. The latter involve most basic properties of semi-continuous and pointwise discontinuous functions, Blumberg’s theorem, Riemann integration, and Volterra’s early work circa 1881. All the aforementioned theorems fall (far) outside of the Big Five of Reverse Mathematics, and we investigate natural restrictions like Baire 1 and quasi-continuity that make these theorems provable again in the Big Five (or similar). Finally, despite the fundamental differences between measure and category, the proofs of our equivalences turn out to be similar.

The uncountability of $\mathbb {R}$ is one of its most basic properties, known far outside of mathematics. Cantor’s 1874 proof of the uncountability of $\mathbb {R}$ even appears in the very first paper on set theory, i.e., a historical milestone. In this paper, we study the uncountability of ${\mathbb R}$ in Kohlenbach’s higher-order Reverse Mathematics (RM for short), in the guise of the following principle:

$$\begin{align*}\mathit{for \ a \ countable \ set } \ A\subset \mathbb{R}, \mathit{\ there \ exists } \ y\in \mathbb{R}\setminus A. \end{align*}$$

An important conceptual observation is that the usual definition of countable set—based on injections or bijections to ${\mathbb N}$—does not seem suitable for the RM-study of mainstream mathematics; we also propose a suitable (equivalent over strong systems) alternative definition of countable set, namely union over ${\mathbb N}$ of finite sets; the latter is known from the literature and closer to how countable sets occur ‘in the wild’. We identify a considerable number of theorems that are equivalent to the centred theorem based on our alternative definition. Perhaps surprisingly, our equivalent theorems involve most basic properties of the Riemann integral, regulated or bounded variation functions, Blumberg’s theorem, and Volterra’s early work circa 1881. Our equivalences are also robust, promoting the uncountability of ${\mathbb R}$ to the status of ‘big’ system in RM.

Reverse Mathematics (RM hereafter) is a program in the foundations of mathematics where the aim is to identify the minimal axioms needed to prove a given theorem from ordinary, i.e., non-set theoretic, mathematics. This program has unveiled surprising regularities: the minimal axioms are very often equivalent to the theorem over the base theory, a weak system of ‘computable mathematics’, while most theorems are either provable in this base theory, or equivalent to one of only four logical systems. The latter plus the base theory are called the ‘Big Five’ and the associated equivalences are robust following Montalbán, i.e., stable under small variations of the theorems at hand. Working in Kohlenbach’s higher-order RM, we obtain two new and long series of equivalences based on theorems due to Bolzano, Weierstrass, Jordan, and Cantor; these equivalences are extremely robust and have no counterpart among the Big Five systems. Thus, higher-order RM is much richer than its second-order cousin, boasting at least two extra ‘Big’ systems.

Cantor’s first set theory paper (1874) establishes the uncountability of ${\mathbb R}$ . We study this most basic mathematical fact formulated in the language of higher-order arithmetic. In particular, we investigate the logical and computational properties of ${\mathsf {NIN}}$ (resp. ${\mathsf {NBI}}$ ), i.e., the third-order statement there is no injection resp. bijection from  $[0,1]$ to ${\mathbb N}$ . Working in Kohlenbach’s higher-order Reverse Mathematics, we show that ${\mathsf {NIN}}$ and ${\mathsf {NBI}}$ are hard to prove in terms of (conventional) comprehension axioms, while many basic theorems, like Arzelà’s convergence theorem for the Riemann integral (1885), are shown to imply ${\mathsf {NIN}}$ and/or ${\mathsf {NBI}}$ . Working in Kleene’s higher-order computability theory based on S1–S9, we show that the following fourth-order process based on ${\mathsf {NIN}}$ is similarly hard to compute: for a given $[0,1]\rightarrow {\mathbb N}$ -function, find reals in the unit interval that map to the same natural number.

ABSTRACT IMPACT: This work will standardize necessary image pre-processing for diagnostic and prognostic clinical workflows dependent on quantitative analysis of conventional magnetic resonance imaging. OBJECTIVES/GOALS: Conventional magnetic resonance imaging (MRI) poses challenges for quantitative analysis due to a lack of uniform inter-scanner voxel intensity values. Head and neck cancer (HNC) applications in particular have not been well investigated. This project aims to systematically evaluate voxel intensity standardization (VIS) methods for HNC MRI. METHODS/STUDY POPULATION: We utilize two separate cohorts of HNC patients, where T2-weighted (T2-w) MRI sequences were acquired before beginning radiotherapy for five patients in each cohort. The first cohort corresponds to patients with images taken at various institutions with a variety of non-uniform acquisition scanners and parameters. The second cohort corresponds to patients from a prospective clinical trial with uniformity in both scanner and acquisition parameters. Regions of interest from a variety of healthy tissues assumed to have minimal interpatient variation were manually contoured for each image and used to compare differences between a variety of VIS methods for each cohort. Towards this end, we implement a new metric for cohort intensity distributional overlap to compare region of interest similarity in a given cohort. RESULTS/ANTICIPATED RESULTS: Using a simple and interpretable metric, we have systematically investigated the effects of various commonly implementable VIS methods on T2-w sequences for two independent cohorts of HNC patients based on region of interest intensity similarity. We demonstrate VIS has a substantial effect on T2-w images where non-uniform acquisition parameters and scanners are utilized. Oppositely, it has a modest to minimal impact on T2-w images generated from the same scanner with the same acquisition parameters. Moreover, with a few notable exceptions, there does not seem to be a clear advantage or disadvantage to using one VIS method over another for T2-w images with non-uniform acquisition parameters. DISCUSSION/SIGNIFICANCE OF FINDINGS: Our results inform which VIS methods should be favored in HNC MRI and may indicate VIS is not a critical factor to consider in circumstances where similar acquisition parameters can be utilized. Moreover, our results can help guide downstream quantitative imaging tasks that may one day be implemented in clinical workflows.

We investigate the connections between computability theory and Nonstandard Analysis. In particular, we investigate the two following topics and show that they are intimately related.

(T.1) A basic property of Cantor space$2^ $ is Heine–Borel compactness: for any open covering of $2^ $, there is a finite subcovering. A natural question is: How hard is it to compute such a finite subcovering? We make this precise by analysing the complexity of so-called fan functionals that given any $G:2^ \to $, output a finite sequence $\langle f_0 , \ldots ,f_n \rangle $ in $2^ $ such that the neighbourhoods defined from $\overline {f_i } G\left( {f_i } \right)$ for $i \le n$ form a covering of $2^ $.

(T.2) A basic property of Cantor space in Nonstandard Analysis is Abraham Robinson’s nonstandard compactness, i.e., that every binary sequence is “infinitely close” to a standard binary sequence. We analyse the strength of this nonstandard compactness property of Cantor space, compared to the other axioms of Nonstandard Analysis and usual mathematics.

Our study of (T.1) yields exotic objects in computability theory, while (T.2) leads to surprising results in Reverse Mathematics. We stress that (T.1) and (T.2) are highly intertwined, i.e., our study is holistic in nature in that results in computability theory yield results in Nonstandard Analysis and vice versa.

Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis with a PRA consistency proof, proposed around 1995 by Patrick Suppes and Richard Sommer. Recently, the author showed the consistency of ERNA with several transfer principles and proved results of nonstandard analysis in the resulting theories (see [12] and [13]). Here, we show that Weak König's lemma (WKL) and many of its equivalent formulations over RCA0 from Reverse Mathematics (see [21] and [22]) can be ‘pushed down’ into the weak theory ERNA, while preserving the equivalences, but at the price of replacing equality with equality ‘up to infinitesimals’. It turns out that ERNA plays the role of RCA0 and that transfer for universal formulas corresponds to WKL.

Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis with a PRA consistency proof, proposed around 1995 by Patrick Suppes and Richard Sommer. It is built on a previous system by Rolando Chuaqui and Patrick Suppes, which was recently reconsidered by Michal Rössler and Emil Jeřábek. A Π1-transfer principle has already been added to ERNA and the consistency of the resulting theory proved in PRA. Here, we equip ERNA with Σ2-transfer and a saturation principle, while keeping the consistency proof inside PRA. We show that the extended theory allows for generalized transfer, a basic tool of nonstandard analysis, and interprets several strong theories, like BΣ2 and IΣ2.

Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis proposed around 1995 by Patrick Suppes and Richard Sommer, who also proved its consistency inside PRA. It is based on an earlier system developed by Rolando Chuaqui and Patrick Suppes, of which Michal Rössler and Emil Jeřábek have recently proposed a weakened version. We add a Πı-transfer principle to ERNA and prove the consistency of the extended theory inside PRA. In this extension of ERNA a Σı-supremum principle ‘up-to-infinitesimals’, and some well-known calculus results for sequences are deduced. Finally, we prove that transfer is ‘too strong’ for finitism by reconsidering Rössler and Jeřábek's conclusions.