Related papers: The reverse mathematics of non-decreasing subseque…
Restriction is a natural quasi-order on $d$-way tensors. We establish a remarkable aspect of this quasi-order in the case of tensors over a fixed finite field -- namely, that it is a well-quasi-order: it admits no infinite antichains and no…
This paper is a contribution to the growing investigation of strong reducibilities between $\Pi^1_2$ statements of second-order arithmetic, viewed as an extension of the traditional analysis of reverse mathematics. We answer several…
In this article, we prove two new versions of a theorem proven by Efron in [Efr65]. Efron's theorem says that if a function $\phi : \mathbb{R}^2 \rightarrow \mathbb{R}$ is non-decreasing in each argument then we have that the function $s…
Suppose you have an uncomputable set $X$ and you want to find a set $A$, all of whose infinite subsets compute $X$. There are several ways to do this, but all of them seem to produce a set $A$ which is fairly sparse. We show that this is…
We study the topological, dynamical, and descriptive set theoretic properties of Hurwitz continued fractions. Hurwitz continued fractions associate an infinite sequence of Gaussian integers to every complex number which is not a Gaussian…
We characterize the computational content and the proof-theoretic strength of a Ramsey-type theorem for bi-colorings of so-called {\em exactly large} sets. An {\it exactly large} set is a set $X\subset\Nat$ such that $\card(X)=\min(X)+1$.…
A classical theorem of Lusin states that all analytic sets are Lebesgue-measurable. In this article we established the reverse mathematical strength of Lusin's theorem, which depends on how precisely it is formalized. By doing so, we answer…
We study the reverse mathematics of pigeonhole principles for finite powers of the ordinal $\omega$. Four natural formulations are presented and their relative strengths are compared. In the analysis of the pigeonhole principle for…
In intuitionistic mathematics, the Brouwer Continuity Theorem states that all total real functions are (uniformly) continuous on the unit interval. We study this theorem and related principles from the point of view of Reverse Mathematics…
Does the $n^{th}$ root of the diagonal Ramsey number converge to a finite limit? The answer is yes. A sequence can be shown to converge if it satifies convergence conditions other than or besides monotonicity. We show such a property holds…
We introduce an interesting method of proving separable reduction theorems - the method of elementary submodels. We are studying whether it is true that a set (function) has given property if and only if it has this property with respect to…
A rotational subset, relative to a continuous transformation $T: \mathbb{T} \to \mathbb{T}$ on the unit circle, is a closed, invariant subset of $\mathbb{T}$ that is minimal and on which $T$ respects the standard orientation of the unit…
We obtain a double exponential bound in Brauer's generalisation of van der Waerden's theorem, which concerns progressions with the same colour as their common difference. Such a result has been obtained independently and in much greater…
No natural principle is currently known to be strictly between the arithmetic comprehension axiom (ACA) and Ramsey's theorem for pairs (RT^2_2) in reverse mathematics. The tree theorem for pairs (TT^2_2) is however a good candidate. The…
Compactness is one of the core notions of analysis: it connects local properties to global ones and makes limits well-behaved. We study the computational properties of the compactness of Cantor space $2^{\mathbb{N}}$ for uncountable covers.…
This paper is about computability. I claim the likely existence of a program DoesHalt(Program, Input) such that DoesHalt( HaltsOnItself, AntiSelf ) halts with resounding 'NO'. HaltsOnItself( Program ) is simply DoesHalt( Program, Program ).…
We consider non oscillatory functions and prove an everywhere Fourier Inversion Theorem for functions of very moderate decrease. The proofs rely on some ideas in nonstandard analysis.
The tree theorem for pairs ($\mathsf{TT}^2_2$), first introduced by Chubb, Hirst, and McNicholl, asserts that given a finite coloring of pairs of comparable nodes in the full binary tree $2^{<\omega}$, there is a set of nodes isomorphic to…
We prove that the statement "there is a $k$ such that for every $f$ there is a $k$-bounded diagonally non-recursive function relative to $f$" does not imply weak K\"onig's lemma over $\mathrm{RCA}_0 + \mathrm{B}\Sigma^0_2$. This answers a…
A Turing degree d bounds a principle P of reverse mathematics if every computable instance of P has a d-computable solution. P admits a universal instance if there exists a computable instance such that every solution bounds P. We prove…