Related papers: Randomness notions and reverse mathematics
A new version of a strong law of large numbers for a ``good'' pairwise independent sequence of random variables (r.v.'s) with a small part of ``bad'' dependent r.v.'s is proposed. The main goal is to relax the assumption on the existence of…
A problem is a multivalued function from a set of \emph{instances} to a set of \emph{solutions}. We consider only instances and solutions coded by sets of integers. A problem admits preservation of some computability-theoretic weakness…
Turing's famous 'machine' framework provides an intuitively clear conception of 'computing with real numbers'. A recursive counterexample to a theorem shows that the theorem does not hold when restricted to computable objects. These…
Solomonoff unified Occam's razor and Epicurus' principle of multiple explanations to one elegant, formal, universal theory of inductive inference, which initiated the field of algorithmic information theory. His central result is that the…
We study algorithmic randomness properties for probability measures on Cantor space. We say that a measure $\mu$ on the space of infinite bit sequences is ML absolutely continuous if the non-ML-random bit sequences form a null set with…
A type-2 computable real function is necessarily continuous; and this remains true for relative, i.e. oracle-based computations. Conversely, by the Weierstrass Approximation Theorem, every continuous f:[0,1]->R is computable relative 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…
In this paper we examine the reverse mathematical strength of a variation of Hindman's Theorem HT constructed by essentially combining HT with the Thin Set Theorem TS to obtain a principle which we call thin-HT. thin-HT says that every…
It is known that several variations of the axiom of determinacy play important roles in the study of reverse mathematics, and the relation between the hierarchy of determinacy and comprehension are revealed by Tanaka, Nemoto, Montalb\'an,…
We survey the Kolmogorov's approach to the notion of randomness through the Kolmogorov complexity theory. The original motivation of Kolmogorov was to give up a quantitative definition of information. In this theory, an object is randomness…
We investigate the statement "the order topology of every countable complete linear order is compact" in the framework of reverse mathematics, and we find that the statement's strength depends on the precise formulation of compactness. If…
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…
Algorithmic randomness theory starts with a notion of an individual random object. To be reasonable, this notion should have some natural properties; in particular, an object should be random with respect to image distribution if and only…
We show that there is a strong connection between Weihrauch reducibility on one hand, and provability in EL_0, the intuitionistic version of RCA_0, on the other hand. More precisely, we show that Weihrauch reducibility to the composition of…
This article critically reappraises arguments in support of Cantor's theory of transfinite numbers. The following results are reported: i) Cantor's proofs of nondenumerability are refuted by analyzing the logical inconsistencies in…
G\"odel's second incompleteness theorem is standardly understood as showing that no sufficiently strong, consistent theory of arithmetic can prove its own consistency, a result typically interpreted against a model-theoretic background in…
Recently it has been proved that, assuming that there is an almost disjoint family of cardinality (2^{\mathfrak c}) in (\mathfrak c) (which is assured, for instance, by either Martin's Axiom, or CH, or even $2^{<\mathfrak c=\mathfrak c$})…
In his first set theory paper (1874), Cantor establishes the uncountability of $\mathbb{R}$. We study the latter in Kohlenbach's higher-order Reverse Mathematics, motivated by the observation that one cannot study concepts like `arbitrary…
We prove that for an arbitrary subtree $T$ of $2^{<\omega}$ with each element extendable to a path, a given countable class $\mathcal{M}$ closed under disjoint union, and any set $A$, if none of the members of $\mathcal{M}$ strongly…
A Ramsey-like theorem is a statement of the form ``For every 2-coloring of $[\mathbb{N}]^2$, there exists an infinite set~$H \subseteq \mathbb{N}$ such that $[H]^2$ avoids some pattern''. We prove that none of these statements are…