Related papers: Goodstein revisited
It is generally accepted that the incompleteness of first-order number theory (PA) is established by an application of Godel's proof. This paper shows that the arithmetization of the syntax of PA implies that the hypothesised class of PA…
In this short note we capitalize on and complete our previous results on the regularity of the homogenized coefficients for Bernoulli perturbations by addressing the case of the Poisson point process, for which the crucial uniform local…
This paper is about the surprising interaction of a foundational result from model theory, about stability of theories, with algorithmic stability in learning. First, in response to gaps in existing learning models, we introduce a new…
In this note, we show that, despite the widespread assumption, the consistency formula for Peano Arithmetic PA, Con(PA), "for all x, x is not a code of a derivation of (0=1)," is not equivalent in PA to the consistency of PA. Specifically,…
A formalisation of G\"odel's incompleteness theorems using the Isabelle proof assistant is described. This is apparently the first mechanical verification of the second incompleteness theorem. The work closely follows {\'S}wierczkowski…
How much dependence is there in the prime factorization of a random integer distributed uniformly from 1 to n? How much dependence is there in the decomposition into cycles of a random permutation of n points? What is the relation between…
Hochster's Monomial Conjecture and Canonical Element Conjecture date back some thirty resp. twenty years. They concern all noetherian commutative local rings, and were proved by their originator right away in equal characteristic. Thanks to…
It is shown that a criterion used to demonstrate realization of the 1935 Einstein-Podolsky-Rosen (EPR) gedanken experiment is sufficient to demonstrate quantum entanglement. A further set of measurable criteria sufficient to demonstrate EPR…
During the early 1830's Bernard Bolzano, working in Prague, wrote a manuscript giving a foundational account of numbers and their properties. In the final section of his work he described what he called `infinite number expressions' and…
Godelian sentences of a sufficiently strong and recursively enumerable theory, constructed in Godel's 1931 groundbreaking paper on the incompleteness theorems, are unprovable if the theory is consistent; however, they could be refutable.…
In an attempt to resolve a folklore conjecture of Erd\H{o}s regarding the non-vanishing at $s=1$ of the $L$-series attached to a periodic arithmetical function with period $q$ and values in $\{-1, 1 \}$, Livingston conjectured the…
We prove decidability results on the existence of constant subsequences of uniformly recurrent morphic sequences along arithmetic progressions. We use spectral properties of the subshifts they generate to give a first algorithm deciding…
In 1978, Apery has given sequences of rational approximations to $\zeta(2)$ and $\zeta(3)$ yielding the irrationality of each of these numbers. One of the key ingredient of Apery's proof are second-order difference equations with polynomial…
The original Goodstein process is based on writing numbers in hereditary $b$-exponential normal form: that is, each number $n$ is written in some base $b\geq 2$ as $n=b^ea+r$, with $e$ and $r$ iteratively being written in hereditary…
This paper deals with the problem of quantifying the approximation a probability measure by means of an empirical (in a wide sense) random probability measure, depending on the first n terms of a sequence of random elements. In Section 2,…
Joinings are fundamental global objects in ergodic theory, yet in compact metric models one naturally observes only finite orbit-distance patterns. We bridge this gap by introducing multi-particle distance arrays, which sample finite orbit…
In this paper, we show that the presence of the Archimedean and the mixture-continuity properties of a binary relation, both empirically non-falsifiable in principle, foreclose the possibility of consistency (transitivity) without…
In this paper, we prove that separation occurs for the stationary Prandtl equation, in the case of adverse pressure gradient, for a large class of boundary data at $x=0$.We justify the Goldstein singularity: more precisely, we prove that…
A Sturmian sequence is an infinite nonperiodic string over two letters with minimal subword complexity. In two papers, the first written by Morse and Hedlund in 1940 and the second by Coven and Hedlund in 1973, a surprising correspondence…
We consider the computation of an approximately stationary point for a Lipschitz and semialgebraic function $f$ with a local oracle. If $f$ is smooth, simple deterministic methods have dimension-free finite oracle complexities. For the…