Related papers: Broad Infinity and Generation Principles
Infinite antichains of permutations have long been used to construct interesting permutation classes and counterexamples. We prove the existence and detail the construction of infinite antichains with arbitrarily large growth rates. As a…
A longstanding goal in deep learning research has been to precisely characterize training and generalization. However, the often complex loss landscapes of neural networks have made a theory of learning dynamics elusive. In this work, we…
Fully analytical dynamical models usually have an infinite extent, while real star clusters, galaxies, and dark matter haloes have a finite extent. The standard method for generating dynamical models with a finite extent consists of taking…
Fixed points are a recurring theme in computer science and are often constructed as limits of suitably seeded fixed point iterations. We present the algebra of iterative constructions (AIC) -- a purely algebraic approach to reasoning about…
In classical set theory, there are many equivalent ways to introduce ordinals. In a constructive setting, however, the different notions split apart, with different advantages and disadvantages for each. We consider three different notions…
Consider a random real tree whose leaf set, or boundary, is endowed with a finite mass measure. Each element of the tree is further given a type, or allele, inherited from the most recent atom of a random point measure…
In this paper I introduce a new and intuitive first-order foundational theory (where the concept of set is not primitive) and use it to show that the power set of an infinite set does not exist. In particular, proofs of uncountability of a…
Invariable generation is a topic that has predominantly been studied for finite groups. In 2014, Kantor, Lubotzky, and Shalev produced extensive tools for investigating invariable generation for infinite groups. Since their paper, various…
In this paper, based on results of exact learning, test theory, and rough set theory, we study arbitrary infinite families of concepts each of which consists of an infinite set of elements and an infinite set of subsets of this set called…
We study generation through the lens of statistical learning theory. First, we abstract and formalize the results of Gold [1967], Angluin [1979], Angluin [1980] and Kleinberg and Mullainathan [2024] in terms of a binary hypothesis class…
Weak selection, which means a phenotype is slightly advantageous over another, is an important limiting case in evolutionary biology. Recently it has been introduced into evolutionary game theory. In evolutionary game dynamics, the…
Systems undergoing an equilibrium phase transition from a liquid state to an amorphous solid state exhibit certain universal characteristics. Chief among these are the fraction of particles that are randomly localized and the scaling…
We propose a model universe, in which the dimension of the space is a continuous variable, which can take any real positive number. The dynamics leads to a model in which the universe has no singularity. The difference between our model and…
Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets.…
The number of spanning trees in the giant component of the random graph $\G(n, c/n)$ ($c>1$) grows like $\exp\big\{m\big(f(c)+o(1)\big)\big\}$ as $n\to\infty$, where $m$ is the number of vertices in the giant component. The function $f$ is…
In most data-scientific approaches, the principle of Maximum Entropy (MaxEnt) is used to a posteriori justify some parametric model which has been already chosen based on experience, prior knowledge or computational simplicity. In a…
Subshifts are sets of colorings of $\mathbb{Z}^d$ defined by families of forbidden patterns. In a given subshift, the extender set of a finite pattern is the set of all its admissible completions. Since soficity of $\mathbb{Z}$ subshifts is…
A famous theorem of Szemer\'edi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a…
We present a self-contained analysis of infinity from two mathematical perspectives: set theory and algebra. We begin with cardinal and ordinal numbers, examining deep questions such as the continuum hypothesis, along with foundational…
Since the time of Darwin, scientists have struggled to reconcile the evolution of biological forms in a universe determined by fixed laws. These laws underpin the origin of life, evolution, human culture and technology, as set by the…