In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph (ℵ).
The Aleph (short story)
"The Aleph" (Spanish: El Aleph) is a short story by Argentine writer and poet Jorge Luis Borges. First published in September 1945, it was reprinted in the short story collection The Aleph and Other Stories in 1949, and revised by the author in 1974.
How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer. | Quanta Magazine
For 50 years, mathematicians have believed that the total number of real numbers is unknowable. A new proof suggests otherwise.
Infinity Category Theory Offers a Bird's-Eye View of Mathematics
Mathematicians have expanded category theory into infinite dimensions, revealing new connections among mathematical concepts
German LLM maker Aleph Alpha pivots to AI support | TechCrunch
Europe doesn't have many large language model (LLM) makers but one of these rare AI beasts -- Germany's Aleph Alpha -- appears to be preparing to rule
the aligned distinct ring · sol mark construction
one ray, repeated ten times. one ring, solved from its center and its air. a construction of whole angles, simple fractions, and φ.
J Notation as a Tool of Thought
Kenneth Iverson’s 1964 language, APL, won him the Turing Award. His award lecture, Notation as a Tool of Thought, argued that better notations would lead people to deeper insights about mathematics. He provided a number of examples ranging across linear algebra, arithmetic, probability, and logic. Unfortunately, most of the mathematics he covers isn’t relevant to programming. However, his core idea still applies, and changing how we describe programs changes how we think about them.
The HoTTest Axiom of math
Narcissistic number
In number theory, a narcissistic number[1][2] (also known as a pluperfect digital invariant (PPDI),[3] an Armstrong number[4] (after Michael F. Armstrong)[5] or a plus perfect number)[6] in a given number base b {\displaystyle b} is a number that is the sum of its own digits each raised to the power of the number of digits.
Homotopy Type Theory: Univalent Foundations of Mathematics
Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and beautiful "univalence axiom" implies that isomorphic structures can be identified. On the other hand, "higher inductive types" provide direct, logical descriptions of some of the basic spaces and constructions of homotopy theory. Both are impossible to capture directly in classical set-theoretic foundations, but when combined in homotopy type theory, they permit an entirely new kind of "logic of homotopy types". This suggests a new conception of foundations of mathematics, with intrinsic homotopical content, an "invariant" conception of the objects of mathematics -- and convenient machine implementations, which can serve as a practical aid to the working mathematician. This book is intended as a first systematic exposition of the basics of the resulting "Univalent Foundations" program, and a collection of examples of this new style of reasoning -- but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant.
A formulation of the simple theory of types
The purpose of the present paper is to give a formulation of the simple theory of types which incorporates certain features of the calculus of λ-conversion. A complete incorporation of the calculus of λ-conversion into the theory of types is impossible if we require that λx and juxtaposition shall retain their respective meanings as an abstraction operator and as denoting the application of function to argument. But the present partial incorporation has certain advantages from the point of view of type theory and is offered as being of interest on this basis (whatever may be thought of the finally satisfactory character of the theory of types as a foundation for logic and mathematics).For features of the formulation which are not immediately connected with the incorporation of λ-conversion, we are heavily indebted to Whitehead and Russell, Hilbert and Ackermann, Hilbert and Bernays, and to forerunners of these, as the reader familiar with the works in question will recognize.The class of type symbols is described by the rules that ı and o are each type symbols and that if α and β are type symbols then (αβ) is a type symbol: it is the least class of symbols which contains the symbols ı and o and is closed under the operation of forming the symbol (αβ) from the symbols α and β.
Sets for Mathematics in nLab
deduction system, natural deduction, sequent calculus, lambda-calculus, judgment
Cohere and Aleph Alpha merge into a $20B transatlantic AI company
Cohere and Aleph Alpha announce a merger creating a ~$20B transatlantic AI company with dual Canadian-German headquarters.
Iconic Math
As an introduction to a forthcoming book (late 2025), this page links to three narrated videos about unary boundary logic:
Kill Math
The power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols.
Newfound Mathematical 'Einstein' Shape Creates a Never-Repeating Pattern
A new shape called an einstein has taken the math world by storm. The craggy, hat-shaped tile can cover an infinite plane with patterns that never repeat.
Mathematics in the Library of Babel — Daniel Litt
Mathematics isn't only about saying true things. It's about asking the right questions, being confused, stumbling about, getting distracted, being wrong, recognizing when you're wrong, being stuck. Mostly being stuck. It's about clinging to a giant edifice and feeling it out until you understand som
