# set theory Published articles for set theory. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## Integer Set Library (ISL) - A Primer DevFeed: [Integer Set Library (ISL) - A Primer]() Original publisher: [Read original article]() Published: 2025-10-19T20:14:22Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Library](), [Optimization](), [compilers](), [Open Source](), [mlir]() Tags: [code](), [compilers](), [isl](), [library](), [loops](), [mathematics](), [mlir](), [open-source](), [optimization](), [polyhedral-optimization](), [programming](), [set-theory]() ### AI overview This primer introduces the Integer Set Library (ISL), an open-source C library that implements core algorithms for polyhedral optimization. It focuses on representing integer sets and relations, manipulating them, and the relationship between ISL and MLIR's Fast Presburger Library. ### Source excerpt Polyhedral optimization is a tool used in compilers for optimizing loop nests. While the major compilers that use this implement polyhedral optimizations from scratch,1 there is a generally-applicable open source C library called the Integer Set Library (ISL) that implements the core algorithms used in polyhedral optimization. This article gives an overview of a subset of ISL, mainly focusing on the representation of sets and relations and basic manipulations on them. ## Methods of Proof -- Direct Implication DevFeed: [Methods of Proof -- Direct Implication]() Original publisher: [Read original article]() Published: 2013-02-16T12:56:53Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Math and Logic](), [Mathematics](), [Learning]() Tags: [contrapositive](), [direct-implication](), [induction](), [mathematics](), [methods-of-proof](), [set-theory](), [syntax](), [techniques]() ### AI overview This tutorial introduces a series on mathematical proof techniques for programmers and begins with direct implication. It presents four basic methods--direct implication, contradiction, contrapositive, and induction--and uses set theory to introduce notation and practice. ### Source excerpt I recently posted an exploratory piece on why programmers who are genuinely interested in improving their mathematical skills can quickly lose stamina or be deterred. My argument was essentially that they don't focus enough on mastering the basic methods of proof before attempting to read research papers that assume such knowledge. Also, there are a number of confusing (but in the end helpful) idiosyncrasies in mathematical culture that are often unexplained. ## Probability Theory -- A Primer DevFeed: [Probability Theory -- A Primer]() Original publisher: [Read original article]() Published: 2013-01-04T13:45:54Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [math](), [Statistics](), [Machine Learning & Artificial Intelligence]() Tags: [expectation](), [mathematics](), [primer](), [probabilistic](), [probability-theory](), [random-variables](), [set-theory](), [statistics](), [theory](), [variables](), [variance]() ### AI overview A primer on finite probability theory that introduces probability spaces, random variables, terminology, and basic results using naive set theory. It emphasizes mathematical formalism rather than real-world applications. ### Source excerpt It is a wonder that we have yet to officially write about probability theory on this blog. Probability theory underlies a huge portion of artificial intelligence, machine learning, and statistics, and a number of our future posts will rely on the ideas and terminology we lay out in this post. Our first formal theory of machine learning will be deeply ingrained in probability theory, we will derive and analyze probabilistic learning algorithms, and our entire treatment of mathematical finance will be framed in terms of random variables. ## Groups -- A Primer DevFeed: [Groups -- A Primer]() Original publisher: [Read original article]() Published: 2012-12-08T23:42:10Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [structure](), [Math and Logic](), [Cryptography]() Tags: [applications](), [group-actions](), [groups](), [lagrange](), [mathematics](), [public-key](), [set-theory](), [structure](), [symmetry](), [theory]() ### AI overview This primer introduces group theory as the study of algebraic structures that can describe operations beyond ordinary numbers. It motivates the subject through examples involving symmetry, squares, and Rubik's Cube, and notes applications including public-key cryptography, error detection, crystal structures, particle physics, and mathematical biology. ### Source excerpt The study of groups is often one's first foray into advanced mathematics. In the naivete of set theory one develops tools for describing basic objects, and through a first run at analysis one develops a certain dexterity for manipulating symbols and definitions. But it is not until the study of groups that one must step back and inspect the larger picture. The main point of that picture (and indeed the main point of a group) is that algebraic structure can be found in the most unalgebraic of settings. ## False Proof--All Numbers are Describable in at Most Twenty Words DevFeed: [False Proof--All Numbers are Describable in at Most Twenty Words]() Original publisher: [Read original article]() Published: 2011-07-28T16:03:27Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Math and Logic](), [Statement]() Tags: [computer](), [false-proof](), [kolmogorov-complexity](), [language](), [mathematics](), [program](), [set-theory](), [turing-machine](), [well-ordering]() ### AI overview The article examines a false proof claiming that every natural number can be described in fewer than twenty words. It explains that the contradiction arises from imprecise notions of description and set construction, connecting the issue to Russell's paradox and the Richard-Berry paradox. ### Source excerpt Problem: Show that every natural number can be unambiguously described in fewer than twenty words. "Solution": Suppose to the contrary that not every natural number can be so described. Let $ S$ be the set of all natural numbers which are describable in fewer than twenty words. Consider $ R = \mathbb{N}-S$, the set of all words which cannot be described in fewer than twenty words. Since $ R$ is a subset of the natural numbers, which is well-ordered, it has a unique smallest element which we call $ r$. ## False Proof--The Reals are Countable DevFeed: [False Proof--The Reals are Countable]() Original publisher: [Read original article]() Published: 2011-07-19T17:10:00Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [math]() Tags: [axiom-of-choice](), [countability](), [false-proof](), [math](), [mathematics](), [set-theory](), [well-ordering]() ### AI overview The article examines a purported proof that the real numbers are countable. It explains that the argument's surjectivity claim is flawed and begins demonstrating the issue using a chosen well-ordering of the integers. ### Source excerpt It seems that false proofs are quickly becoming some of the most popular posts on Math ∩ Programming. I have been preparing exciting posts on applications of graph coloring, deck stacking, and serial killers. Unfortunately, each requires resources which exist solely on my home desktop, which is currently dismantled in California while I am on vacation in Costa Rica. Until I return from the tropics, I will continue with more of the ever -popular false proofs. ## Set Theory--A Primer DevFeed: [Set Theory--A Primer]() Original publisher: [Read original article]() Published: 2011-07-09T18:14:59Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Math and Logic]() Tags: [axiom-of-choice](), [bijections](), [cardinality](), [countability](), [example](), [functions](), [mathematics](), [numbers](), [power-set](), [primer](), [set](), [set-theory](), [symbols](), [theory](), [variable]() ### AI overview This primer introduces set theory by defining sets, elements, membership, cardinality, notation, and several ways to construct sets. It uses numerical examples and introduces natural numbers, integers, and rational numbers while noting that unrestricted operations can lead to paradoxes. ### Source excerpt It's often that a student's first exposure to rigorous mathematics is through set theory, as originally studied by Georg Cantor. This means we will not treat set theory axiomatically (as in ZF set theory), but rather we will take the definition of a set for granted, and allow any operation to be performed on a set. This will be clear when we present examples, and it will be clear why this is a bad idea when we present paradoxes.