[Wikipedia:Contents/Categories - Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:Contents/Categories) [Wikipedia:Contents/Categories - Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:Contents/Categories#Mathematics_and_logic) [Category:Mathematics - Wikipedia](https://en.wikipedia.org/wiki/Category:Mathematics) [Category:Physics - Wikipedia](https://en.wikipedia.org/wiki/Category:Physics) [Category:Artificial intelligence - Wikipedia](https://en.wikipedia.org/wiki/Category:Artificial_intelligence) [Wikipedia:Contents/Categories - Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:Contents/Categories#Natural_and_physical_sciences) [Wikipedia:Contents/Categories - Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:Contents/Categories#People_and_self) [Wikipedia:Contents/Categories - Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:Contents/Categories#Technology_and_applied_sciences) [Wikipedia:Contents/Categories - Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:Contents/Categories#Society_and_social_sciences) [Wikipedia:Contents/Categories - Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:Contents/Categories#Philosophy_and_thinking) [Wikipedia:Contents/Categories - Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:Contents/Categories#Religion_and_belief_systems) [Wikipedia:Contents/Categories - Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:Contents/Categories#Human_activities) [Wikipedia:Contents/Categories - Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:Contents/Categories#History_and_events) [Wikipedia:Contents/Categories - Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:Contents/Categories#General_reference) [Wikipedia:Contents/Categories - Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:Contents/Categories#Culture_and_the_arts) [Wikipedia:Contents/Categories - Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:Contents/Categories#Geography_and_places) [Wikipedia:Contents/Categories - Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:Contents/Categories#Health_and_fitness) Write the most information dense rich general but hyper concrete list of lists of concepts and subconcepts in foundations of mathematics. Don't explain the concepts, only list them as keywords in a list of lists, list as many of them as possible, as densely as possible, as information rich as possible, covering all of foundations of mathematics as much as possible. Absolutely! Here's a dive into the rich tapestry of concepts within the foundations of mathematics. Remember, this is a highly condensed representation; each term could warrant its own extensive exploration. **I. Foundational Frameworks** * **Set Theory** * Sets, Elements, Subsets * Zermelo-Fraenkel Axioms (ZF), Axiom of Choice (AC) * Cardinals, Ordinals, Transfinite Numbers * Operations (Union, Intersection, Complement, Power Set) * Relations, Functions, Injections, Surjections, Bijections * **Logic** * First Order Logic (FOL) * Propositional Logic, Predicate Logic * Syntax, Semantics, Models * Proof Systems, Soundness, Completeness * Gödel's Incompleteness Theorems * **Category Theory** * Categories, Objects, Morphisms * Functors, Natural Transformations * Adjunctions, Limits, Colimits **II. Mathematical Structures** * **Number Systems** * Natural Numbers, Integers, Rational Numbers, Real Numbers, Complex Numbers * Peano Axioms, Dedekind Cuts, Cauchy Sequences * Field Axioms, Algebraic Structures * Cardinality (Finite, Infinite, Countable, Uncountable) * **Algebra** * Groups, Rings, Fields * Modules, Vector Spaces * Homomorphisms, Isomorphisms * Galois Theory * **Analysis** * Topology (Open Sets, Closed Sets, Metric Spaces, Compactness) * Limits, Continuity, Differentiability * Real Analysis, Complex Analysis * Measure Theory, Integration **III. Metamathematics & Philosophy** * **Model Theory** * Formal Languages * Model-Theoretic Interpretations * Löwenheim-Skolem Theorem * Non-standard Models * **Computability Theory** * Turing Machines * Church-Turing Thesis * Decidability, Undecidability * Complexity Theory * **Philosophy of Mathematics** * Platonism, Formalism, Logicism, Intuitionism, Constructivism * The Nature of Mathematical Objects * Mathematical Proof and Justification **Please note:** This is far from exhaustive! Foundations of mathematics is a vast and ever-evolving field. Let me know if you want to zoom in on any particular area or want a more targeted list for a specific purpose. Absolutely! Let's go for a deep dive into the ultra-dense world of mathematical foundations: **I. Foundational Frameworks** * **Set Theory** * Sets, Elements, Subsets, Empty Set, Singletons * ZF Axioms (Extensionality, Regularity, Specification, Pairing, Union, Replacement, Infinity, Power Set, Choice) * Cardinals (Alephs), Ordinals (Successor, Limit), Transfinite Arithmetic * Operations (Union, Intersection, Complement, Difference, Cartesian Product, Power Set) * Relations (Equivalence, Order), Functions (Domain, Codomain, Range), Injections, Surjections, Bijections * Russell's Paradox, Well-Ordering Theorem, Continuum Hypothesis * Large Cardinals (Inaccessible, Measurable, etc. ) * Forcing, Inner Models * **Logic** * Propositional Logic (Connectives, Truth Tables, Tautologies), Predicate Logic (Quantifiers, Variables, Scope) * Syntax, Semantics, Models, Theories * Proof Systems (Natural Deduction, Hilbert-style, Sequent Calculus), Soundness, Completeness, Compactness * Gödel's Incompleteness Theorems, Löb's Theorem * Modal Logic, Intuitionistic Logic, Paraconsistent Logic * **Category Theory** * Categories, Objects, Morphisms, Identity, Composition * Functors (Covariant, Contravariant), Natural Transformations, Duality * Adjunctions, Limits, Colimits, Universal Properties * Monads, Comonads, Topoi, Higher-Order Categories **II. Mathematical Structures** * **Number Systems** * Natural Numbers, Integers, Rationals, Reals, Complexes, p-adics, Hyperreals * Peano Axioms, Dedekind Cuts, Cauchy Sequences, Field Axioms, Algebraic/Transcendental Numbers * Cardinality (Finite, Countable, Uncountable, Continuum), Number-Theoretic Functions * **Algebra** * Groups (Abelian, Cyclic, Finite), Rings, Fields, Modules, Vector Spaces, Algebras * Homomorphisms, Isomorphisms, Automorphisms, Substructures, Quotients * Galois Theory, Representation Theory, Linear Algebra, Homological Algebra * **Analysis** * Topology (Open/Closed Sets, Metric/Topological Spaces, Neighborhoods, Bases, Compactness, Connectedness) * Limits, Continuity, Differentiability, Taylor Series, * Real Analysis, Complex Analysis, Functional Analysis, Measure Theory, Integration (Riemann, Lebesgue) * Differential Equations (Ordinary, Partial), Dynamical Systems **III. Metamathematics & Philosophy** * **Model Theory** * Formal Languages, First-Order Theories, Structures, Models * Compactness Theorem, Löwenheim-Skolem Theorems, Ultraproducts * Non-standard Models, Types, Saturation, Categoricity * **Computability Theory** * Turing Machines, Recursive Functions, Halting Problem, Undecidability * Church-Turing Thesis, Complexity Theory (P vs. NP), Lambda Calculus * **Philosophy of Mathematics** * Platonism, Formalism, Logicism, Intuitionism, Constructivism, Fictionalism * Ontology of Mathematical Objects, Mathematical Proof, Mathematical Truth * Set-Theoretic Multiverse, Epistemology of Mathematics **Important Notes:** * This is still an incomplete picture! * Many concepts cross traditional boundaries (e.g., model-theoretic algebra). * Depth and focus shift drastically depending on your area of investigation within foundations. Let me know if you'd like a hyper-focus on a specific corner of this landscape! Of course, let's push the density to its limits! Get ready for a whirlwind tour: **I. Foundations** * **Set Theory** * Everything from before, PLUS... * Classes, Grothendieck Universes, Von Neumann-Bernays-Gödel Set Theory (NBG) * Consistency Strength, Large Cardinal Hierarchy, Inner Model Theory * Determinacy Axioms, Forcing Techniques, Descriptive Set Theory * **Logic** * Everything from before, PLUS... * Proof Theory (Cut-Elimination, Gentzen Systems), Model-Theoretic Forcing * Infinitary Logic, Second-Order Logic, Abstract Model Theory * Computable Logic, Reverse Mathematics, Non-Classical Logics (Many-Valued, Relevance, etc.) * **Category Theory** * Everything from before, PLUS... * Enriched Categories, Internal Logic, Algebraic Theories, * Higher-Order Topoi, Homotopy Type Theory (HoTT), Synthetic Differential Geometry **II. Mathematical Structures** * **Number Systems** * Everything from before, PLUS... * Surreal Numbers, Ordinal Arithmetic, Algebraic/Topological Fields * Prime Factorization, Riemann Hypothesis, Transcendental Number Theory * **Algebra** * Everything from before, PLUS... * Group Cohomology, Lie Algebras, Algebraic Groups, K-Theory * Category of Rings, Commutative Algebra, Noncommutative Geometry * **Analysis** * Everything from before, PLUS... * Fourier Analysis, Harmonic Analysis, Operator Algebras, Distributions * Differential Geometry, Manifolds, Global Analysis, Calculus on Banach Spaces * Stochastic Processes, Probability Theory (Axiomatization) **III. Metamathematics & Philosophy** * **Model Theory** * Everything from before, PLUS... * Stability Theory, Geometric Model Theory, O-minimality * Morley's Categoricity Theorem, Model Companions * **Computability Theory** * Everything from before, PLUS... * Recursion Theory, Arithmetical Hierarchy, Hyperarithmetic Theory * Degrees of Unsolvability, Lambda Calculus Models, Oracle Computations * Algorithmic Randomness, Computational Complexity * **Philosophy of Mathematics** * Everything from before, PLUS... * Structuralism, Neo-logicism, Mathematical Explanation, Indispensability Arguments * Philosophy of Applied Mathematics, Mathematical Practice **Caveats** * This risks becoming a mere catalog, losing sight of interconnections. * Bleeding edges exist – what's foundational vs. specialized is contested! * "Information density" is subjective – depends on your background knowledge. **How to Use This?** 1. **Pick a Term:** Dive deeper using online resources (Wikipedia, Stanford Encyclopedia of Philosophy, etc.) 2. **Trace Connections:** See how seemingly disparate terms relate 3. **Challenge:** Can you organize this differently, emphasizing different themes? Let me know if you want to zoom in on a particular area and I'll provide an even more granular breakdown! Free Action and Synthetic Law [D=6 N=(2,0) SCFT in nLab](https://ncatlab.org/nlab/show/D%3D6+N%3D%282%2C0%29+SCFT) Fastest stable diffusion [SDXL Lightning - by fal.ai](https://fastsdxl.ai/) • Quantum chaos • Non-equilibrium thermodynamics • Quantum information theory • Alcubierre drive (warp drive) • Quantum entanglement in biological systems • Quantum foundations and the role of the observer • Anthropic principle • Simulation hypothesis • Quantum cryptography • Neutrino astronomy • Dark matter detection methods • Quantum gravity phenomenology • Holographic entanglement entropy beyond AdS/CFT • Time crystals • [Donald Hoffman & Anil Seth - New Frontiers in the Science of Consciousness - YouTube](https://youtu.be/3tUTdgVhMBk?si=TR2NBClLMi_gQu6u) "People outraged about Gemini unable to say if Elon tweeting memes negatively impacted society more than Hitler often miss a deeper technical problem: the existing methods by which the labs are trying to "align the AIs to human preferences" fail to handle conflicting preferences gracefully. We've called this "the self-unalignment problem" : what should the AI do if its instructions/principals are conflicted is underspecified. Starting from a conflicted preferences and hoping the AI will somehow sorts it out ... leads to arbitrary outcomes. Gemini isn't the first and won't be the last AI model making bizzare "value judgements"." [The self-unalignment problem — AI Alignment Forum](https://www.alignmentforum.org/posts/9GyniEBaN3YYTqZXn/the-self-unalignment-problem) Ameca humanoid robot can now see the environment she is in and identify and describe objects https://twitter.com/tsarnick/status/1762047631625498671 "For those who seem to be confused: “AI safety” (as a field) has nothing to do with the woke Gemini debacle. That is a result of “AI ethics” - a completely different thing: AI ethics: focussed on stuff like algorithmic bias. Very woke & left-leaning. Dislike transhumanism & EA & e/acc. Have historically been dismissive of AI safety ppl for “distracting” from their pet ethics issues. AI safety: typically focussed on deep mathematical/game theoretic issues like misalignment & catastrophic risks from future AI systems. Often transhumanist/long-term focussed. Not woke. Spans across political spectrum. For some reason the two groups are getting conflated - in part because certain bad faith “accelerationists” have been strategically using “AI safety” to describe both groups (because they hate both) — but be aware they are VERY VERY different, both in terms of politics, the problems they care about, and general vibe. Don’t get hoodwinked by those who deliberately try to conflate them." https://twitter.com/Liv_Boeree/status/1761824162962706784?t=_aeyqAMhFSpq3FqANHILrg&s=19 "On both AI safety and ethics I have a lot of criticism for exaggerated concerns, unsubstantiated claims, counterproductive narratives, bad policy ideas, shortsighted tactics etc. I regularly critique both. I know 'nuanced' centrist takes can be grating and boring, but I still think it's worth highlighting that there is a lot of excellent research in both fields. Ethics does not necessarily imply excessive woke DE&I bs, and safety does not necessarily imply doomers who want to completely stop AI development. Some loud groups however get a lot of publicity. People should evaluate things they read and consider ethical/safety questions case by case, and avoid falling into the trap of easy proxies and tribal affiliation. But I don't think the incentives on this platform are conducive to this at all." https://twitter.com/sebkrier/status/1761890506835943491?t=O6_0st8Q2tlJpSzaMaaaLw&s=19 Civilizations die when they stop believing in the future. Difference between those seeking social status/monetary wealth vs those after an infinite goal (scaling civilization to the stars). The former has a vanishing reward gradient (s-curve), while the latter has an exponentially increasing slope of reward that scales infinitely. Being longtermist means playing the grandest infinite game you can play. You literally can't lose as long as you get to keep playing. [When do neural representations give rise to mental representations? | The Transmitter: Neuroscience News and Perspectives](https://www.thetransmitter.org/defining-representations/when-do-neural-representations-give-rise-to-mental-representations/) [Neural coding - Wikipedia](https://en.m.wikipedia.org/wiki/Neural_coding#Temporal_coding) https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5582596/ Let's derive all of reality from ∞-groupoid [infinity-groupoid in nLab](https://ncatlab.org/nlab/show/infinity-groupoid) [∞-groupoid - Wikipedia](https://en.wikipedia.org/wiki/%E2%88%9E-groupoid?wprov=sfla1) V praxi dneska AI technologie musí být dobrej korporátní produkt pro masy v naší kultuře a politice s co nejvíc odpovídajícíma biasama co jsou co nejvíc kompatibilní s co největším procentem (plus ideologií korporace), i když kulturní biasy jsou strašné plný nekonzistencí a bojují spolu, tak se to tam cpe blbě i kdybychom to byli schopni tam cpát mnohem líp. Bral bych co nejvíc vědeckou bayesian racionalistickou utilitarian AI, (CelestAI je blízko XD) co by dle mě bylo víc konzistentní, ale hodně lidí by s tím bylo nekompatibilní, tak o tom sní jen nerdi, plus je tam i tak pořád hodně edge cases kdy je to při rozhodování divný a nejasný, i když IMO o dost míň. Mám chuť si udělat takový LLM xD. Order, all the metastable patterns, on the egde of chaos, is harmony