The hottest Foundations Substack posts right now

And their main takeaways
Category
Top Business Topics

Natural Ordinal Addition

Infinitely More • 7 implied HN points • 15 Mar 26
🔬 Science Foundations
  1. The natural sum and product (Hessenberg operations) make the ordinals into a commutative semiring, contrasting with standard ordinal arithmetic where addition and multiplication are not commutative.
  2. The natural ordinal operations match the operations on surreal numbers, so the ordinals under natural addition and multiplication form a subsemiring of the surreals.
  3. There are five independent, self-contained ways to define the natural sum and product—order-theoretic, computational, proof-theoretic, and others—and all five are equivalent, giving complementary perspectives and routes to generalization.

Unreal numbers

lcamtuf’s thing • 4489 implied HN points • 15 Feb 26
🔬 Science Foundations
  1. Natural numbers can be built from a base element (zero) and a successor rule, and addition and multiplication follow from simple recursive definitions.
  2. Integers and rationals are formed by ordered pairs and equivalence classes so subtraction and division have in-system representations, and these extended sets remain countable.
  3. Computable numbers are those a Turing machine can approximate and are still countable, but the real numbers are uncountable (by diagonalization), so most reals cannot be computed.

Counting to Epsilon Naught

Infinitely More • 35 implied HN points • 04 Mar 26
🔬 Science Foundations
  1. Counting ordinals continues past the finite numbers to ω, then ω+1, ω+2, and onward through blocks like ω·2, ω·3, … so that each new limit ordinal begins a new ω-long era.
  2. By iterating these constructions and forming longer and longer exponential towers—ω, ω^ω, ω^(ω^ω), …—we reach ever higher ordinals, and the supremum of all finite such towers is the ordinal ε0.
  3. ε0 is the first ordinal fixed point of exponentiation by ω (so ω^ε0 = ε0), and there is a computable notation system for all ordinals below ε0 with important applications like Goodstein’s theorem and the Hydra game.

Cantor Normal Form

Infinitely More • 23 implied HN points • 12 Feb 26
🔬 Science Foundations
  1. Cantor normal form gives every ordinal a unique, canonical representation, acting like a numeral system built on base ω instead of base ten.
  2. The notation is as powerful and convenient for working with ordinals as the decimal system is for ordinary numbers, so it makes representing and comparing ordinals systematic and clear.
  3. Using Cantor normal form simplifies ordinal arithmetic because many terms cancel or "disappear," and it provides a foundation for further topics like the surreal numbers.

Indecomposable Ordinals

Infinitely More • 38 implied HN points • 01 Feb 26
🔬 Science Foundations
  1. An ordinal λ>0 is additively indecomposable when the sum of any two smaller ordinals is still below λ; equivalently, it cannot be expressed as a sum of two smaller ordinals.
  2. Concrete examples: ω is additively indecomposable, ω·2 is not, and the next indecomposable after ω is ω^2, so such ordinals must exceed every finite multiple of ω.
  3. There are analogous notions for multiplication and exponentiation — multiplicative or exponential indecomposability raises similar questions about whether indecomposable equals irreducible and motivates a full characterization via tools like Cantor normal form.
Get a weekly roundup of the best Substack posts, by hacker news affinity:

Ordinal arithmetic

Infinitely More • 30 implied HN points • 22 Jan 26
🔬 Science Foundations
  1. The series develops the basics of ordinal arithmetic—standard addition, multiplication, and exponentiation—and then moves on to topics like indecomposable and irreducible ordinals, Cantor normal form, and binary ordinal representation.
  2. It introduces the natural (Hessenberg) ordinal operations, which are commutative and make the ordinals into a commutative semiring, and it will study the natural ring of ordinals ⟨Ord⟩ inside the surreal numbers, asking about expressions, algebraic properties, and unique factorization.
  3. This essay first lays a rigorous foundation by giving order-theoretic and recursive definitions of the standard ordinal operations, which the later, deeper investigations will rely on.

Anthropomorphizing the Russell paradox

Infinitely More • 41 implied HN points • 05 Jan 26
📖 Philosophy Foundations
  1. Cantor’s diagonal argument shows that for any set there are always more subsets than elements. You can see this intuitively by imagining people and their committees or fruits and their salads.
  2. Applying the same diagonal idea produces Russell’s paradox, which shows that allowing every property to define a set leads to a contradiction, so there can be no single universal set and set formation must be restricted.
  3. Modern axiomatic set theory (like ZFC) provides a robust foundation that achieves much of the logicist goal of grounding mathematics in logical principles, though there is still debate about whether every axiom is purely logical.

Ultrafinitism

Infinitely More • 48 implied HN points • 12 Dec 25
📖 Philosophy Foundations
  1. Ultrafinitism is the view that only relatively small or computationally accessible numbers truly exist, and extremely large numbers conventionally discussed by mathematicians are denied.
  2. This stance is different from general anti-realism because it accepts small numbers as unproblematic while treating very large numbers as ontologically different or nonexistent.
  3. A central challenge is the 'draw the line' objection: it’s hard to specify where feasible numbers stop and huge ones begin, and this makes concrete questions about enormous expressions difficult or undecidable.

Ultrafinitism with a largest number

Infinitely More • 25 implied HN points • 19 Dec 25
📖 Philosophy Foundations
  1. Ultrafinitism holds that only comparatively small or ‘feasible’ numbers exist, and finite arithmetic (FA) formalizes this by axiomatizing arithmetic with a single largest natural number.
  2. The full theory true in all finite truncation models is not computably axiomatizable, so FA is a distinct and simply stated theory rather than that inexpressible common truncation theory.
  3. Any model of FA can be interpreted inside a strictly taller FA-model where the former largest number attains much larger values (making previously undefined sums and products defined), revealing a potentialist hierarchy that, when iterated, yields models arising from truncations of bounded induction.

Infinitesimals revisited

Infinitely More • 38 implied HN points • 04 Jul 25
📖 Philosophy Foundations
  1. Infinitesimals were once thought to be nonsense in calculus but actually led to important mathematical breakthroughs. They help us understand changes in functions in a very effective way.
  2. Nonstandard analysis, introduced in the 1960s, provides a solid way to use infinitesimals rigorously through hyperreal numbers. This helps to connect the old and modern approaches in calculus.
  3. Different perspectives on nonstandard analysis can lead to various mathematical ideas and research directions, showing that there's not just one correct way to approach mathematical concepts.

The Swiss Foundation is dead. Long live the Swiss Association - now onchain on OtoCo!

The Otonomist • 79 implied HN points • 27 Apr 23
💼 Business Foundations
  1. OtoCo now allows users to create Swiss Associations natively on blockchain, decentralizing governance.
  2. Swiss Associations offer benefits like limited liability, tax efficiency, and ease of operation for decentralized projects.
  3. Creating a Swiss Association on OtoCo is affordable and simple, requiring only two initial members to start.

Final Session Invitation!

Quantum Formalism • 0 implied HN points • 28 Jan 21
🚌 Education Foundations
  1. The final session of the foundation module will cover topics such as Compatible & Incompatible Operators, Time-Translation Symmetry, Quantum Integrable Systems, Quantum Axioms, Composite Quantum Systems, and Quantum Entanglement.
  2. Newcomers are welcome to join the final session, but it's recommended to have covered the previous sessions or have some background in university-level physics/mathematics to fully follow along.
  3. The registration link for the final session can be accessed at https://www.crowdcast.io/e/lecture-17-quantum-formalism.