Cardinals vs ordinals algebraically

comparison of algebraic properties of cardinals and ordinals

Nice — here’s a compact, useful comparison of the algebraic properties of cardinals vs. ordinals, with definitions, the key laws, concrete examples, and the important caveats.

Quick definitions

• Ordinal: an order-type of a well-ordered set. Ordinal arithmetic depends on the order (how elements are arranged).
• Cardinal: a size (equivalence class under bijection) of a set. Cardinal arithmetic depends only on size, not on order.

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Table of core properties (side-by-side)

Feature / Operation	Cardinals (cardinal arithmetic)	Ordinals (ordinal arithmetic)
Addition	Commutative, associative; identity 0; defined by disjoint union up to bijection. Example: ℵ0 + 1 = 1 + ℵ0 = ℵ0.	Not commutative in general; associative; identity 0. Example: 1 + ω = ω but ω + 1 > ω.
Multiplication	Commutative, associative; identity 1; distributive over addition. For infinite κ, κ·λ = max(κ,λ) in the usual cases (e.g. one factor infinite and nonzero).	Not commutative; associative; identity 1. Left-distributive over addition: α·(β+γ) = α·β + α·γ. Right-distributivity can fail: (α+β)·γ ≠ α·γ + β·γ in general. Example: 2·ω = ω, but (1·ω)+(1·ω)=ω+ω>ω.
Exponentiation	Commutativity no longer meaningful; κ^λ is size of functions λ→κ. Can behave wildly; many values independent of ZFC (e.g. continuum hypothesis concerns 2^{ℵ0}).	Noncommutative and order-sensitive; defined by transfinite recursion. Very different qualitative behavior from cardinals.
Commutativity of operations	Addition & multiplication commute (cardinal algebra is commutative).	Addition & multiplication do not commute in general.
Distributivity	Multiplication distributes over addition (both sides).	Multiplication is left-distributive (α(β+γ)=αβ+αγ) but not necessarily right-distributive.
Cancellativity	Fails in general for infinite cardinals: e.g. κ + μ = κ + ν need not imply μ = ν; also κ·μ = κ·ν doesn't imply cancellation.	Fails strongly: e.g. ordinal cancellation is impossible in many cases (ω + 1 = ω + 1 trivial, but different summands can produce same sum in other contexts).
Identities / inverses	Additive identity 0. No additive inverses (no subtraction in general). Multiplicative identity 1. No multiplicative inverses except 1.	Additive identity 0, multiplicative identity 1. No inverses (no subtraction or division in general).
Order vs size sensitivity	Only size matters. Rearranging disjoint pieces does not change sum/product.	Order matters — e.g. appending on left vs right gives different ordinals.
Algebraic structure viewpoint	Forms a commutative semiring (no negatives). Many regular laws hold; but cancellation and inverse laws fail for infinite cardinals.	A noncommutative semiring-like structure: associative +,· with identities, but many familiar ring laws fail (noncommutative, distribution only one-sided, no cancellation).
Continuity at limits	Cardinal arithmetic often stabilizes for infinite cardinals (κ+κ = κ, κ·κ = κ for infinite κ).	Ordinal operations are sensitive to limit ordinals; definitions use transfinite recursion and require separate rules at limit stages.
Finer variants	—	Natural (Hessenberg) sum/product: alternate ordinal operations that are commutative and associative — useful for algebraic manipulations; coincide with ordinary sum/product for finite ordinals in simple ways.
Independence / set-theory sensitivity	Cardinal exponentiation (e.g. 2^{ℵ0}) is highly set-theoretic (CH independence). Some cardinal equalities require AC to get canonical representatives.	Ordinal arithmetic is more absolute (less independent); facts about ordinals are generally settled in ZF (order-type theory).

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Concrete examples (to build intuition)

• Ordinal asymmetry: 1 + ω = ω (put a single element before ω — the tail still looks like ω) ω + 1 > ω (put element after ω — you get a new largest element)
• Cardinal symmetry: as sets, ℵ0 ∪ {new} has cardinality ℵ0, so ℵ0 + 1 = 1 + ℵ0 = ℵ0.
• Multiplication contrast: Ordinals: 2·ω = ω (ω many copies of 2) but ω·2 = ω + ω (two blocks of size ω) — not equal. Cardinals: 2·ℵ0 = ℵ0 and ℵ0·2 = ℵ0 — both equal (multiplication commutes).
• Natural (Hessenberg) sum/product: These ordinal variants are commutative. For many algebraic manipulations with ordinals one uses them, but ordinary ordinal addition/multiplication are the canonical order-type operations.

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Why the differences arise (intuitive reasons)

• Cardinals collapse all order information — only bijections matter. That forces commutativity: |A ∪ B| = |B ∪ A|.
• Ordinals encode how the elements are arranged (first, second, limit positions). Arithmetic is defined by concatenation / repeated concatenation of well-ordered blocks, so left/right placement matters → noncommutativity and directional distributivity.

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Algebraic language summary

• Cardinal numbers: a commutative semiring (with limits/behaviors that simplify for infinite sizes); many familiar algebraic laws hold but cancellation and inverses fail for infinite sizes; cardinal exponentiation is set-theory sensitive.
• Ordinal numbers: a noncommutative semiring-like structure under the usual ordinal + and ·; associative but noncommutative; multiplication distributes only on one side; sensitive to order and limit stages; additional commutative “natural” operations exist (Hessenberg) that restore commutativity.

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Small checklist for proofs / exercises

• Prove ordinal addition is associative but not commutative (use concatenation / order-type arguments; give 1+ω vs ω+1).
• Prove cardinal addition is commutative by giving explicit bijection between A ⊔ B and B ⊔ A.
• Show κ + λ = max(κ,λ) for infinite cardinals under the usual hypotheses (exercise: construct injections in both directions).
• Exhibit failure of right-distributivity for ordinals via the 2·ω vs (1·ω)+(1·ω) example.
• Explore Cantor Normal Form (CNF) for ordinals: uniqueness + how CNF helps compute ordinal addition/multiplication.

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If you want, I can:

• produce a one-page cheat sheet (LaTeX or printable) with the table + sample proofs, or
• walk through worked proofs of any specific law (e.g. why ordinal multiplication is left-distributive but not right-distributive), or
• show how the natural (Hessenberg) sum/product are defined and give examples.

Which follow-up would you like?