TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)verified 2026-07-29 (claude-fable-5)‡ rests on unproved material (inherited)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Hessenberg: $\kappa \otimes \kappa = \kappa$ for every infinite cardinal $\kappa$ , proved in ZF from the canonical well-order of $\kappa \times \kappa$

Statement

Let $\kappa$ be an infinite cardinal, that is a cardinal (Cardinal (initial ordinal) and cardinality) with $\omega \le \kappa$ . Then

$\kappa \otimes \kappa = \kappa, \qquad \text{equivalently} \qquad \lvert \kappa \times \kappa \rvert = \kappa$

(Cardinal sum $\kappa \oplus \lambda$ , product $\kappa \otimes \lambda$ and exponentiation $\kappa^{\lambda}$ , and why they are written apart from the ordinal operations).

This is a theorem of ZF and uses no choice principle. The well-order that carries the proof is written down from the ordinal order on $\kappa$ ; nothing is selected anywhere. That matters for this page: Hessenberg's theorem is exactly the part of "an infinite set is the same size as its square" that survives without choice.

Facts & Assumptions

Given: An infinite cardinal $\kappa$ , in ZF. No choice principle is assumed. For ordinals $\xi, \eta$ write $\max(\xi,\eta)$ for the $\subseteq$ -larger of the two, which exists by comparability.

[L1]

For a well-orderable $X$ : $X \approx \lvert X \rvert$ , the value is a cardinal, equinumerous sets receive the same one, $\lvert \alpha \rvert \le \alpha$ , and $\lvert \alpha \rvert = \alpha$ exactly when $\alpha$ is a cardinal (A set equinumerous with some ordinal has a least such ordinal, that ordinal is a cardinal, and equinumerous sets get the same one; no choice principle is used).

[L2]

$\times$ respects $\approx$ , and $\alpha \times \beta$ carries an explicit well-order for ordinals $\alpha, \beta$ (Disjoint union, cartesian product, function space and power set respect equinumerosity, and for ordinals $\alpha, \beta$ the sets $\alpha \sqcup \beta$ and $\alpha \times \beta$ carry explicit well-orders, so their cardinalities exist in ZF).

[L3]

For cardinals, $\kappa \le \lambda$ iff $\kappa \preceq \lambda$ ; and $A \preceq B$ with both well-orderable gives $\lvert A \rvert \le \lvert B \rvert$ (claim (a) of Commutativity, associativity, distributivity and monotonicity of $\oplus$ and $\otimes$ , the unit laws, the two exponent laws, and $\kappa \le \lambda$ if and only if $\kappa$ injects into $\lambda$ ).

[L4]

$\omega$ is a cardinal, every natural number is a cardinal, every infinite cardinal is a limit ordinal, and for $m, n \in \omega$ the value $m \otimes n$ is again a natural number (Every natural number and $\omega$ are cardinals, every infinite cardinal is a limit ordinal, and on the natural numbers the cardinal operations are the published finite counting operations, with $\lvert A \rvert$ in the finite sense equal to $\lvert A \rvert$ in the cardinal sense, Successor and limit ordinals).

[L5]

$\mathbb{N} \times \mathbb{N} \approx \mathbb{N}$ , that is $\omega \times \omega \approx \omega$ ( $\mathbb{N} \times \mathbb{N} \approx \mathbb{N}$ , Finite, countably infinite, countable, uncountable).

[L6]

Every well-order is order isomorphic to exactly one ordinal, its order type; an order isomorphism is a bijection and carries the initial segment below a point onto the initial segment below its image (Every well-order has a unique order type, Order embedding and order isomorphism, Initial segment of a well-order).

[L7]

If $S \subseteq W$ for a well-order $(W,<)$ satisfies " $W_{<a} \subseteq S$ implies $a \in S$ " for every $a \in W$ , then $S = W$ (Transfinite induction).

[L8]

Ordinals: elements of ordinals are ordinals, $\alpha \notin \alpha$ , $\alpha \subseteq \beta$ iff $\alpha \in \beta$ or $\alpha = \beta$ , trichotomy holds, every nonempty set of ordinals has an $\in$ -least element, and every set of ordinals is well ordered by $\in$ (Basic closure properties of ordinals, Trichotomy and well-ordering of the ordinals, Ordinal (von Neumann), Well-order and well-ordered set).

[L9]

$\omega$ is the least limit ordinal and is an ordinal ( $\omega$ is the least limit ordinal); a bijection witnesses $\approx$ and a subset inclusion is an injection (Equinumerous sets, $A \approx B$ and $A \preceq B$ , Injection, surjection, bijection).

Proof

technique · direct

1.1

For an ordinal $\alpha$ define $(\xi,\eta) \lhd (\xi',\eta')$ on $\alpha \times \alpha$ to hold when $\max(\xi,\eta) \in \max(\xi',\eta')$ , or the two maxima are equal and $\xi \in \xi'$ , or the two maxima are equal, $\xi = \xi'$ and $\eta \in \eta'$ ; this is the lexicographic order on the triple $(\max(\xi,\eta), \xi, \eta)$ of ordinals, hence irreflexive, transitive and trichotomous by [L8], and a nonempty $S \subseteq \alpha \times \alpha$ has a $\lhd$ -least element obtained by taking in turn the $\in$ -least maximum occurring in $S$ , then the $\in$ -least admissible $\xi$ , then the $\in$ -least admissible $\eta$ , each of which is the least element of a nonempty set of ordinals and so is determined rather than chosen; therefore $\lhd$ well-orders $\alpha \times \alpha$ .

1.2

For $(\xi,\eta) \in \alpha \times \alpha$ put $\gamma = \max(\xi,\eta) \cup \{\max(\xi,\eta)\}$ , the successor of the maximum; then every $(\xi',\eta') \lhd (\xi,\eta)$ has $\max(\xi',\eta') \subseteq \max(\xi,\eta) \in \gamma$ , so $\xi', \eta' \in \gamma$ , and the $\lhd$ -initial segment of $\alpha \times \alpha$ below $(\xi,\eta)$ is contained in $\gamma \times \gamma$ .

1.3

Base value: $\omega \times \omega \approx \omega$ by [L5], so $\lvert \omega \times \omega \rvert = \lvert \omega \rvert = \omega$ by [L1] and [L4].

L1L4L5

1.4

Lower bound: for any ordinal $\mu$ with $0 \in \mu$ the map $\xi \mapsto (\xi, 0)$ is an injection $\mu \to \mu \times \mu$ , so $\lvert \mu \rvert \le \lvert \mu \times \mu \rvert$ by [L3], and $\lvert \mu \rvert = \mu$ when $\mu$ is a cardinal.

L1L3L9

2.1

Now let $\mu$ be an infinite cardinal with $\omega \in \mu$ , and assume the induction hypothesis that $\lvert \nu \times \nu \rvert = \nu$ for every infinite cardinal $\nu \in \mu$ ; for $(\xi,\eta) \in \mu \times \mu$ and $\gamma$ as in step 1.2 we have $\gamma \in \mu$ , because $\mu$ is a limit ordinal by [L4] and $\max(\xi,\eta) \in \mu$ , and moreover $\lvert \gamma \times \gamma \rvert \in \mu$ : if $\gamma \in \omega$ then $\lvert \gamma \times \gamma \rvert = \gamma \otimes \gamma \in \omega \subseteq \mu$ by [L4], while if $\omega \subseteq \gamma$ then $\nu = \lvert \gamma \rvert$ satisfies $\omega \le \nu \le \gamma \in \mu$ by [L1] and [L3], so $\nu$ is an infinite cardinal in $\mu$ and $\gamma \times \gamma \approx \nu \times \nu$ by [L1] and [L2], whence $\lvert \gamma \times \gamma \rvert = \lvert \nu \times \nu \rvert = \nu \in \mu$ .

step 1.2L1L2L3L4

3.1

Under the same hypothesis, the $\lhd$ -initial segment $I$ of $\mu \times \mu$ below any $(\xi,\eta)$ has order type in $\mu$ : $I \subseteq \gamma \times \gamma$ by step 1.2 and $I$ is well ordered by the restriction of $\lhd$ by step 1.1, so $\lvert I \rvert \le \lvert \gamma \times \gamma \rvert \in \mu$ by [L3] and step 2.1; and if the order type $\theta$ of $I$ satisfied $\mu \subseteq \theta$ then $\mu \preceq \theta \approx I$ by [L6] and [L9], giving $\mu \le \lvert I \rvert$ by [L3], which contradicts $\lvert I \rvert \in \mu$ .

step 1.1step 1.2step 2.1L3L6L9

4.1

Under the same hypothesis, $\lvert \mu \times \mu \rvert = \mu$ : let $\delta$ be the order type of $(\mu \times \mu, \lhd)$ and $g : \delta \to \mu \times \mu$ the inverse of the collapsing isomorphism ([L6]); if $\mu \in \delta$ then $g$ carries the initial segment of $\delta$ below $\mu$ , which is $\mu$ itself, onto the $\lhd$ -initial segment below $g(\mu)$ , whose order type would then be $\mu$ , contradicting step 3.1; so $\delta \subseteq \mu$ and $\lvert \mu \times \mu \rvert = \lvert \delta \rvert \le \delta \le \mu$ by [L1], while step 1.4 gives $\mu \le \lvert \mu \times \mu \rvert$ .

step 3.1step 1.4L1L6L8

5.1

Apply [L7] to the well-order $(\kappa \cup \{\kappa\}, \in)$ of [L8] and to $S = \{\mu \in \kappa \cup \{\kappa\} : \mu$ is not an infinite cardinal, or $\lvert \mu \times \mu \rvert = \mu\}$ : a $\mu$ below which everything lies in $S$ is in $S$ , trivially if $\mu$ is not an infinite cardinal, by step 1.3 if $\mu = \omega$ , and by step 4.1 otherwise; hence $S = \kappa \cup \{\kappa\}$ , so $\kappa \in S$ and $\kappa \otimes \kappa = \lvert \kappa \times \kappa \rvert = \kappa$ .

step 1.3step 4.1L7L8∎

Remarks

Why the maximum comes first. Under the plain lexicographic order of Disjoint union, cartesian product, function space and power set respect equinumerosity, and for ordinals $\alpha, \beta$ the sets $\alpha \sqcup \beta$ and $\alpha \times \beta$ carry explicit well-orders, so their cardinalities exist in ZF, the initial segment below $(1,0)$ in $\omega \times \omega$ is the whole of $\{0\} \times \omega$ , which is already infinite; the order type of $\omega \times \omega$ is then $\omega \cdot \omega$ , far above $\omega$ . Ordering by the maximum first bounds every initial segment inside a square $\gamma \times \gamma$ with $\gamma < \kappa$ , and the induction hypothesis then says that square is small. The whole proof is that one change of order.

No choice, and it is worth saying why. Every place that invites a selection avoids it: the $\lhd$ -least element of a nonempty set is found by three successive minimisations, the order type of a well-order is unique, and the bijection $\gamma \approx \lvert \gamma \rvert$ is used only through Disjoint union, cartesian product, function space and power set respect equinumerosity, and for ordinals $\alpha, \beta$ the sets $\alpha \sqcup \beta$ and $\alpha \times \beta$ carry explicit well-orders, so their cardinalities exist in ZF, which quantifies over existing bijections rather than picking one for each $\gamma$ at once.

What the theorem does not say. It is about cardinals, that is about well-orderable sets. "Every infinite set $A$ satisfies $A \times A \approx A$ " is a strictly stronger statement, and Tarski: the Axiom of Choice is equivalent to the statement that $A \times A \approx A$ for every infinite set $A$ , so extending Hessenberg's theorem from the alephs to arbitrary sets is exactly as strong as choice shows it is equivalent to the Axiom of Choice. So Hessenberg's theorem is not a weaker version of Tarski's with a cheaper proof; it is the exact fragment that ZF proves.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 108 results over 34 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

P. Koellner, Set Theory: The Independence Phenomenon, Theorem 3.15 (standard reference, not scraped)
Cardinal number — cardinal arithmetic (Wikipedia) (standard reference, not scraped)
Aleph number (Wikipedia) (standard reference, not scraped)
T. Jech, Set Theory, 3rd millennium ed., Ch. 3 (Cardinal numbers) (standard reference, not scraped)

Hessenberg: κ⊗κ=κ\kappa \otimes \kappa = \kappaκ⊗κ=κ for every infinite cardinal κ\kappaκ, proved in ZF from the canonical well-order of κ×κ\kappa \times \kappaκ×κ