Comparing a partition’s size does not recover the Axiom of Choice

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Family 244 · Mathematical logic
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-21

1. The question

Suppose a set is partitioned into nonempty pieces. Choosing one representative from each piece gives an injection from the collection of pieces into the original set. The Axiom of Choice guarantees that this can always be done.

The Partition Principle asks for less. It requires an injection from the collection of pieces into the original set, but the image of a piece need not lie inside that piece. Does this weaker cardinal comparison nevertheless imply the full Axiom of Choice?

Family 244 contains a manuscript dated 24 September 2026 claiming a negative answer. Assuming the consistency of ordinary ZF set theory, it constructs a consistent theory in which the Partition Principle holds and the Axiom of Choice fails. It separately gives a transitive-model construction with a countable-sequence preservation property.

2. Background and essential definitions

Injections, surjections and sections

An injection $j:Y\to X$ never sends two different elements to the same value. A surjection $f:X\to Y$ reaches every element of its target.

The Partition Principle, abbreviated PP, says

$$\forall X\,\forall Y, \quad (\exists f:X\twoheadrightarrow Y) \Longrightarrow (\exists j:Y\hookrightarrow X).$$

The fibers $f^{-1}(\{y\})$ form a partition of $X$, so this is the partition formulation in function language.

A section of $f$ is stronger: it is a map $s:Y\to X$ satisfying

$$f\circ s=\mathrm{id}_Y.$$

It selects an element from the correct fiber. PP does not impose this equation. For a finite illustration, if $f$ partitions $\{1,2,3,4\}$ into $\{1,2\}$ and $\{3,4\}$, an injection may send the first piece to $3$ and the second to $1$. It compares sizes but chooses no representative from the assigned piece.

Finite families do have choice functions in ZF. The foundational issue is whether the weaker condition for arbitrary infinite sets forces the stronger one.

Choice and wellordering

The Axiom of Choice, AC, says that every family of nonempty sets has a function choosing one element from each set. Over ZF it is equivalent to the assertion that every set can be wellordered. A wellorder is a linear order in which every nonempty subset has a least element.

Choice for wellorderable families, written $\mathrm{AC}_{\mathrm{WO}}$, restricts the indexing family to an ordinal:

$$(X_\xi)_{\xi<\eta},\quad X_\xi\ne\varnothing \quad\Longrightarrow\quad \exists c\ \forall\xi<\eta\ [c(\xi)\in X_\xi].$$

The sets being chosen from need not themselves be wellorderable. This includes countable choice, but does not mean that all possible index sets are covered.

Relative consistency and transitive models

ZF is set theory without AC; ZFC adds AC. A relative-consistency theorem says that if one theory has no contradiction, neither does another. It does not prove the consistency of ZF from nothing.

A transitive model has the property that members of its sets are themselves in the model. Such models let membership be interpreted literally. Assuming a countable transitive model exists is stronger than merely assuming syntactic consistency, so the manuscript carefully states its two model-theoretic conclusions separately.

3. Development history and previous partial results

The Partition Principle is traced to Beppo Levi in 1902. It belongs to the study of cardinal arithmetic without Choice, where familiar comparisons between surjections, injections and wellorderings can separate.

Banaschewski and Moore studied its relation to the dual Cantor–Bernstein principle. Ordinary Cantor–Schröder–Bernstein, provable in ZF, turns injections in both directions into a bijection. The dual version asks for a bijection from surjections in both directions. PP would convert those surjections into injections, so it implies the dual principle.

PP already contains substantial choice. The implication $\mathrm{PP}\Rightarrow\mathrm{AC}_{\mathrm{WO}}$ is classical; the manuscript cites the account by Ryan-Smith, with the argument attributed through Pelc to Pincus. Thus the ordinal-indexed choice clause in the new theorem is not an additional strength beyond PP. It is listed separately because the construction proves it before using it to establish full PP.

Blass’s theory of small violations of choice provides a local-to-global framework. One version says that a fixed set $A$ is large enough that every nonempty set is a surjective image of $A\times\eta$ for some ordinal $\eta$. An injective version asks instead that every set embed into such a product. These directions cannot be exchanged without an argument.

Ryan-Smith clarified which local comparison principles suffice to recover full PP under suitable presentation assumptions. Reverse injections for quotients of one set are not automatically the same as reverse injections for surjections between arbitrary subsets of that set.

The model-building method uses symmetric forcing extensions, which preserve ZF while withholding some choices available in a larger forcing extension. Pincus’s methods and Holy–Schilhan’s work on ordering and dependent choice supply predecessors for the support and locality machinery.

The manuscript notes that contemporary work still listed PP versus AC as open, and that corrections to several earlier proposed constructions withdrew their model claims. The new theorem is presented with an ordinary consistency deduction and a separate transitive construction, rather than treating a stronger model-existence assumption as if it followed from consistency alone.

4. What the new manuscripts claim

The main assertion is

$$\operatorname{Con}(\mathrm{ZF}) \Longrightarrow \operatorname{Con}\bigl( \mathrm{ZF}+\mathrm{PP}+\mathrm{AC}_{\mathrm{WO}}+\neg\mathrm{AC} \bigr).$$

Here $\operatorname{Con}$ means syntactic consistency. The target is ordinary first-order set theory with pure sets, Foundation and the full Separation and Replacement schemes. It is not a theory of atoms or a fragment omitting those axioms.

The separate theorem starts with any countable transitive model $V$ of ZFC. It constructs a set-forcing extension $V[G]$ and a transitive symmetric submodel $W$ with

$$V\subseteq W\subseteq V[G],\qquad \operatorname{Ord}^W=\operatorname{Ord}^V,$$

such that $W$ satisfies the displayed target theory. Every countable sequence of ground-model elements that belongs to $W$ already belongs to $V$.

The preservation statement concerns sequences of elements of the ground model. It does not say that $W$ contains no new sets, nor that every countable sequence of newly added objects belongs to the ground.

In the constructed model, a set $A$ cannot be wellordered. The family of all nonempty subsets of $A$ has no choice function. The reverse injections promised by PP and this failure-of-choice family exist in the same model, so the separation is internal to one universe of sets.

5. Why the advance matters

Importance: 5/5, provisional. The result would settle a longstanding question about whether cardinal comparison alone recovers arbitrary selection.

It identifies a precise gap between saying that a quotient has no more elements than its source and choosing one element from every fiber. The two operations agree under AC but need not have the same strength over ZF.

The separation is robust enough to retain ordinal-indexed choice and, in the transitive version, to avoid adding countable sequences of ground elements. Thus failure of full Choice is not produced merely by discarding every familiar form of countable selection.

The rating concerns the foundational separation and the scope of its consistency assumption. It does not mean PP is weak in every respect: it still implies significant cardinal-comparison principles and $\mathrm{AC}_{\mathrm{WO}}$.

6. The main idea

Concentrate the failure of Choice in one set

The construction adds a set $A$ of distinct generic subsets of the ground model’s $\omega_1$. Permutations of their labels prevent $A$ from having a wellordering inside the symmetric submodel.

At the same time, the model is arranged so that every nonempty set, of arbitrary rank, is a surjective image of $A\times\eta$ for some ordinal $\eta$. This makes $A$ a common source for controlling all sets, rather than proving PP only for low-rank examples.

The forcing also provides enough wellorderable support to meet each member of an ordinal-indexed family, yielding $\mathrm{AC}_{\mathrm{WO}}$.

The central extra property is much stronger than merely injecting quotients back into $A$:

Every nonwellorderable surjective image of $A$ is bijective with $A$.

The challenge is to make these bijections belong to the symmetric model. Choosing arbitrary bijections in the surrounding choice universe would not suffice.

Make local bijections coherent

The manuscript organizes labels using diagrams indexed by a specially constructed left-cancellative monoid. A monoid is an associative system with an identity; left cancellation means $xy=xz$ forces $y=z$. Its translated subdiagrams provide a controlled notion of support.

For a fixed quotient of $A$, an availability pattern records exactly which subdiagrams can name a given quotient element. Labels have corresponding patterns. Copying and support-descent arguments give equal cardinalities for the label fibers and quotient fibers with each exact pattern.

Equal cardinalities locally are still insufficient. The chosen bijections must agree on overlaps and commute with changes of coordinates. The paper organizes the fibers into components with countable cofinal chains of diagrams.

Along each chain, a preliminary bijection is invariant under permutations fixing a sufficiently late diagram. The remaining symmetries reduce to a countable group. Bounded initial segments of the generic subsets distinguish those residual symmetries and select compatible transported bijections.

Their union becomes one hereditarily symmetric graph, hence a bijection inside $W$. This coherence step is what turns external cardinal comparisons into the internal quotient-classification property.

Assemble reverse injections for arbitrary sets

Take any surjection $f:X\to Y$. Present $X$ as an image of $A\times\eta$. By taking the first ordinal slice that reaches each target element, partition $Y$ into disjoint pieces $Y_i$, and choose disjoint source pieces $S_i$ mapping onto them. The $S_i$ need not cover all of $X$.

If $Y_i$ is wellorderable, ordinal-indexed choice selects a representative in each fiber, giving an injection $Y_i\to S_i$.

If $Y_i$ is not wellorderable, neither is $S_i$. Both are images of $A$, so the special quotient property makes each bijective with $A$. Composing those bijections gives an injection $Y_i\to S_i$, although it need not follow the fibers of $f$.

For each slice, form the set of all suitable injections. These are nonempty sets indexed by an ordinal. A final use of $\mathrm{AC}_{\mathrm{WO}}$ chooses one per slice. Their union is injective because the source pieces are disjoint, proving PP for the original arbitrary sets.

Keep the consistency assumptions separate

For ordinary relative consistency, the paper passes from consistency of ZF to a countable model of a constructible choice ground. That model may be externally ill-founded. Symmetric forcing is interpreted through names and forcing relations, without pretending that external well-founded recursion is available.

For a given countable transitive ground, names can instead be evaluated by ordinary well-founded recursion. This gives the literal inclusions and ordinal preservation in the second theorem. The two routes share the construction but justify different conclusions under different premises.

7. Scope, limitations, and open questions

The reverse injection in PP is not generally a section of the original surjection. In the nonwellorderable slices, the construction deliberately uses bijections with $A$ without requiring fiber preservation. Replacing that freedom by a section would recover full Choice.

The principal statement is relative consistency, conditional on consistency of ZF. It does not assert that ZF proves its own consistency or that ordinary consistency alone yields a transitive model.

The model also satisfies the dual Cantor–Schröder–Bernstein principle and the Weak Partition Principle, as consequences of PP. This does not collapse all weak choice principles into one; their other implications remain separate questions.

The nonwellorderable set $A$ cannot have a choice function on all its nonempty subsets: repeatedly selecting from the remaining subset would wellorder $A$. This explains how full Choice can fail even while every ordinal-indexed family has a choice function.

8. Sources and further reading

Banaschewski and Moore, The dual Cantor–Bernstein theorem and the partition principle, Notre Dame Journal of Formal Logic 31 (1990), 375–381, gives the cardinal-comparison background. Blass, Injectivity, projectivity, and the axiom of choice, Transactions of the American Mathematical Society 255 (1979), 31–59, introduces small violations of choice.

Ryan-Smith, Local reflections of choice, Acta Mathematica Hungarica 176 (2025), 244–257, explains the local-to-global distinctions used here. Holy and Schilhan, The ordering principle and higher dependent choice, is a predecessor for the symmetric support construction.

For the method, Karagila’s Forcing & symmetric extensions gives background. Blass and Kulshreshtha, Cardinal well-foundedness and choice, records the surrounding comparison questions in the version cited by the source.

Source manuscripts

• M1. OpenAI, The Partition Principle does not imply Choice (September 24, 2026).

Sources consulted on 21 October 2026 at repository revision 1581d4c. Download the manuscript citations.