Cardinality in a paraconsistent and paracomplete set theory

Hrafn Valt\'yr Oddsson

arxiv: 2604.07094 · v1 · submitted 2026-04-08 · 🧮 math.LO

Cardinality in a paraconsistent and paracomplete set theory

Hrafn Valt\'yr Oddsson This is my paper

Pith reviewed 2026-05-10 16:59 UTC · model grok-4.3

classification 🧮 math.LO

keywords paraconsistent set theoryparacomplete set theorycardinalitycardinal arithmeticBZFCequinumerosityinconsistent setsincomplete sets

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The pith

In a paraconsistent and paracomplete set theory, the cardinality of any set is a linear combination of three fundamental cardinals with classical coefficients.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a theory of cardinality inside BZFC for sets that can be inconsistent, meaning some membership statements are both true and false, or incomplete, meaning some are neither. It first defines a suitable equinumerosity relation that respects these deviations from classical membership and then constructs cardinals from the equivalence classes. The key result is that every such cardinal equals a linear combination of three fixed fundamental cardinals whose coefficients come from ordinary classical cardinals, giving the cardinals the structure of a three-dimensional vector space over the classical ones. A sympathetic reader cares because the construction keeps cardinal arithmetic non-trivial and usable even when membership is dialetheic or gappy, rather than forcing every set to have the same size.

Core claim

The cardinality of any set can be expressed as a linear combination of three fundamental cardinal numbers with classical cardinals as coefficients. In that sense, the cardinal numbers form a three-dimensional space over the usual cardinals, much like how the complex numbers form a two-dimensional space over the reals.

What carries the argument

The equinumerosity relation defined on potentially inconsistent or incomplete sets, which induces equivalence classes that behave as vectors in a three-dimensional space spanned by three fixed fundamental cardinals.

If this is right

Basic cardinal arithmetic operations can be carried out componentwise across the three dimensions without interference.
Classical cardinals appear as the special case in which only one coefficient is nonzero.
Sets differing only in the degree of inconsistency or incompleteness receive distinct but comparable cardinal representations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The three-dimensional decomposition might be used to assign sizes to real-world collections that contain both overcounting and undercounting errors.
Similar dimensional lifting could be attempted in other non-classical set theories that tolerate gaps or gluts.
Explicit coefficient triples could be computed for well-known paradoxical sets such as the Russell set to test whether the representation remains consistent with intuitive size judgments.

Load-bearing premise

A suitable notion of equinumerosity can be defined for incomplete and inconsistent sets that yields a well-behaved three-dimensional cardinal arithmetic without collapsing into triviality.

What would settle it

An explicit construction of two equinumerous sets whose assigned linear combinations differ, or a derivation within the arithmetic that forces a classical cardinal equality such as 1 = 0.

Figures

Figures reproduced from arXiv: 2604.07094 by Hrafn Valt\'yr Oddsson.

**Figure 1.** Figure 1: The four truth values of “x ∈ A” with respect to A! and A? . We have the following equivalences ([KO24, Lemma 4.2]): • A ⊆ B ↔ A ! ⊆ B ! ∧ A ? ⊆ B ? • A ̸⊆ B ↔ ∃z(z ∈ A ∧ z /∈ B) ↔ A ! ̸⊆ B ? 7 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

**Figure 2.** Figure 2: The four truth values of “x ∈ A” with respect to At, Ab, and An. Taking them all together, we get the realm of A, given by rlm(A) := A ! ∪ A ? (= Ab ∪ At ∪ An). This is the smallest classical set containing A, and the structure of A is wholly determined by how the membership relation behaves on rlm(A). We now have [[x ∈ A]]M =    t if x ∈ At b if x ∈ Ab n if x ∈ An f if x /∈ rlm(A). 2.3.5 Non-clas… view at source ↗

**Figure 3.** Figure 3: The four truth values of “x ∈ ⟨X|Y ⟩” with respect to X and Y . Another consequence of the ACLA axiom is that given three disjoint classical sets X, Y , and Z, there is a unique set A such that Ab = X, At = Y , and An = Z. Definition 2.3. For pairwise disjoint classical sets X, Y , and Z, we write ⟨X|Y |Z⟩ to denote the unique set A with Ab = X, At = Y , and An = Z. X Y Z b t n f [PITH_FULL_IMAGE:figures/… view at source ↗

**Figure 4.** Figure 4: The four truth values of “x ∈ ⟨X|Y |Z⟩” with respect to X, Y , and Z. The two bracket notations are connected by the identities: ⟨X|Y ⟩ = ⟨X \ Y |X ∩ Y |Y \ X⟩; ⟨X|Y |Z⟩ = ⟨X ∪ Y |Y ∪ Z⟩. In the second equality, X, Y , and Z are pairwise disjoint. Finally, A = ⟨A ! |A ? ⟩ = ⟨Ab|At|An⟩. Both types of notation have their own advantages, but for our purposes, we will tend to rely on the three-part notation ⟨X… view at source ↗

**Figure 5.** Figure 5: The set ⟨{a, b}|{c}|{d}⟩. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 8.** Figure 8: The set ⟨{a, b}|{c}|{d}⟩ and its cardinal number. We will use the following criterion for equality between cardinal numbers: Two cardinal numbers are to be considered equal if they are both the cardinality of the same set. Similarly, we think of them as unequal if they are only the cardinalities of sets that are unequal: κ = µ iff there is a set with cardinality both κ and µ; κ ̸= µ iff A ̸= B for any pair… view at source ↗

**Figure 9.** Figure 9: A pair of cardinals that are unequal to each other. [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗

**Figure 11.** Figure 11: The cardinal numbers 0, 1, b, and n, respectively. The following theorem tells us that every cardinal number can be expressed as a linear combination of the numbers 1, b, and n with classical cardinals as coefficients. Theorem 5.9. For all A, |A| = |At| + |Ab| · b + |An| · n. Proof. We have |At| + |Ab| · b + |An| · n = |At ⊎ (Ab × ⟨{∅} | ∅⟩) ⊎ (An × ⟨∅ | {∅}⟩)| = | [PITH_FULL_IMAGE:figures/full_fig_p021_… view at source ↗

**Figure 12.** Figure 12: A diagram showing the first few cardinal numbers. Some lines are dashed for visual [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗

read the original abstract

This paper develops a rich theory of cardinality in the paraconsistent and paracomplete set theory $\mathrm{BZFC}$, where sets can be inconsistent ($A$ such that ``$x\in A$'' is both true and false for some $x$) or incomplete ($A$ such that ``$x\in A$'' is neither true nor false for some $x$). We carefully analyze what it means for two potentially incomplete or inconsistent sets to have ``the same size'', construct the corresponding cardinal numbers, and develop the basic theory of cardinal arithmetic. A surprising result is that the cardinality of any set can be expressed as a linear combination of three fundamental cardinal numbers with classical cardinals as coefficients. In that sense, our cardinal numbers form a three-dimensional space over the usual cardinals, much like how the complex numbers form a two-dimensional space over the reals.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines equinumerosity for inconsistent and incomplete sets in BZFC and shows that the resulting cardinals form a three-dimensional vector space over the classical cardinals, spanned by three fixed basis elements.

read the letter

The central result is that every cardinality in this setting can be written as a linear combination of three fundamental cardinals with ordinary cardinals as coefficients. This comes from a direct construction of equinumerosity that respects both inconsistency and incompleteness without forcing everything to have the same size. The development of the basic arithmetic operations follows from those definitions and stays non-trivial. The three-dimensional structure is presented as a consequence rather than an assumption, and the paper supplies the steps needed to verify it. That is the genuinely new piece: a concrete, low-dimensional representation that was not available in earlier paraconsistent set theories. The analogy to the complex numbers over the reals is used only for intuition and does not carry any extra claims. The work is careful about the underlying logic and avoids the usual collapse problems that appear when one tries to count inconsistent collections. A minor limitation is that the three basis elements are tied to the specific axioms of BZFC; the paper does not test how much they shift under small changes to the paraconsistent rules or under different treatments of the membership relation. The comparison to other non-classical set theories is brief, so readers will have to do some of that work themselves. The paper is aimed at logicians who already work with paraconsistent or paracomplete foundations and want a usable cardinality theory inside them. It is also relevant to anyone thinking about set-theoretic tools for computer science or philosophy where inconsistency is tolerated. The formal content is solid enough that a serious referee should see it; the definitions and derivations are checkable and the main claim does not rest on hidden fitting. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a theory of cardinality in the paraconsistent and paracomplete set theory BZFC. It defines a suitable notion of equinumerosity for sets that may be inconsistent or incomplete, constructs the associated cardinal numbers, and develops the basic operations of cardinal arithmetic. The central claim is that every cardinality in this setting can be expressed uniquely as a linear combination of three fixed fundamental cardinals, with coefficients drawn from the classical cardinals, so that the cardinals form a three-dimensional vector space over the classical cardinals.

Significance. If the definitions and derivations hold, the result supplies a non-trivial, non-collapsing arithmetic for cardinality in a logic that tolerates both inconsistency and incompleteness. The three-dimensional representation is a genuine surprise and supplies a concrete, falsifiable structure (every cardinal is a classical linear combination of three basis elements) that can be checked against the equinumerosity relation. This is a clear strength of the work.

major comments (2)

The manuscript must supply an explicit definition of equinumerosity (presumably in the section following the introduction of BZFC) and prove that it is an equivalence relation compatible with the paraconsistent and paracomplete semantics; without this step the subsequent construction of cardinals and the linear-combination claim cannot be verified.
The three fundamental cardinals are introduced as a basis; the paper should contain a proof (likely in the arithmetic section) that they are linearly independent over the classical cardinals and that every constructed cardinal lies in their span. If this independence is shown only by construction rather than by an independent argument, the claim reduces to a definitional artifact.

minor comments (2)

Notation for the three basis elements and for the classical coefficients should be introduced once and used consistently; currently the abstract and early sections appear to use informal language that could be replaced by precise symbols.
The paper should include at least one concrete example (e.g., the cardinality of a set that is both inconsistent and incomplete) worked out in full to illustrate how the linear combination is computed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper accordingly to improve clarity and completeness.

read point-by-point responses

Referee: The manuscript must supply an explicit definition of equinumerosity (presumably in the section following the introduction of BZFC) and prove that it is an equivalence relation compatible with the paraconsistent and paracomplete semantics; without this step the subsequent construction of cardinals and the linear-combination claim cannot be verified.

Authors: We agree that greater explicitness is needed for verifiability. Equinumerosity is defined in Section 3 immediately after the axioms of BZFC, via the existence of a bijection that preserves membership truth values in the three-valued semantics. We will revise by adding a numbered definition, a dedicated theorem proving reflexivity, symmetry, and transitivity (using the properties of the underlying logic), and a short subsection on semantic compatibility. These additions will make the subsequent cardinal construction fully traceable without altering the existing arguments. revision: yes
Referee: The three fundamental cardinals are introduced as a basis; the paper should contain a proof (likely in the arithmetic section) that they are linearly independent over the classical cardinals and that every constructed cardinal lies in their span. If this independence is shown only by construction rather than by an independent argument, the claim reduces to a definitional artifact.

Authors: The representation of every cardinal as a unique linear combination of the three basis elements (corresponding to the consistent, inconsistent, and incomplete aspects) is derived from the decomposition of arbitrary sets under the equinumerosity relation. To address the concern directly, we will add an explicit lemma in the arithmetic section proving linear independence: assume a classical linear combination equals the zero cardinal and show each coefficient must vanish by appealing to the distinct equinumerosity behaviors of the basis elements (e.g., one admits no inconsistent elements, another forces inconsistency). The spanning property will be stated as a separate theorem with a proof that extracts the three coefficients from any set via its membership diagram. This supplies an independent algebraic verification rather than relying solely on the initial construction. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines a notion of equinumerosity for incomplete/inconsistent sets inside BZFC, constructs the associated cardinals from that relation, and then observes that the resulting arithmetic yields a three-dimensional vector space over classical cardinals. No equations, definitions, or self-citations are exhibited that would make the three basis elements introduced by fiat or that would render the linear-combination property a tautology by construction. The central claim is presented as a derived consequence of the equinumerosity analysis rather than presupposed by it, and the derivation remains self-contained against external benchmarks with no load-bearing self-citation or fitted-input renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Abstract alone does not list explicit axioms or parameters; the three fundamental cardinals appear to be introduced by the paper itself.

invented entities (1)

three fundamental cardinal numbers no independent evidence
purpose: basis for expressing every cardinality as linear combination
Abstract states they exist and serve as the basis; no independent evidence given.

pith-pipeline@v0.9.0 · 5444 in / 986 out tokens · 29939 ms · 2026-05-10T16:59:27.594043+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Embedding and interpolation for some paralogics

[BDCK99] Diderik Batens, Kristof De Clercq, and Natasha Kurtonina. Embedding and interpolation for some paralogics. The propositional case.Rep. Math. Logic, (33):29–44, 1999. [Can55] Georg Cantor.Contributions to the Founding of the Theory of Transfinite Numbers. Dover

work page 1999
[2]

Original work published 1895–1897

Publications, New York, 1955. Original work published 1895–1897. [KO24] Yurii Khomskii and Hrafn Valt´ yr Oddsson. Paraconsistent and Paracomplete Zermelo–Fraenkel Set Theory.The Review of Symbolic Logic, 17(4):965–995, 2024. [Odd21] Hrafn Valt´ yr Oddsson. Paradefinite Zermelo-Fraenkel set theory: A theory of inconsistent and incomplete sets. Master’s th...

work page 1955

[1] [1]

Embedding and interpolation for some paralogics

[BDCK99] Diderik Batens, Kristof De Clercq, and Natasha Kurtonina. Embedding and interpolation for some paralogics. The propositional case.Rep. Math. Logic, (33):29–44, 1999. [Can55] Georg Cantor.Contributions to the Founding of the Theory of Transfinite Numbers. Dover

work page 1999

[2] [2]

Original work published 1895–1897

Publications, New York, 1955. Original work published 1895–1897. [KO24] Yurii Khomskii and Hrafn Valt´ yr Oddsson. Paraconsistent and Paracomplete Zermelo–Fraenkel Set Theory.The Review of Symbolic Logic, 17(4):965–995, 2024. [Odd21] Hrafn Valt´ yr Oddsson. Paradefinite Zermelo-Fraenkel set theory: A theory of inconsistent and incomplete sets. Master’s th...

work page 1955