A Note on the Possibility of Self-Reference in Mathematics

Arieh Lev

REVIEW 4 major objections 3 minor 2 references

A Note on the Possibility of Self-Reference in Mathematics

T0 review · 4 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper argues that a proposition about its own provability, when encoded as a set-theoretic function, forces an infinite membership chain and contradicts ZF's axiom of foundation.

desk verdict The paper's central lemma assumes the non-well-founded set it claims to derive—useful as a philosophical thought experiment, but not a formal result. read the letter →

arxiv 1908.02539 v1 pith:2YUQRZT4 submitted 2019-08-07 math.LO

Arieh Lev This is my paper · ORCID

classification math.LO MSC 03E3003F40

keywords self-referenceaxiomoffoundationfirstincompletenesstheoremmeta-modelarithmeticstrongcircularitythree-valuedlogicprovability

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that one specific kind of self-reference—a sentence whose entire content is a claim about its own truth or provability—cannot coexist with ZF's axiom of foundation in the informal 'meta-model' of arithmetic, $N^{*}$, that mathematicians use when working with number theory. The argument encodes each such proposition as a function from the encoded form of the proposition it speaks about to $\{0,1\}$; a self-referential sentence then becomes a function that lies in its own domain, and the standard set-theoretic encoding of that function produces an infinite $\in$-descending chain. That chain contradicts the axiom of foundation, so accepting such sentences as legitimate meaningful claims turns $N^{*}$ into an inconsistent model. The same reasoning is applied to the sentence built in the first incompleteness theorem, under a 'referential' interpretation in which the sentence's content loops back on itself. The paper then proposes a three-valued treatment under which such sentences receive the value 'no meaning' rather than true or false.

What carries the argument

The engine of the argument is the natural map from a set of propositions of type $P$ to a set of functions $F_P$: each proposition '$L$: $M$ is provable/unprovable' becomes a function $f_L$ with domain $\{f_M\}$ and value $1$ or $0$ according to what $L$ claims. For the strongly self-referential proposition $H$, this produces a function whose domain contains itself, $f_H \in \mathrm{dom}(f_H)$. Encoding $f_H$ as its graph $\{(f_H,0)\}$, with ordered pairs built by the standard set-theoretic pairing definition, gives the membership cycle $f_H \in B \in A \in f_H$ and hence an infinite $\in$-descending sequence that violates the axiom of foundation. A second piece of machinery is the distinction between a formal interpretation of the incompleteness sentence, which refers only to a syntactic string and involves no regress, and a referential interpretation, under which asking what the sentence's content refers to leads to infinite regress and turns the sentence into a strongly self-referential proposition.

What would settle it

Take the same self-referential proposition $H$ and replace the 'natural map' with an arithmetized syntactic coding, so that $f_H$ is represented by a natural number rather than by the set $\{(f_H,0)\}$; if the graph of $f_H$ is then a set of pairs of numbers, the chain $f_H \in B \in A \in f_H$ disappears and no infinite $\in$-descending sequence can be derived, showing that the contradiction depends on the encoding rather than on self-reference alone.

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Extended reading notes

Core claim

The paper's central claim is that, in the meta-model $N^{*}$, accepting a strongly self-referential proposition as a meaningful sentence contradicts ZF's axiom of foundation. The proof works by representing every proposition devoted entirely to the provability or unprovability of another proposition as a function whose domain is the singleton containing the representation of that other proposition. A proposition $H$ of the form '$H$ is not provable' is therefore represented by a function $f_H$ with $f_H$ in its own domain; writing $f_H$ as its graph, $\{(f_H,0)\}$, and expanding the ordered pair by the standard set-theoretic definition yields a membership cycle $f_H \in B \in A \in f_H$. From this cycle an infinite $\in$-descending sequence follows, contradicting the axiom of foundation. The paper records this as Lemma 1, derives Lemma 2 (every set of type $P$ is empty in $N^{*}$), and argues that the incompleteness sentence of the first incompleteness theorem, under a referential interpretation, is strongly self-referential and so triggers the same contradiction.

Load-bearing premise

The whole argument rests on assuming that a proposition's meaning can be naturally encoded as a function whose domain contains the encoded form of the proposition it speaks about; without that 'natural coding' assumption, the self-referential loop that produces the contradiction never gets off the ground.

Editorial extensions

If this is right

In any consistent meta-model that keeps the axiom of foundation, strongly self-referential propositions cannot be assigned a truth value; they must be treated as meaningless or excluded.
Under the referential interpretation, the sentence constructed in the first incompleteness theorem is not merely unprovable and irrefutable but inconsistent with foundation in $N^{*}$; the theorem itself remains valid, while its usual self-referential gloss becomes a source of paradox.
Assigning such sentences the value NM ('no meaning') and adopting the extended law of excluded middle restores consistency, because NM-sentences can appear neither as axioms nor inside proofs.
With the revised definition of completeness, the first incompleteness theorem no longer forces incompleteness in the meta-model: the unprovable self-referential sentence is NM, so it falls outside the completeness requirement entirely.
For a consistent theory $T$ containing enough of ZF including foundation, the paper rephrases the first incompleteness theorem as the claim that there exists a sentence that has no meaning in $T$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The contradiction is conditional on the 'natural coding' assumption that the meaning of a proposition can be literally identified with a set-theoretic function whose graph contains the encoded referent; if propositional content is coded instead by an arithmetized syntactic code, the membership cycle $f_H \in B \in A \in f_H$ does not form, and no contradiction with foundation is forced.
If working mathematicians tacitly use the referential interpretation when they say the incompleteness sentence 'says of itself that it is unprovable,' then the paper's inconsistency result applies to ordinary informal mathematical reasoning, not only to a specially constructed semantics; which interpretation is actually operative is an empirical question about mathematical practice.
The same function-encoding move could in principle be run on other diagonal self-referential arguments, such as the halting problem or a diagonal proof of uncountability, yielding a foundation-type contradiction only if their semantic contents are encoded the same way; the paper names these as directions for future work but does not itself derive those contradictions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The punchline: the core argument is circular. Lemma 1 defines f_H as a function with f_H = {(f_H,0)}. In ZF, the axiom of foundation proves that no such set exists, so the proof assumes the exact failure of foundation it purports to establish. The natural map from propositions to functions is stipulated, not derived, and swapping in a standard Gödel numbering makes the membership cycle disappear. The contradiction is an artifact of the chosen encoding.

Now the credit. The paper is clearly written and patient, and it draws a genuinely useful distinction between a formal reading of Gödel's sentence and a referential reading that generates an infinite regress. The author also openly flags a risk of circularity in Section 4, which is more honesty than many papers show—though the risk actually materializes at the encoding step, not where the author seems to think. The analogy to Herzberger's grounding semantics is apt and the three-valued logic proposal in Section 7 is coherent.

The soft spots are proportional: Lemma 2 inherits the flaw from Lemma 1, and the later claims about Gödel's sentence are unsupported as formal statements because they lean entirely on that flawed lemma. The paper would work as a conditional observation: under a semantics that identifies self-referential propositions with self-membered functions, strong self-reference conflicts with foundation. But that is close to tautological.

Who is this for? A reader interested in the philosophical analogy between ungroundedness and the foundation axiom might find the framing thought-provoking, and the exposition is accessible enough for a graduate seminar. But anyone looking for a rigorous result about self-reference and consistency will be disappointed.

My recommendation: I would not send this to a serious referee for the central claim. The flaw is basic and load-bearing. It could be a useful discussion piece in the right setting, but only if the author reframes it as a conditional remark rather than a theorem.

Referee Report

4 major / 3 minor

Summary. The paper proposes an interpretation of self-referential propositions in an informal 'meta-model' N* of ZF, and argues that strongly self-referential propositions (e.g., 'H: H is unprovable') lead to a contradiction with the axiom of foundation in N*. It extends this claim to Gödel's sentence G under a 'referential interpretation' that yields an infinite regress of meaning, and it discusses consequences including a three-valued logic (with value NM) for handling such sentences. Sections 4 and 5 present the central technical argument; Sections 6-8 draw philosophical and metamathematical implications.

Significance. If the central claim were correct, it would constitute a substantive philosophical result about the limits of self-reference in mathematics and would offer a new perspective on the Gödel sentence and the incompleteness theorems. The paper also makes an interesting connection between the axiom of foundation and the notion of ungroundedness. However, the central argument is circular: the proof of Lemma 1 assumes the existence of a non-well-founded set in order to derive a violation of foundation. Because the key encoding step is not justified and is not forced by standard metamathematical practice, the main results are unsupported. The paper does engage seriously with relevant literature on the liar paradox, ungroundedness, and three-valued logics, but the technical foundation does not hold.

major comments (4)

[Section 4, Lemma 1] The proof of Lemma 1 defines the function f_H as a set satisfying f_H = {(f_H, 0)}. In ZF, the axiom of foundation (together with the Kuratowski ordered pair) entails that no set x satisfies x = {(x, 0)}: such a set would give the membership cycle f_H ∈ {f_H} ∈ (f_H, 0) ∈ f_H. The proof therefore posits the existence of the very object whose impossibility it purports to derive. The step from 'H is a legitimate proposition' to 'f_H exists as a set' is an assumption, not a consequence established within ZF. This makes the derivation of Lemma 1 circular and invalid.
[Section 4, definition of F_P and the natural map] The paper assumes a 'natural map' from a set of type P to a set of functions F_P, where each proposition 'L: M is (un)provable' is represented by f_L: {f_M} → {0,1}, and the map is one-to-one and onto. This representation is not forced: if one uses a standard Gödel numbering of formulas inside N* (as is done in the actual proof of the first incompleteness theorem), the membership cycle f_H ∈ {f_H} does not arise, and no contradiction with foundation follows. The paper itself notes an analogy with Gödel coding (Section 4, final paragraph) but states that accepting the analogy is not necessary for the proof; however, the proof of Lemma 1 depends essentially on this particular set-theoretic encoding, so the alleged contradiction is an artifact of the chosen semantics rather than a property of self-referential propositions.
[Section 4, Lemma 2] The proof of Lemma 2 is abbreviated and inherits the circularity of Lemma 1. Even setting that aside, the claim that every set of type P is empty does not follow from the impossibility of self-referential propositions in such a set. A type-P set could consist of an infinite sequence of propositions H_1, H_2, ... where each H_i refers to H_{i+1}; the argument's step from such a sequence to an infinite ∈-decreasing sequence of functions again depends on the unjustified 'natural' encoding as functions with singleton domains. Without a proof that the encoding can be defined for all propositions in A_P, the contradiction with foundation is not established.
[Section 5, referential interpretation of G] The paper claims that under the 'referential interpretation', the Gödel sentence G is strongly self-referential and that combining this with the previous section shows that the existence of G contradicts the axiom of foundation in N*. This conclusion relies entirely on Lemma 1, which is not proven. The description of the infinite regress in the meaning of G is a philosophical point about interpretation, but it does not by itself produce a set-theoretic contradiction; the transition from 'regress in content' to 'violation of foundation' requires the same dubious encoding as in Section 4. Thus result #2 of Section 6 is unsupported.

minor comments (3)

[Throughout] There are numerous typographical and formatting errors, including garbled bibliography entries (e.g., the Chaitin entry and the Internet Encyclopedia entry appear corrupted) and inconsistent notation (e.g., 'N' vs. 'N*' and 'TG' vs. 'G'). The paper would benefit from careful proofreading.
[Section 2] The definition of 'extensional property' is informal; the paper states it will not rigorously define extensional properties of propositions. Since the argument in Section 4 relies on the 'natural map' from propositions to functions, a clearer specification of what counts as an extensional property would be helpful.
[Section 4] The definition of a set of type P is explicitly circular, and the paper notes this possibility. However, the note says that accepting H as legitimate ensures the existence of non-empty type-P sets; this is true only if the encoding step is valid, which is exactly what is at issue.

Circularity Check

3 steps flagged · score 8.0 of 10

Lemma 1 builds the non-well-founded set it claims to derive: the natural map forces f_H = {(f_H,0)}, making the contradiction an artifact of the chosen encoding.

self definitional [Section 4, construction of F_P and proof of Lemma 1]
"Now let us assume the existence of the following self-referential proposition in the set A_p: 'H : H is not provable.' This leads us to conclude that for the function f_H : {f_H} → {0,1}, we have f_H(f_H)=0, and specifically, we have that f_H is a member of its own domain. ... Representing the function f_H as a set would produce f_H = {(f_H,0)}."

The 'natural map' is stipulated to send H to a function whose domain contains f_H itself; encoding functions as graphs then yields the membership cycle f_H ∈ {f_H} ∈ (f_H,0) ∈ f_H. Since ZF proves no set x satisfies x = {(x,0)}, the construction has assumed the existence of the very non-well-founded set Lemma 1 claims to derive. The contradiction is an artifact of this graph encoding; a Gödel-number coding of the same proposition produces no membership cycle.
self definitional [Section 4, definition of type-P sets and the remarks following Lemma 1]
"Definition: A set A of propositions will be called a set of type P, if every proposition in A is a proposition wholly devoted to a statement about the provability (or non-provability) of a proposition belonging to A. ... There is a chance of circularity in the definition of these sets, which might lead these sets to be empty. However, assuming the existence of self-referential propositions like proposition H ... ensures that there will be non-empty sets like these."

The non-emptiness of A_P is obtained by assuming the existence of a self-referential proposition H, exactly the premise whose legitimacy Lemma 1 is supposed to test. Thus the derivation of a foundation contradiction presupposes the self-referential proposition as a legitimate object, and the paper's own note concedes the definition may be circular. The lemma therefore does not establish that self-reference contradicts foundation; it only shows that if one posits a set-theoretic solution to H's self-reference, that solution is non-well-founded.

1 more flagged steps

self definitional [Section 5 and Section 6, result #2 concerning Gödel's sentence]
"We may ascribe two meta-interpretations to the sentence G: a 'formal interpretation' and a 'referential interpretation.' ... combining the second interpretation for G with the results of the previous section shows that the very existence of the Gödel's sentence causes a contradiction to the axiom of foundation in the model N*."

The contradiction for Gödel's sentence is not a consequence of the incompleteness theorem; it is obtained by adopting the 'referential interpretation,' in which G is treated as strongly self-referential. That is precisely the premise required to apply Lemma 1, so result #2 inherits Lemma 1's circularity rather than adding independent support.

full rationale

The central Lemma 1 is not self-contained: its proof defines the map from propositions to functions so that the self-referential H is represented by a set f_H whose domain contains f_H, then encodes f_H as {(f_H,0)} and reads off the ∈-cycle. This assumes the existence of the non-well-founded set it purports to derive; ZF with Kuratowski pairs proves no such set exists, so the 'natural map' cannot exist for H in N*. The paper's own Note concedes the definition of type-P sets may be circular. Result #2 inherits the same problem, since the 'referential interpretation' is chosen to make G strongly self-referential. No self-citation is load-bearing here; the defect is constructional, not bibliographic.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The central claim rests on two ad hoc assumptions: the natural map from propositions to self-membered functions (Section 4) and the referential interpretation of Gödel's sentence (Section 5). Both are informal choices, not theorems of ZF. The meta-model N* itself is an invented entity with no independent evidence. No free parameters are fitted to data.

assumptions (4)

domain assumption There exists an informal meta-model N* of arithmetic that includes ZF, natural language, and interpretations for sentences.
The entire argument is framed inside N*, which is introduced in Section 1 as the informal model mathematicians use when working with number theory. Its existence and properties are stipulated rather than derived.
ad hoc to paper A proposition whose only claim is about its own truth value or provability is 'strongly self-referential', and its truth value or provability is an extensional property independent of phrasing.
Defined in Section 2. This notion selects which propositions fall under Lemma 1. It is not a standard formal definition and is not justified beyond examples.
ad hoc to paper Every proposition of the form 'M is (un)provable' in a set of type P can be naturally interpreted as a function f_L: {f_M} -> {0,1}, and the map from propositions to functions is one-to-one and onto.
Introduced in Section 4 to construct the set F_P. This is the load-bearing step that turns a proposition H into a set f_H containing itself, because for H the domain is {f_H}. The paper does not prove this interpretation exists within ZF.
ad hoc to paper Under the 'referential interpretation' of Gödel's sentence G, one may repeatedly ask what 'the diagonalization of U' says, producing an infinite regress and making G strongly self-referential.
Introduced in Section 5 as a second meta-interpretation of G. The paper explicitly contrasts it with the 'formal interpretation' that yields no regress and does not contradict foundation. The inconsistency claim depends on adopting this optional reading.

invented entities (1)

The meta-model N*
purpose: Serves as an informal semantic frame where ZF, natural language, and interpretations coexist, and where the alleged contradictions with the axiom of foundation are located.
N* is defined introspectively in Section 1 as the model mathematicians 'generally use'. It has no mathematical characterization or independent check, so it is a postulated entity that hosts the inconsistency.

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Cite this review

Pith. "Pith review of A Note on the Possibility of Self-Reference in Mathematics." pith.science (2026). https://pith.science/paper/2YUQRZT4

@misc{pith2026190802539,
  author       = {Pith},
  title        = {Pith review of: A Note on the Possibility of Self-Reference in Mathematics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2YUQRZT4}},
  note         = {Machine review of arXiv:1908.02539}
}

read the original abstract

In this paper we propose an interpretation for self-referential propositions in a "meta-model" N* of ZF. This meta-model N* is considered as an informal model of arithmetic that mathematicians often use when working with number theory. Specifically, we assume that within this meta-model, the axiom system ZF is applied, interpretations for sentences can be offered, and natural language can be used. We show that under the proposed interpretation, some types of self-referential propositions that are considered legitimate in mathematics turn N* into an inconsistent model, and examine the connection of this result to a certain interpretation of godel's first incompleteness theorem. Some general problems which follow from the above discussion are then addressed.

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

[B] Bochvar D. A. 1981, ‘On a three-valued logical calculus and its applications to the analysis of the paradoxes of the classical functional calculus’, History and Philosophy of Logic 2(1-2), pp. 87-112. Translation by Merrie Bergmann to the original paper published in

work page 1981
[1938]

S., Burgess J

[BBJ] Boolos G. S., Burgess J. P. and Jeffry R. C. 2007, Computability and Logic, 5th ed. (Cambridge: Cambridge University Press) [Cha] Chaitin G. J. 1982, ‘Gödel’s Theorem and information.’ International Journal 954.-, pp. 94122sics yPhl Theoretica of [Chi] Chihara C. S. 1973, Ontology and the Vicious Circle Principle (Cornell University Press) , hilosop...

work page 1970

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

[B] Bochvar D. A. 1981, ‘On a three-valued logical calculus and its applications to the analysis of the paradoxes of the classical functional calculus’, History and Philosophy of Logic 2(1-2), pp. 87-112. Translation by Merrie Bergmann to the original paper published in

work page 1981

[2] [1938]

S., Burgess J

[BBJ] Boolos G. S., Burgess J. P. and Jeffry R. C. 2007, Computability and Logic, 5th ed. (Cambridge: Cambridge University Press) [Cha] Chaitin G. J. 1982, ‘Gödel’s Theorem and information.’ International Journal 954.-, pp. 94122sics yPhl Theoretica of [Chi] Chihara C. S. 1973, Ontology and the Vicious Circle Principle (Cornell University Press) , hilosop...

work page 1970