Citation inventory: Russell (1907), On Some Difficulties in the Theory of Transfinite Numbers and Order Types
Target: Bertrand Russell, On Some Difficulties in the Theory of Transfinite Numbers and Order
Types, Proceedings of the London Mathematical Society, series 2, volume 4, pp. 29–53
(DOI 10.1112/plms/s2-4.1.29; publisher year 1907, often cited as 1906).
Machine-readable records for every citing work found: russell-citations.json.
Why this inventory exists
This paper is a foundational reference for our counting model, not an algorithmic one: it is where Russell works through which collections a definition may legitimately form — the same discipline our counting contract applies when it fixes a scope, a language, and an equivalence before assigning a cardinality. The inventory below records who has built on that work, so the evidence register can cite the line of descent rather than a single 1907 data point.
Coverage and provenance
Retrieved 2026-09-28, by direct public-API queries (reproducible from the URLs below):
| Index | Query | Records | Notes |
|---|---|---|---|
| OpenAlex | cites:W2143162796, 2 cursor pages at 200/page |
301 | all pages exhausted; terminal cursor null |
| Semantic Scholar | citations of paper 6a8e5460f95b92c81c264e94dbffe4d3b1ed904c, limit 1000 |
212 | one page, no next |
| Crossref | 10.1112/plms/s2-4.1.29 |
70 | count only; Crossref exposes no citing list |
Union under a DOI-or-title+year merge key: 376 records (132 in both indexes, 169 OpenAlex only, 75 Semantic Scholar only). An earlier working pass with a more conservative merge reported 368 — the union size is a function of the deduplication rule, not a disagreement between indexes.
What "all papers that reference it" can honestly mean here: every citing work indexed by these two databases as of the retrieval date, deduplicated as above. It is not an exhaustive census of every paper that cites Russell 1907 (index coverage differs, and citations without a DOI in either database are invisible to this merge), and it is not a full-text review of 376 works. Verification levels are tracked per record below.
Classification by title keywords
Titles are the sole classification basis except where a verification level is stated. Counts do not sum to coverage totals across indexes because of the union.
| Class | Records |
|---|---|
| Definability and paradox (definable, Richard, Berry, diagonal, Cantor) | 59 |
| Russell and history (Russell, Frege, Principia, logicism) | 50 |
| Set-theory foundations (axioms, Zermelo, foundations) | 37 |
| Type theory and polymorphism (predicativity, impredicativity, lambda) | 17 |
| Cardinality and order (cardinal, ordinal, transfinite, continuum) | 8 |
| Other / unclassified | 205 |
Most relevant citers
These are the records a reviewer of our counting model is most likely to be asked about, curated from the classes above. Each states how far it was verified.
Bibliography-verified (the citing reference was inspected)
| Work | Cites Russell via | What it contributes |
|---|---|---|
| Fan, Hobson's Conception of Definable Numbers, Hist. & Phil. of Logic 41(2), 2020, DOI | Crossref reference list entry 10.1112/plms/s2-4.1.29 |
Language-relative definability in the Hobson–Richard tradition; connects the diagonal generation of definitions to later computability. Full text paywalled. |
Abstract-inspected (abstract read this session; citation database-reported)
| Work | Index evidence | What the abstract contributes |
|---|---|---|
| Luna & Taylor, Cantor's Proof in the Full Definable Universe, Australasian J. of Logic 9, 2010, DOI | both indexes | Cantor's proof restricted to the definable universe yields a Richard-style paradox: that universe "seems to be countable on one account and uncountable on another". The way out is that definitional contexts restrict the scope of quantifiers — precisely our rule that a count is only meaningful once language, scope and equivalence are fixed. |
| The entanglement of logic and set theory, constructively, Inquiry, 2019, DOI | both indexes | Intuitionistic/predicative treatment of infinite quantification; relevant to predicative readings of our environment. |
| In Praise of Impredicativity (meta-programming formalization), J. Logic Comput., 2019, DOI | both indexes | Where impredicativity is useful despite the paradoxes; documents the boundary we draw when we refuse to let a declaration supply its own implementation. |
| Basic cardinal arithmetic (chapter), 2004, DOI | OpenAlex | Set-theoretic sum/product/exponent identities — the same shapes our Size algebra uses, in their home setting. |
Title-screened candidates (title and index match only; not yet read)
| Work | Index evidence | Why it may matter |
|---|---|---|
| Polymorphism and the obstinate circularity of second order logic, Bull. Symb. Logic, 2017, DOI | Semantic Scholar only | Second-order quantification's circularity; the predicativity boundary for polymorphic counting. |
| Predicativity and parametric polymorphism of Brouwerian implication, 2017 | Semantic Scholar only (arXiv:1710.07704) | A constructive/predicative parametricity; closest modern heir to our uniformity assumptions. |
| Polymorphism and the free bicartesian closed category, 2019 | Semantic Scholar only (arXiv:1907.03481) | Categorical semantics of the type shapes we count. |
| Russell's 1903–1905 Anticipation of the Lambda Calculus, Hist. & Phil. of Logic, 2003, DOI | both indexes | Direct historical bridge from Russell's substitutional theory to the calculus our fragment restricts. |
The remaining 360+ records are in russell-citations.json with year, DOI where available, index flags, and title class. Nothing in the machine-readable file should be read as verified beyond "database-reported".
Access gaps
The publisher full text of Russell 1907 is behind a bot-blocked 403 (Wiley); the contemporary review is a different work, not the paper. Secondary supporting material used in place of the primary text is recorded per claim in the evidence register.