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LIT-1454: Representation of a preference ordering by a numerical function

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StatusUnavailable — registered 2026-10-23, not read: no lawful full text could be reached. Looked for on 2026-10-23. The Cowles Foundation's page for it (cowles.yale.edu/node/148338) lists it as a Discussion Paper, document control number 2083, with no file. The old Cowles Foundation Paper URL cited elsewhere (cowles.econ.yale.edu/P/cp/p00b/p0097.pdf) redirects to the Foundation's home page. Its archive of Cowles Commission discussion papers lists the economics series only up to Econ 245, and guessed paths for the paper on its file server were not found. The only Crossref record is the 1983 reprint in Debreu's Mathematical Economics (Cambridge University Press, ch. 6, pp. 105–110), which is paid, and Unpaywall reports no open copy of it. OpenAlex was not consulted (it refused requests without a key that day). Its theorem is read first-hand in Debreu's Theory of Value, which the Cowles Foundation does post (LIT-1435, §4.6 and ch. 4 note 2). There Debreu proves the connected-separable case, credits that form to Eilenberg, and cites this paper for removing connectedness on second-countable spaces. It stays Unavailable until a copy is supplied, not on merit.
Version2
Tagsmathematics, social-science
Extended byLIT-1435 — Theory of Value: An Axiomatic Analysis of Economic Equilibrium

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Gerard Debreu (1954), “Representation of a preference ordering by a numerical function”, in R. M. Thrall, C. H. Coombs & R. L. Davis (eds), Decision Processes, Wiley, pp. 159–165. Reprinted in G. Debreu, Mathematical Economics (Cambridge University Press, 1983), ch. 6 — DOI-10.1017/CCOL052123736X.007

Key takeaways

Registered, not read. What is said here comes from Debreu’s restatement in Theory of Value (LIT-1435), not from this paper.

  • The theorem as Debreu later credits it to this paper. On a space satisfying the second axiom of countability, a continuous complete preorder has a continuous utility function, and connectedness is not needed. Theory of Value, p. 73: “the assumption of connectedness has been removed for spaces satisfying the second axiom of countability in G. Debreu [3]”.
  • The lexicographic counterexample. Theory of Value gives the lexicographic ordering of R² as a complete preordering with no real-valued representation (p. 72). Whether this paper also gives it is not known here.

Standing in the record

Filed on 2026-10-23 because CLAIM-436 rests its third part on “Debreu’s representation theorem”. That part states the theorem for “a connected, separable topological space”. In Theory of Value Debreu credits that form to Eilenberg (“essentially a result of S. Eilenberg”), and keeps this paper for the second-countable form without connectedness. The claim’s statement is correct, but the attribution should be “Eilenberg (1941); Debreu (1954)”, or the claim should use the second-countable form if it means to cite this paper. See NOTE-1112.

No instruction for machine-learning practice; nothing here belongs in the anthology.

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