Contribution
For this record, one section of the founding text of axiomatic general equilibrium. §4.6 gives Debreu’s statement and proof of the existence of a continuous utility function for continuous preferences, and its notes give the counterexample and the history.
Key insight
Representability is not automatic. It needs a topological condition on the preferences (continuity) and a countability condition on the space (a countable dense subset). The lexicographic order shows what goes wrong without them. The reals cannot hold uncountably many disjoint intervals (pp. 72–73).
Assumptions
- Preferences are a complete preorder (“completely preordered by ≾”).
- (a): for every x′ the sets {x | x ≾ x′} and {x | x ≿ x′} are closed.
- The consumption set is connected, and either a subset of Rˡ or, in the note’s generalisation, a separable topological space.
Key results
- Theorem (1), p. 56: “Let Xᵢ be a connected subset of Rˡ, completely preordered by ≾. Under assumption (a) there is on Xᵢ a continuous utility function.”
- Its converse, p. 56: “If there is a continuous utility function on Xᵢ, preferences satisfy assumption (a)”.
- The generalisation, p. 73: the proof used Rˡ “only to obtain a countable dense subset”, so the theorem holds on “a connected, separable topological space”. This is “essentially a result of S. Eilenberg [1]” (American Journal of Mathematics 63:39–45, 1941). The 1954 paper removes connectedness for second-countable spaces.
- The counterexample, pp. 72–73: the lexicographic ordering of R², (a, b) ≺ (a′, b′) if (1) a < a′, or (2) a = a′ and b < b′. A representation f would give each a a non-degenerate interval Iₐ, pairwise disjoint, so “a one-to-one correspondence a ↔ Iₐ is established between the set of real numbers (which is uncountable) and a set of pairwise disjoint, non-degenerate, real intervals (which is countable), a contradiction.”
- Ordinality, p. 56: “Given a preference preordering, a corresponding utility function is thus arbitrary to a large extent.”
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | A continuous complete preorder on a connected subset of Rˡ has a continuous utility function | strong | proof, pp. 56–59 |
| C2 | The same holds on any connected separable space | strong | the proof uses only a countable dense subset, p. 73 |
| C3 | The lexicographic order on R² has no real-valued representation | strong | proof, pp. 72–73 |
| C4 | A utility function is fixed only up to increasing transformation | strong | immediate, p. 56 |
Concepts
- complete preorder — a complete, transitive relation ≾ on the consumption set.
- utility function — “an increasing function from Xᵢ preordered by ≾ to R” (p. 56).
- continuity assumption (a) — closed upper and lower contour sets.
- lexicographic ordering — order by the first coordinate, and by the second only on ties.
Connections
Restates Debreu 1954 (LIT-1454) and Eilenberg 1941 (not in the record), and names Wold as an earlier rigorous special case. Note 2 points to Herstein and Milnor for weaker continuity assumptions. The record’s LIT-067 is Debreu’s 1952 equilibrium theorem.
Bearing on the record
The record’s proposal reads Veit’s “common currency” (THEORY-101) and ALARM’s override (THEORY-090) as continuous and lexicographic potential orderings (CLAIM-435). It rests the difference on CLAIM-436: an ordering is a currency where it has a numerical representation, and the lexicographic order has none. §4.6 bears on this as follows.
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CLAIM-436’s mathematics is right. Its part 3 states the theorem for “a connected, separable topological space whose upper and lower contour sets are closed”. That is exactly the form on p. 73. Its part 2 is Debreu’s lexicographic argument, moved from R² to [0, 1]². Its part 4, that successive currencies need not share exchange rates, follows Debreu’s remark that a utility function is “arbitrary to a large extent”. The claim’s stated defeater, “a misstatement of Debreu’s theorem’s hypotheses”, does not fire.
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The attribution should change. Debreu credits the connected-separable form to Eilenberg 1941, and keeps the 1954 paper for the second-countable form without connectedness. The claim should cite “Eilenberg (1941); Debreu (1954)”. Or, if it means the 1954 theorem, it should state that form: a continuous complete preorder on a second-countable space has a continuous representation. Both forms cover the option spaces the claim has in mind.
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Continuity is sufficient, not necessary, for a currency. Debreu’s theorem gives a continuous representation for continuous orderings, and the converse ties continuity to continuous representation. It does not say a discontinuous ordering has no representation. Only some discontinuous orderings, the lexicographic one among them, have none. This matters for CLAIM-435’s test. For ALARM, that claim predicts “a step at a threshold beyond which no resource buys the cost”. For Veit it predicts a continuous indifference curve. A step shows the ordering is discontinuous. It does not show it is lexicographic in Debreu’s sense, or that it lacks a currency. Three cases need telling apart.
- A two-class override. Every option over the threshold ranks below every option under it, and the options are otherwise traded off. The first key then takes two values. Such an ordering has a representation: add a large constant penalty above the threshold to a trade-off function. It is a currency, a discontinuous one, and it is what “no resource buys the cost” describes.
- A cost-only override. Above the threshold, options are ranked only by cost and the resource is ignored. This is also representable.
- The non-representable case. The cost prevails at every level across a continuum, and the resource breaks ties only between equal costs. This is Debreu’s R² example, and only this has no currency.
So the test as worded separates continuous from discontinuous potential, not currency from no currency. CLAIM-436’s phrase “a lexicographic override lies in the gap” is true only of the third case. ALARM’s “100% weight”, as CLAIM-435 operationalises it, is the first case and has a currency. Either the two claims should say “discontinuous” where they say “lexicographic”, or CLAIM-435 needs a further mark: whether the cost overrides the resource at every cost level above the threshold, not only across the threshold.
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Completeness is presupposed. Debreu’s theorem starts from a complete preorder. The proposal’s ambivalence, “no ordering at all” (CLAIM-440, consequence 4), lies outside the theorem’s domain, as the record already says (THEORY-027). Debreu offers nothing on orders that fail completeness.
Limitations
Only one section of the book was read. The OCR is imperfect, and quoted text was checked against the page images on the three pages quoted most. Elsewhere the quotations are from OCR checked by script, with symbols transcribed by eye.
Open questions
Whether the trade-off experiments Veit’s criterion relies on can tell a discontinuous but representable ordering from a lexicographic one, given finite stimulus sets. On a finite set every complete preorder is representable (CLAIM-436, part 1), so the distinction is only empirical in the limit.