RemarkRemark: AI-generatedProof: Not applicableverified 2026-09-24 (gpt-6-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Choice ledger for the Riemann integral: the page results are proved in ZF

The results on this page are proved in ZF. Their proofs use finite witness selection, explicit formulas, completeness of the real numbers, and finite subcovers. The only countable union in the proof of the Lebesgue criterion is a union of compact content-zero sets; its interval covers are chosen by least natural-number codes, so no countable-choice premise is spent.

The ledger, item by item

item	choice used	where the apparent choice is resolved
Partition of $[a, b]$ as a finite strictly increasing list $a = t_{0} < t_{1} < \dots < t_{n} = b$ , its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions	none	finite lists and recursion
For bounded $f$ on $[a, b]$ and a partition $P$ : the infimum $m_{i}$ and supremum $M_{i}$ of $f$ on the $i$ -th subinterval, and the lower and upper Darboux sums $L (f, P) = \sum_{i} m_{i} Δ_{i}$ and $U (f, P) = \sum_{i} M_{i} Δ_{i}$	none	canonical suprema and infima
Refining a partition raises the lower Darboux sum and lowers the upper one, and every lower sum is at most every upper sum: $L (f, P) \leq L (f, P^{'}) \leq U (f, P^{'}) \leq U (f, P)$ when $P^{'}$ refines $P$ , and $L (f, P) \leq U (f, Q)$ for arbitrary partitions $P$ and $Q$ ; moreover the two changes are at most $2 M (n^{'} - n) ∥ P ∥$	none	finite induction
The lower and upper Darboux integrals of a bounded $f$ on $[a, b]$ as $\sup_{P} L (f, P)$ and $\inf_{P} U (f, P)$ , Darboux integrability as their equality, and the notation $\int_{a}^{b} f$	none	supremum and infimum over the set of partitions
If $m \leq f \leq M$ on $[a, b]$ then $m (b - a) \leq L (f, P) \leq \underline{\int_{a}^{b}} f \leq \overline{\int_{a}^{b}} f \leq U (f, P) \leq M (b - a)$ for every partition $P$ ; in particular every constant function is integrable, with $\int_{a}^{b} c = c (b - a)$	none	ordered-field bounds
Riemann's criterion: a bounded $f$ on $[a, b]$ is Darboux integrable if and only if for every real $ε > 0$ there is a partition $P$ with $U (f, P) - L (f, P) < ε$	none	finitely many existential instantiations
Tagged partitions of $[a, b]$ , with a tag $ξ_{i}$ in each subinterval, and the Riemann sum $S (f, P, ξ) = \sum_{i} f (ξ_{i}) Δ_{i}$	none	tags exhibited by formulas
[The Darboux and Riemann definitions agree: a bounded $f$ on $[a, b]$ is Darboux integrable with integral $I$ if and only if for every real $ε > 0$ there is a real $δ > 0$ such that $	S(f,P,\xi) - I	< \varepsilon $f o r e v e r y t a g g e d p a r t i t i o n o f m e s h b e l o w$ \delta$](/physics/item/thm-darboux-equals-riemann)
A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion	none	current Heine-Cantor in $R$ : a continuous real function on a compact subset of $R$ is uniformly continuous proof uses a finite subcover
[A monotone function on $[a, b]$ is Riemann integrable: for the uniform partition into $N$ parts the upper minus lower sum telescopes to $	f(b) - f(a)	,(b-a)/\iota(N)$](/physics/item/thm-monotone-implies-integrable)
A bounded function on $[a, b]$ that is continuous except at finitely many points is Riemann integrable	none	choice-free Heine–Cantor supplier
A countable union of compact content-zero subsets of the line is null in ZF	none	least-coded rational finite covers
Lebesgue's criterion for Riemann integrability: a bounded $f$ on $[a, b]$ is Riemann integrable if and only if its set of discontinuities has measure zero	none	compact content-zero superlevel sets and the preceding lemma
A bounded function on $[a, b]$ whose set of discontinuities is at most countable is Riemann integrable	none	direct countable-set null cover
FALSE: every bounded function on $[a, b]$ is Riemann integrable	none	explicit Dirichlet witness
FALSE: a bounded function on $[a, b]$ is Riemann integrable exactly when its set of discontinuities is nowhere dense	none	explicit fat-Cantor witness
FALSE: a nonnegative Riemann integrable function on $[a, b]$ with $\int_{a}^{b} f = 0$ is identically zero	none	explicit Thomae witness
FALSE: a pointwise limit of a sequence of Riemann integrable functions on $[a, b]$ is Riemann integrable	none	explicit pointwise-limit witness

The three potentially misleading steps

Finite tags. In The Darboux and Riemann definitions agree: a bounded $f$ on $[a, b]$ is Darboux integrable with integral $I$ if and only if for every real $ε > 0$ there is a real $δ > 0$ such that $∣ S (f, P, ξ) - I ∣ < ε$ for every tagged partition of mesh below $δ$ , a fixed partition has finitely many subintervals. The required list of tags exists by finite induction (Every natural-number-indexed list of nonempty sets has a choice function on its family of values); it is not an instance of countable choice.

Partitions for each tolerance. Riemann's criterion: a bounded $f$ on $[a, b]$ is Darboux integrable if and only if for every real $ε > 0$ there is a partition $P$ with $U (f, P) - L (f, P) < ε$ asserts that for each positive tolerance a suitable partition exists. Its proof never assembles these partitions into a sequence. The later Lebesgue-criterion proof likewise uses one partition for each fixed tolerance and oscillation threshold; it does not choose all such partitions simultaneously.

The discontinuity set. In the forward half of Lebesgue's criterion for Riemann integrability: a bounded $f$ on $[a, b]$ is Riemann integrable if and only if its set of discontinuities has measure zero, each oscillation superlevel set is compact and has content zero. The set of discontinuities is their countable union. The choice-free lemma A countable union of compact content-zero subsets of the line is null in ZF encodes finite rational covers by naturals, takes the least valid code for each superlevel set, and combines those finite covers with a fixed pairing map. The converse implication uses the finite-cover and Cousin partition argument.

Scope of this accounting

The general union of an arbitrary sequence of null sets is a separate claim whose standard proof selects one cover for each set. This page uses the more specific compact content-zero lemma, so it makes no assertion here about the choice strength or independence of the general claim. The ledger records the premises of the proofs now on disk, rather than an optimality theorem for every possible proof.

Choice ledger for the Riemann integral: the page results are proved in ZF

The ledger, item by item

The three potentially misleading steps

Scope of this accounting

Depends on

Dependency tree · 2 levels

Sources