Numerical choice, Riemann integration, and Reverse Mathematics

Abstract: Riemann integration remains a well-known part of mathematics for both historical and conceptual reasons. We study basic properties like boundedness of Riemann integrable functions and related classes in mathematical logic. On one hand, weak logical systems already establish that a Riemann integrable function on the unit interval is bounded or dominated by a continuous function. On the other hand, the following slight generalisation already implies a rather strong logical system, namely the `Big Five' system ATR$<em>{0}$ which accommodates transfinite recursion. $$ \text{For $f:\mathbb{R}\rightarrow \mathbb{R}$Riemann integrable on any interval$[-a, a]$for$a>0$, there is continuous$g:\mathbb{R}\rightarrow \mathbb{R}$with$f(x)\leq g(x)$for all$x\in \mathbb{R}$.} $$ As part of the \emph{Reverse Mathematics} program, we obtain equivalences for the centred statement and variations involving the axiom of numerical choice. A central result is that numerical choice for $Π</em>{1}<sup>{1}$-formulas is equivalent to ATR$_{0}$. We also obtain equivalences involving basic properties of metric spaces and establish connections to Kohlenbach's generalisations of weak König's lemma, Cousin's lemma, and the representation of open sets.

• The paper proves that, over suitable higher-order base systems, domination of everywhere Riemann-integrable functions on ℝ by continuous functions is equivalent to higher-order numerical bounding choice and corresponds in second-order arithmetic to Π¹₁-comprehension.
• It establishes a sharp contrast between domains: Riemann-integrable functions on [0,1] are bounded in weak systems, whereas extending domination and integration-modulus principles to all of ℝ requires principles strong enough to support transfinite recursion.
• The paper connects numerical choice with metric compactness, Cousin’s lemma, generalized König’s lemmas, open-set coding, and effective Baire-2 functions, while identifying unresolved questions about measure, open choice, and higher-order conservation.

This paper by Sanders and Taranovsky studies the logical strength of a deceptively simple statement about Riemann integration: that an everywhere Riemann integrable function $f:\mathbb{R}\to\mathbb{R}$ is dominated by some continuous $g:\mathbb{R}\to\mathbb{R}$. Working in Kohlenbach's higher-order Reverse Mathematics (RM) over the base theory $\mathsf{RCA}_0^\omega$, the authors show that weak systems prove boundedness of Riemann integrable functions on the unit interval, while the domination statement on all of $\mathbb{R}$ is equivalent to relatively strong principles — culminating in the central result that numerical choice for $\Pi^1_1$-formulas is equivalent to the "Big Five" system $\Pi^1_1\text{-}\mathsf{CA}_0$, which accommodates transfinite recursion.

The central contrast

The paper is organized around the juxtaposition of two facts. On one hand, over $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}+(\exists^2)$$-style bases, a Riemann integrable $f:[0,1]\to\mathbb{R}$ is bounded, and one direction of the Vitali–Lebesgue theorem holds: continuous almost everywhere plus bounded implies Riemann integrable. Patrick Uftring's proof (credited in a footnote) shows Riemann integrable functions are bounded already in weak systems. A sharper characterization is also established: over $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}$$ plus a measure-theoretic tree principle, Riemann integrability is equivalent to boundedness together with being $\varepsilon$-$g:\mathbb{R}\to\mathbb{R}$0-almost continuous almost everywhere.

On the other hand, the statement (RB) — for $g:\mathbb{R}\to\mathbb{R}$1 Riemann integrable on every interval $g:\mathbb{R}\to\mathbb{R}$2, there is continuous $g:\mathbb{R}\to\mathbb{R}$3 with $g:\mathbb{R}\to\mathbb{R}$4 on $g:\mathbb{R}\to\mathbb{R}$5 — turns out to be very strong. This asymmetry between the unit interval and the whole real line is the driving theme of the paper.

Numerical choice and $g:\mathbb{R}\to\mathbb{R}$6

The first main theorem establishes, over $g:\mathbb{R}\to\mathbb{R}$7, equivalences among:

• $g:\mathbb{R}\to\mathbb{R}$8 itself,
• the $g:\mathbb{R}\to\mathbb{R}$9-separation principle,
• bounding numerical choice ($\mathsf{RCA}_0^\omega$0): from $\mathsf{RCA}_0^\omega$1 with $\mathsf{RCA}_0^\omega$2, obtain $\mathsf{RCA}_0^\omega$3 with $\mathsf{RCA}_0^\omega$4,
• exact numerical choice ($\mathsf{RCA}_0^\omega$5): obtain $\mathsf{RCA}_0^\omega$6 with $\mathsf{RCA}_0^\omega$7,
• the statement that every $\mathsf{RCA}_0^\omega$8-function which is the lim sup of continuous functions and non-zero at most once per unit interval is dominated by a continuous function.

The proof of the reversal proceeds by using numerical choice to compute iterated Turing jumps along arbitrary recursive well-orderings, converting transfinite recursion into bounded recursion; the complexity analysis is claimed optimal via a lim inf argument. A companion conservation result shows that $\mathsf{RCA}_0^\omega$9 is conservative over $\mathbb{R}$0, adapting Hunter's term-model construction. However, both $\mathbb{R}$1 and $\mathbb{R}$2 are unprovable in strong systems such as $\mathbb{R}$3: they imply the enumeration of countable sets of reals ($\mathbb{R}$4) and the uncountability of $\mathbb{R}$5 ($\mathbb{R}$6), and $\mathbb{R}$7 even proves full $\mathbb{R}$8-comprehension. The authors conjecture that $\mathbb{R}$9 cannot prove that countable sets have measure zero, a statement intermediate between $\Pi^1_1$0 and $\Pi^1_1$1.

Equivalences with Riemann integration and related classes

Over $\Pi^1_1$2, the higher-order bounding choice principle $\Pi^1_1$3 is shown equivalent to each of:

• domination of sub-continuous functions by continuous ones on $\Pi^1_1$4;
• domination of everywhere Riemann integrable functions (statement RB);
• existence of moduli of Riemann integration, globally or restricted to $\Pi^1_1$5, including for continuous ae functions bounded on closed intervals;
• the same statements restricted to locally bounded, simply continuous, or Baire measurable functions.

The reversals use a Cantor-set coding device: a function supported on rational translates of the Cantor set is simultaneously sub-continuous, simply continuous, Baire measurable, measurable, and Riemann integrable everywhere with zero integral, so any of these regularity hypotheses suffices to recover the choice function. Notably, the restriction of these statements to the unit interval collapses to weak provability, confirming the sharpness of the global formulation. Several robustness observations accompany this: $\Pi^1_1$6 can often be replaced by induction axioms; the modulus-of-integration statements cannot be weakened to mere existence of the Riemann integral as a function (the minimal model containing all hyperarithmetic reals is a counterexample); and bounded variation functions admit trivial moduli, so the results cannot extend to that class.

Metric spaces

Analogous equivalences are obtained for basic properties of metric spaces $\Pi^1_1$7 with $\Pi^1_1$8: sequential compactness implying uniform boundedness across sequences of spaces, Heine–Borel finite subcoverings for countable unions, uniform bounds for sequences of continuous (even Lipschitz) functions, semi-uniform Dini convergence, and — perhaps most strikingly — the statement that a countable union of finite sets (cuf set) has a uniform size bound. The latter appears much weaker than enumerability yet still yields $\Pi^1_1$9. The constructions code arithmetical formulas into metrics on Baire space whose level sets are compact.

Pushing down to second-order RM

A methodological contribution is the "there and back again" technique: higher-order equivalences for $\Pi^1_1\text{-}\mathsf{CA}_0$0 can be pushed down to second-order equivalences for $\Pi^1_1\text{-}\mathsf{CA}_0$1, hence for $\Pi^1_1\text{-}\mathsf{CA}_0$2. Concretely, over $\Pi^1_1\text{-}\mathsf{CA}_0$3, domination of sub-continuous or Riemann integrable effectively Baire 2 functions by continuous functions is equivalent to $\Pi^1_1\text{-}\mathsf{CA}_0$4 and to $\Pi^1_1\text{-}\mathsf{CA}_0$5; the same holds for $\Pi^1_1\text{-}\mathsf{CA}_0$6-functions and for variants defined via effective representations (EB2, effectively measurable, effectively sico, effectively Baire measurable). This directly connects second- and higher-order RM, since the witness functions constructed are effectively Baire 2 whenever the parameter formula is arithmetical with only second-order parameters.

Connections: Cousin's lemma, König's lemma, and open sets

Three further threads tie numerical choice into the broader higher-order RM landscape.

Cousin's lemma. Over $\Pi^1_1\text{-}\mathsf{CA}_0$7, $\Pi^1_1\text{-}\mathsf{CA}_0$8 implies Cousin's lemma $\Pi^1_1\text{-}\mathsf{CA}_0$9 (finite subcovers of uncountable covers of $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}+(\exists^2)$$0), via the Lebesgue number lemma; only a very weak fragment of choice is needed.

Kohlenbach's generalised weak König's lemma. The paper proves $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}+(\exists^2)$$1 (Kohlenbach's version allowing arithmetical tree membership) over $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}+(\exists^2)$$2, improving Kohlenbach's earlier negative result: $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}+(\exists^2)$$3 is not provable in $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}+(\exists^2)$$4, and in fact implies $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}+(\exists^2)$$5. A notable "splitting" result decomposes exact numerical choice:

$$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}+(\exists^2)$$6

where $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}+(\exists^2)$$7 extends to finitely branching trees; splittings of this kind are described as rare in RM. Relatedly, $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}+(\exists^2)$$8 (the existential dual) is provable already in $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}+(\exists^2)$$9, and equivalences with Ramsey's theorem for $f:[0,1]\to\mathbb{R}$0-classes and $f:[0,1]\to\mathbb{R}$1-versions of ADS are noted, though the authors regard those as less natural since they mention $f:[0,1]\to\mathbb{R}$2-classes explicitly.

Open sets. Assuming the weak principle $f:[0,1]\to\mathbb{R}$3 (every open set is $f:[0,1]\to\mathbb{R}$4), $f:[0,1]\to\mathbb{R}$5 implies the coding principle $f:[0,1]\to\mathbb{R}$6 — every open set has an RM-code — one of the strongest natural third-order principles known, previously shown equivalent to the supremum principle for semi-continuous functions, Urysohn's lemma, and Tietze extension. Moreover $f:[0,1]\to\mathbb{R}$7 implies the Baire category theorem for $f:[0,1]\to\mathbb{R}$8. A hierarchy of fragments involving Lindelöf's lemma and the open-choice principle $f:[0,1]\to\mathbb{R}$9 is established, with item (c.2) — uniform separation bounds for disjoint sequences of closed sets — identified as the strong component, itself implying $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}$$0.

Limitations and open questions

The paper is explicit about several boundaries. The conservation result covers only $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}$$1; extensions to other principles (existence of $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}$$2 sets, generalized choice forms) are deferred to future work. Whether $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}$$3 proves that countable sets have measure zero remains open, as does whether $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}$$4 implies $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}$$5. The role of $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}$$6 in RM — its possible connection to the Baire category theorem or Tao's pigeonhole principle for measure — is posed as a question. Generalizations from metric to second-countable topological spaces, and equivalences for $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}$$7 at the level of $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}$$8 and beyond that avoid mentioning $$\mathsf{RCA}_0^\omega+\mathsf{QF}\text{-}\mathsf{AC}^{0,1}$$9-classes, are left unresolved. Foundational remarks note that third-order RM resists a small canonical list of systems because logical strength depends on two independent parameters: the complexity of guaranteed sets of reals, and the relative complexity of reals guaranteed to exist given such a set; combining both can produce "explosive" behavior such as $\varepsilon$0 proving $\varepsilon$1-comprehension.

Conclusion

The paper establishes that numerical choice for $\varepsilon$2-formulas is exactly $\varepsilon$3 in second-order arithmetic, and that its higher-order analogue is captured by natural analytic statements — above all, the domination of everywhere Riemann integrable functions by continuous ones — as well as by basic metric-space properties, Cousin's lemma, Kohlenbach's generalised König's lemma, and (modulo the weak $\varepsilon$4 assumption) the coding of open sets. The resulting picture places a textbook-level fact about Riemann integration at the strength of transfinite recursion, while the unit-interval versions remain provable in weak systems, delineating precisely where the logical cost of quantifying over the entire real line arises.