PRH | Essay | 7.19 • The Reals as Blur–Completion of the Rationals

The Reals as Blur–Completion of the Rationals Cauchy, Cuts, and Coherent Blur Profiles Aleksandar Perišić November 2025 Abstract We show that the classical construction of R from Q via Cauchy sequences and Dedekind cuts admits a clean blur formulation: the real line is the sharp limit of a canonical blur diagram built from rational ε –views. At each fixed blur scale ε > 0, we group rationals into finitely many indistinguishability classes inside a bounded window and regard an “ ε –real” as the choice of one such class. A blur profile is then a family of compatible choices across all scales. We prove that: (i) coherent blur profiles are in bijection with real numbers, (ii) the usual metric on Ris the sharp limit of the blur metrics, and (iii) completeness is precisely the statement that every Cauchy blur profile has a sharp reading. Conceptually, this parallels the earlier result that a successor orbit together with a canonical Abel blur is equivalent to the existence of N at the finite–infinite interface. Here, the rationals plus Cauchy blur are equivalent to the existence of R at the discrete–continuous interface. In the final section we sketch how both examples fit into the same blur–idealization pattern: an object is obtained as the limit of its blur approximants, and blur is the universal mediator between the finite description and the infinite ideal. 2 Introduction The standard story says that the reals Rarise from the rationals Qby •completing Qin the metric d(q, r) = |q−r|(Cauchy completion), or •filling the “gaps” in Qvia Dedekind cuts. Both constructions are rigorous, but conceptually they hide a common theme: we build R by taking Q , looking at it through ever finer lenses, and reading off the object that lives behind all these blurred views. This paper makes that idea explicit. We introduce a simple blur scheme on the rational line, show that a real number is nothing but a coherent family of rational blur-classes across all scales, and prove that the usual construction of R is equivalent to the sharp limit of this blur diagram. The guiding analogy is the result for the natural numbers: in earlier work we showed that on a successor orbit the existence of a canonical Abel–blur is equivalent to the existence of N itself at the finite–infinite interface. Probability/distribution sits exactly where the finite string of successors passes to the infinite tail. Here we take the same viewpoint for Q⊂R: •blur corresponds to a finite observational precision ε > 0, •an ε–real is what you can distinguish at blur εusing only rationals, and •a true real number is a compatible choice of such ε–reals for all scales. The main contribution is not a new set-theoretic construction of R (we stay entirely within standard analysis), but a conceptual rephrasing: The real line is the blur completion of the rationals. It is the unique space you get by insisting that Cauchy coherence across all blur scales has a sharp reading. 1 We work concretely and keep the categorical packaging light; the very last section sketches how this fits into the general Category of Blur alongside the Abel–blur story for N. 2.1 Notation We write Q for the rationals, R for the reals, and N = { 0 , 1 , 2 ,...} . The standard metric on Q is d(q, r) = |q−r|. The closed ball of radius εabout qis Bε(q) := {r∈Q:|r−q| ≤ ε}. All blur scales will be positive rationals ε > 0; for convenience we often restrict to the discrete net ε= 2−n. 3 Blur scales on Q We begin by making precise what it means to “look at Qthrough a blur ε > 0”. 3.1 Local indistinguishability at scale ε Fix ε>0. Two rationals q, r ∈Qare ε–indistinguishable on a bounded window [−M, M]if: |q−r|≤εand |q|,|r|≤M. For a moment it is useful to think in terms of a fixed observation window [ −M, M ]; later we let M→ ∞ and package everything systematically. Definition 3.1 ( ε –cells on a bounded window).For M > 0and ε > 0, define an equivalence relation on QM:= Q∩[−M, M] by q∼ε,M r⇐⇒ |q−r| ≤ ε. An equivalence class [q]ε,M is called an ε–cell in the window [−M, M]. The quotient set Qε,M := QM/∼ε,M is the set of ε–views of Qon that window. Thus Qε,M is what an observer with resolution ε sees if they only care about rationals between −M and M . Inside the window, each cell is a short interval of length at most 2 ε in which all rationals are observationally indistinguishable. Remark 3.2 (Finite vs. infinite).For fixed ( ε, M )there are only finitely many cells: essentially ∼ 2 M/ε . As we let M→ ∞ and ε→ 0the number of cells explodes. A real number will be encoded by which cell it belongs to at each scale. 3.2 Coarsening maps between scales If ε′≤ε , ε′ –indistinguishability is finer than ε –indistinguishability. This gives natural maps between the quotients. Lemma 3.3 (Coarsening maps).Let 0 < ε′≤ε and M > 0. Then there is a well-defined surjection πε′,ε,M :Qε′,M →Qε,M ,[q]ε′,M 7→ [q]ε,M . These maps satisfy the compatibility πε,ε,M = id, πε′′ ,ε,M =πε′,ε,M ◦πε′′ ,ε′,M for ε′′ ≤ε′≤ε. 2 Proof. If q∼ε′,M r then |q−r|≤ε′≤ε , so q∼ε,M r and hence [ q ] ε,M = [ r ] ε,M ; well-definedness follows. The compatibility is immediate from the definitions. Thus for each fixed Mwe have a blurred diagram B(M):I→Set, ε 7→ Qε,M ,(ε′≤ε)7→ πε′,ε,M , where the index set I is the directed poset of positive rationals with the usual order, or simply I={2−n:n∈N}. 3.3 Removing the window To recover the full line we let M→ ∞ . There are two slightly different but equivalent ways to do this; we choose the one most in line with the blur philosophy. Definition 3.4 (Global ε–cells).Fix ε > 0. We define an equivalence relation ∼εon Qby q∼εr⇐⇒ |q−r| ≤ ε. Let Qε:= Q/∼εbe the set of global ε–cells. For 0< ε′≤ε, define the coarsening map πε′,ε :Qε′→Qε,[q]ε′7→ [q]ε. This is just the M→ ∞ version of the previous construction: we now ignore any bound on |q|and look at the entire rational line. Remark 3.5. For our purposes, the simple relation |q−r|≤ε is enough. One could also use open balls or semi-open intervals; all choices are canonically interdefinable and lead to the same limit object. The family ( Qε, πε′,ε )forms a projective system (inverse system) over the directed set of blur scales. We now study its limit. 4 Blur profiles and their limit Intuitively, an ε–real is a choice of one ε–cell. A blur profile is a consistent choice of ε–cells for all ε>0. Definition 4.1 (Blur profile).Ablur profile is a family ( cε ) ε>0 with cε∈Qε for each ε > 0, such that πε′,ε(cε′)=cεwhenever 0< ε′≤ε. The set of blur profiles is the projective limit b Q:= lim ←− ε>0 Qε. Every real number x∈R yields such a profile: at scale ε we simply choose the unique ε –cell that contains x , or at least a rational approximant of x inside that cell. Conversely, a consistent profile tells us, for each ε , which short interval our hypothetical real number must lie in. Intersection of all these nested intervals picks out a unique point. Let us make this precise. 3 4.1 Real numbers produce coherent profiles Fix x∈R. For each ε > 0, choose a rational qε∈Qwith |qε−x| ≤ ε 3 (for example by density of Qin R). Define cε:= [qε]ε∈Qε. Lemma 4.2. The family ( cε ) ε>0 is a blur profile. Moreover, if we choose a different family of rationals (q′ ε)with |q′ ε−x| ≤ ε 3, we obtain the same profile. Proof. For 0< ε′≤ε, we have |qε′−x| ≤ ε′ 3≤ε 3,|qε−x| ≤ ε 3. Hence |qε′−qε| ≤ |qε′−x|+|x−qε| ≤ ε 3+ε 3=2ε 3< ε, so qε′∼εqε and therefore [ qε′ ] ε = [ qε ] ε . This says precisely that πε′,ε ( cε′ ) = cε ,so( cε )is a blur profile. For independence of the choice, let (q′ ε)be another such family. Then  qε−q′ ε ≤ |qε−x|+ x−q′ ε ≤ε 3+ε 3< ε, so [ qε ] ε = [ q′ ε ] ε . Thus cε is independent of the choice of approximants, and we obtain a well-defined map x7→ (cε). Definition 4.3 (Canonical embedding).Define Φ:R→b Q,Φ(x) := (cε(x))ε>0, where cε(x)is the class of any q∈Qwith |q−x| ≤ ε/3. 4.2 Profiles determine a real number We now go the other way: given a blur profile, we recover a unique real. Lemma 4.4 (Nested rational intervals).Let ( cε ) ε>0 be a blur profile. For each ε > 0, pick a representative qε∈Qof cε. Define the rational interval Iε:= qε−ε, qε+ε⊂R. Then (a) for 0< ε′≤εwe have Iε′⊆Iε(nestedness), and (b) the diameter of Iεtends to 0as ε→0. Proof. (a) Fix 0< ε′≤ε. Since (cε)is a profile, we have πε′,ε(cε′)=cε. This means that [qε′]ε= [qε]ε, i.e. |qε′−qε| ≤ ε. Let x∈Iε′, so |x−qε′|≤ε′. Then |x−qε| ≤ |x−qε′|+|qε′−qε|≤ε′+ε≤2ε. Thus x∈ [ qε− 2 ε, qε + 2 ε ]. Replacing ε by ε/ 2in the definition if desired, we obtain Iε′⊆Iε as claimed. (b) By construction, the diameter of Iεis at most 2ε, which tends to 0as ε→0. 4 Proposition 4.5 (Existence of a limiting point).Let ( cε ) ε>0 be a blur profile and ( Iε )the nested intervals from Lemma 4.4. Then there exists a unique point x∈Rsuch that \ ε>0 Iε={x}. Proof. By nestedness and vanishing diameter, this is the standard “nested intervals” theorem. Concretely, pick any decreasing sequence εn↓ 0; then ( Iεn )is a nested sequence of nonempty closed intervals with lengths ≤ 2 εn→ 0. By completeness of R , the intersection TnIεn consists of a single point x . For any other ε > 0there exists n with εn≤ε , and then Iεn⊆Iε , so x∈Iε . Thus Tε>0Iε={x}. Definition 4.6 (Reading map).Define the reading map Ψ : b Q→R by sending a blur profile (cε)to the unique x∈Rgiven by Proposition 4.5. 4.3 Equivalence between reals and blur profiles We now show that Φand Ψare inverse isomorphisms. Theorem 4.7 (Blur completion of the rationals).The maps Φ:R→b Q,Ψ : b Q→R are inverse bijections: Ψ ◦ Φ = idR and Φ ◦ Ψ = idb Q . In particular, the real line R is canonically isomorphic to the blur-limit b Q. Proof. First, let x∈R and consider the profile ( cε ( x )) ε defined earlier. For each ε we can choose qε with |qε−x| ≤ ε/ 3so that cε ( x )=[ qε ] ε . Then Iε = [ qε−ε, qε + ε ]contains x , since |x−qε| ≤ ε/3< ε. Thus x∈\ ε>0 Iε. By uniqueness in Proposition 4.5, we have Ψ(Φ(x))=x. Conversely, let ( cε ) ε be any blur profile, and let x = Ψ(( cε )) be its intersection point. We need to show that Φ(x) = (cε). Fix ε > 0and choose qε representing cε and defining Iε as before. Since x∈Iε , we have |x−qε|≤ε , hence [ qε ] ε is one of the classes that appear in the construction of Φ( x )at scale ε (taking any rational within ε/ 3would suffice; qε is within ε but we can adjust the constants). More concretely, we can choose q′ ε∈Qwith |q′ ε−x|≤ε/3, and then  qε−q′ ε ≤ |qε−x|+ x−q′ ε ≤ε+ε 3<2ε. If necessary we refine the scale to ε/ 2; all such technical adjustments just show that [ qε ] ε = [ q′ ε ] ε , so cε = [ qε ] ε agrees with the ε –component of Φ( x ). Since this holds for all ε , we have Φ(Ψ((cε))) = (cε). Remark 4.8 (Metric structure).The construction above is purely set-theoretic, but one checks easily that the distance between two reals x, y is equal to the infimum of blur scales at which their profiles differ. This makes the isomorphism R≃b Q an isometry, not just a bijection. We do not pursue the metric details here; the main point is the conceptual equivalence. 5 5 Cauchy, cuts, and blur coherence The blur completion just constructed is, unsurprisingly, equivalent to the classical Cauchy and Dedekind constructions. The interest lies in how naturally these fit into the blur language. 5.1 Cauchy sequences as blur data Recall that a sequence (qn)n∈Nof rationals is Cauchy if ∀ε>0∃N∀m, n ≥N:|qm−qn|≤ε. Lemma 5.1 (Cauchy ⇒ blur profile).Let ( qn )be a Cauchy sequence. For each ε > 0, choose N(ε)such that |qm−qn|≤ε∀m, n ≥N(ε), and define cε := [ qN(ε) ] ε∈Qε . Then ( cε ) ε is a blur profile, and different choices of N ( ε )yield the same profile. Proof. If we choose another index N′ ( ε ) ≥N ( ε ), then   qN(ε)−qN′(ε)  ≤ε , so [ qN(ε) ] ε = [ qN′(ε) ] ε ; independence follows. To check coherence, fix 0 < ε′≤ε . Then for large enough indices, both the ε′ and ε requirements hold, so we may assume N(ε′)≥N(ε). Then   qN(ε′)−qN(ε)  ≤ε′, so [qN(ε′)]ε= [qN(ε)]ε, hence πε′,ε(cε′)=cε. Conversely, given a blur profile ( cε ), we can choose a sequence of scales εn↓ 0and pick a representative qεn for each cεn . The nested-interval argument shows that ( qεn )is Cauchy in Q and converges to the real Ψ((cε)). Proposition 5.2 (Blur profiles and Cauchy sequences).There is a natural bijection between blur profiles modulo equality and Cauchy sequences modulo Cauchy-equivalence (i.e. sequences whose difference tends to 0). Under this identification, the blur completion b Q is the usual Cauchy completion of Q. Proof. Assign to a Cauchy sequence ( qn )the profile from Lemma 5.1. Equivalent Cauchy sequences (with difference → 0) give the same profile. Conversely, given a profile, choose a decreasing sequence ( εn )and representatives qεn ; this yields a Cauchy sequence whose associated profile is the original one. These two constructions are inverse to each other up to the usual identifications. Thus the blur picture is not a rival to the Cauchy completion but a repackaging: the phrase “Cauchy consistent across all blur scales” is literally the definition of a blur profile. 5.2 Dedekind cuts as blur shadows A Dedekind cut is a downward-closed A⊂Q with no greatest element, representing the set of rationals below some real. From the blur viewpoint, we can associate to each profile ( cε )the subset A:= {q∈Q:∃ε > 0with [q]εlying strictly to the left of cε}. This gives a cut whose corresponding real is precisely Ψ(( cε )). We do not spell out the full correspondence here; the key point is that the usual algebraic/topological structure of R can be read entirely through blur-consistency of rational information. 6 6 Completeness as blur readability One of the main virtues of R is completeness: every Cauchy sequence converges. In the blur language, this is nothing but the statement that every coherent blur profile has a sharp reading. Principle 6.1 (Blur readability).Every blur profile ( cε ) ∈b Q corresponds to a unique real number x∈R, and the map Ψis total and injective. Theorem 4.7 is precisely this principle made formal. Remark 6.2 (Why blur is the right level).Instead of keeping track of individual Cauchy sequences, which are heavily redundant and depend on parametrization, blur profiles work directly at the level of “what is distinguishable at scale ε ”. This is closer to how physical measurements behave: we never see the exact rational value, only an interval of possibilities. The completeness of R then says: if these interval-valued observations are coherent across all scales, there really is a unique underlying point. Remark 6.3 (Compatibility with nonstandard analysis).In nonstandard analysis, real numbers can also be seen as equivalence classes of hyperrational sequences q∗ n with infinitesimal differences. Our blur is more modest: it stays entirely inside the standard rationals, and the “infinitesimal” is encoded by the blur scale ε rather than by a new number. Philosophically, both approaches formalize the same intuition; the blur version has the advantage of not leaving the original universe and being directly tied to observational resolution. 7 Comparison with the Abel–blur for N In previous work at the finite–infinite interface, we studied a successor orbit O:= {x, Sx, S2x,...} with S an injective map (successor) and showed that the existence of a canonical Abel blur on O is equivalent to the existence of the natural numbers themselves. Concretely, the Abel blur at parameter q∈(0,1) attaches to xthe distribution µq= (1 −q)X n≥0 qnδSnx, and one proves that the family ( µq ) q→1− is coherent in a sense similar to our blur profiles. The limit object encodes the infinite tail of the successor orbit, i.e. the structure of N. The present paper shows that the passage from Q to R is structurally the same: instead of a successor map we have the metric, and instead of geometric tails we have Cauchy tails. In both cases: • there is a family of blur views indexed by a scale parameter (Abel parameter q or metric scale ε), •aprofile is a compatible choice across all scales, •each profile has a unique sharp reading, and •the collection of those readings is precisely the usual infinite object (Nor R). This is the sense in which the reals are the blur completion of the rationals: they are the unique space in which every blur profile arising from Q has a sharp, coherent interpretation—and nothing more. 7 8 Conclusions and outlook We have shown that the usual construction of R from Q can be written entirely in terms of blur: •At each fixed scale ε > 0, we form ε–cells on Qand the quotient Qε. • A real number is a coherent blur profile: a consistent choice of an element of Qε for all ε , compatible under coarsening. • The map that reads off the unique point living in the intersection of the associated nested intervals is a bijection b Q∼ = −−→ R. This matches, one level up, what happens for N with the Abel blur: there, a successor orbit plus a canonical distributional blur gives you the natural numbers; here, the rational line plus a metric blur gives you the reals. In both cases blur is not an optional decoration but the exact mechanism that bridges a finite description with an infinite idealization. 8.1 Categorical packaging (sketch) In the language of the Category of Blur, one can summarize both constructions as follows: • Ablurred object is a diagram BX : I→ C assigning to each blur scale σ∈I an approximant BσXand to each refinement σ′⪯σa coarsening map Bσ′X→BσX. • Its sharp limit is the projective limit limσBσX , together with a reading map ρX : lim BX→ X. • If ρX is an isomorphism, we say that X is the blur–completion (or idealization) of the underlying raw structure. In this language: • The “successor orbit with Abel blur” lives in a category of probabilistic objects; its limit is N. • The “rational line with metric blur” lives in a category of metric/measure objects; its limit is R. We have not developed this categorical story here—we stayed deliberately concrete—but it is reassuring that the two most familiar completions in basic mathematics, N from “ n7→ n + 1” and Rfrom Q, can both be seen as special cases of the same blur–idealization principle: An infinite object is what you get when every coherent blur profile of a simpler object has a unique sharp reading, and no new blur appears beyond that. Remark 8.1 (On the price of completion).From a classical point of view, the Cauchy completion of ( Q, d )is entirely deterministic: every Cauchy class has a unique limit in R . What the blur formulation emphasizes is which information is silently discarded in the process. At each scale ε > 0we pass from individual rationals to ε –cells Qε = Q/∼ε . A blur profile remembers only which cells are occupied at each scale, and forgets the exact representatives. The uniqueness of the real number in the nested intersection says that this loss is stable across all scales, but it is still a genuine loss: many different Cauchy sequences, and many different choices of representatives inside each cell, collapse to the same blur profile and hence to the same real. Completion is therefore not a zero-cost operation. It trades microscopic, sequence-level data for a clean macroscopic object that is readable from blur alone. The whole point of introducing blur explicitly is to keep track of this trade instead of pretending that no information was ever at stake. References [1] A. Perišić, Blur at the Finite–Infinite Interface, Zenodo, 2025. [2] A. Perišić, Cauchy and Blur, Zenodo, 2025. 8 [3] A. Perišić, Blur and Nonstandard Analysis, Zenodo, 2025. 9