Beyond Convergence: Zeno’s Dichotomy and the Physical Constitution of Motion

Scott LeBrun

Independent Researcher

Abstract

On the standard mathematical treatment, convergence resolves Zeno’s dichotomy as a problem about completing an indefinitely subdivided continuous trajectory in finite time: no final halfway stage is required. This paper isolates a further physical question: does the arbitrary divisibility of a successful continuum representation imply corresponding divisibility of the physical history represented? I distinguish descriptive divisibility from physical divisibility and use representational surplus, in a restricted sense, for interpolating structure in an effective description that need not correspond one-to-one with fundamental physical stages. The Finite Constitution Hypothesis proposes that every physically realised causal interval contains only finitely many fundamental events, while elementary succession occurs when no event lies causally between two related events. A Zeno Reconstruction Test uses nested refinements to introduce ontological saturation: mathematical subdivision may continue after further refinement ceases to distinguish additional fundamental events. A toy reconstruction and a two-axis comparison separating event constitution from spacetime geometry show what the test can and cannot establish. The proposal is not presented as established physics, but as a framework for separating continuum representation from physical constitution.

Keywords: Zeno’s dichotomy; motion; physical divisibility; finite constitution; representational surplus; ontological saturation; causal structure; continuum representation

1. Zeno after convergence

Zeno’s dichotomy asks how motion can be completed if reaching a destination requires first reaching halfway, then halfway through the remainder, and so on without end. The same structure appears in an ordinary movement as small as moving a finger one inch. Before reaching the endpoint, the finger passes the halfway position; before that, a quarter; before that, an eighth; and the coordinate description can be subdivided indefinitely.

1/2 + 1/4 + 1/8 + 1/16 + ··· = 1 (1)

Convergence establishes that an indefinitely subdivided continuous trajectory can have finite total extent and duration. A continuous trajectory can therefore include every intermediate coordinate required by the continuum and nevertheless reach the endpoint in finite time. For uniform motion through distance D in duration T,

x(t) = D t / T, 0 ≤ t ≤ T (2)

gives x(T) = D. There is no final halfway point that must be completed before arrival. Modern analysis therefore removes the mathematical contradiction that the dichotomy appears to generate for a continuous trajectory (Huggett 2024).

The remaining question is not whether continuous motion is mathematically possible. It is whether every subdivision available in the mathematical representation corresponds to another physically realised stage of the movement. The distinction is easy to overlook because a continuum supplies a coordinate for every fraction of the path. But the existence of a coordinate in a model and the existence of an additional fundamental element in the physical history are different claims. The paper does not argue that calculus fails, that continuous motion is inconsistent, or that time is known to be discrete. It asks whether continuum divisibility must be fundamental and develops a conceptual framework for separating the divisibility of a representation from the constitution of the process represented.

1.1 Descriptive and physical divisibility

I will call a trajectory descriptively divisible when its mathematical representation admits further proper subdivision of any non-trivial represented interval. A represented interval is physically divisible when the fundamental ontology contains an additional physically distinct element corresponding to an intermediate distinction within that interval. For an event ontology ordered by causal precedence, this specializes as follows: between comparable fundamental events A and B, physical divisibility requires a distinct event C such that A → C → B. This definition is deliberately ontological: a theory represents the history, but the mere availability of additional mathematical coordinates in that representation is not itself the criterion for additional physical constitution.

The central distinction is therefore:

Descriptive divisibility does not, by itself, entail physical divisibility.

Zeno’s sequence unquestionably supplies indefinitely many descriptive subdivisions. The physical question is whether each subdivision must correspond to another fundamental stage in the history, rather than merely to additional resolution within a successful representation.

This is not the claim that earlier philosophy ignored the distinction. Aristotle already treats continuous motion as potentially divisible without requiring an actual sequence of stopped half-runs, and atomistic responses to Zeno have a long history (Huggett 2024). The present proposal asks a narrower question: whether a successful continuum representation may contain more interpolating structure than the physical history it represents. The contribution is therefore not a new discrete-spacetime theory, but a framework for separating mathematical divisibility, physical constitution, and reconstruction.

2. Where the proposal sits among existing accounts

Several established positions already reject the picture of motion as an infinite checklist of completed tasks. On an at-at account, motion consists in occupying different locations at different times in the appropriate order; no additional instantaneous “moving” property is required. Arntzenius (2000) compares this with impetus and no-instants alternatives and shows that each carries distinct commitments. An at-at analysis, however, does not by itself decide whether every time in a continuum is a distinct member of the fundamental physical ontology. To avoid building that stronger commitment into the label, the comparison below uses “continuous-state account” for a theory in which the physical history itself is defined at continuum-many times or spacetime locations.

A second nearby position is causal finitism. Pruss (2018) argues that causal finitism can motivate, but does not require, discrete time; continuous time may coexist with discrete causal structure. That distinction is important here. Finite constitution is not identified with a universal chronon or a regular time lattice. The proposal concerns whether fundamental causal histories are locally finite and how smooth motion is represented when they are.

A still closer precedent is Okada’s (2020) discussion of Hilbert’s finitist distinction between indefinitely divisible mathematical continua and finite descriptions of real processes. Okada explicitly contrasts the standard analytic resolution of Zeno with a no-infinite-divisibility principle and finite-state transition modelling. This substantially narrows the novelty available here: the present paper does not claim to originate the idea that real processes might lack infinite divisibility. Its proposed contribution is the descriptive–physical divisibility distinction, the restricted notion of representational surplus, and a refinement-based reconstruction test.

Causal-set theory already takes local finiteness of causal order as a fundamental structural principle (Bombelli et al. 1987; Surya 2019). Likewise, work on emergent spacetime has emphasized the general explanatory burden faced by theories whose fundamental ontology differs from observed spacetime: the empirical spatiotemporal world must be recovered from the underlying structure (Huggett and Wüthrich 2013). Nothing in this paper claims local finiteness or continuum recovery as new physical ideas. The narrower proposal is to use Zeno-style indefinite refinement to ask exactly when additional structure in a continuum description corresponds to additional physical constitution and when it belongs to reconstruction.

3. The Finite Constitution Hypothesis

Let A → B denote that fundamental physical event A causally precedes fundamental physical event B. For any comparable pair define the open causal interval

I(A,B) = { C | A → C → B }. (3)

The Finite Constitution Hypothesis is:

For every physically realised pair A → B, |I(A,B)| < ∞. An elementary succession A ⇒ B occurs when A → B and I(A,B) = ∅.

The finiteness condition mirrors local finiteness in causal-set theory, where an irreducible causal relation with no intervening element is called a link (Surya 2019). The formal condition is therefore prior art. The present hypothesis interprets such local finiteness as one possible constitution of physical motion and asks how a smooth trajectory could represent a history with this structure.

A particular causal chain may therefore contain irreducible physical successions even though its coordinate representation admits no last mathematical subdivision. An irreducible succession A ⇒ B is one for which no fundamental physical event lies causally between A and B. In this restricted sense one may speak of a “last physical subdivision” along that particular transition, but not of a universal smallest duration, length, or scale. The coordinate interval used to represent the succession may still be divided indefinitely.

3.1 Events, states, and elementary change

The counting condition in the hypothesis is a condition on fundamental event occurrences. A developed theory may associate dynamical state information with individual events, causal relations, finite collections of events, or more global structures. “State” should therefore not be treated as a second kind of countable event unless the underlying theory makes it one. This distinction matters because a physical state may contain position, momentum, internal variables, quantum amplitudes, correlations, or quantities with no direct classical analogue.

An elementary transition should not be pictured as two frozen photographs with an unexplained gap between them. The elementary change is the lawful continuation from one admissible physical condition to a successor. It is not a rapid journey through unlisted intermediate positions. That picture would simply insert a smaller continuous process inside the transition and reproduce the original question at a finer scale.

This is a structural schema for an elementary-change theory, not a proposed microscopic law determining actual successors. A physical theory must still specify which continuations are allowed, how interactions alter them, and—where several outcomes are possible—how probabilities or amplitudes are assigned. Quantum-causal-history work provides an instructive precedent for treating causal structure, state information, and evolution rules as distinct ingredients (Hawkins, Markopoulou, and Sahlmann 2003).

3.2 Order is not duration

The hypothesis deliberately does not identify elementary succession with a universal tick. A ⇒ B provides causal order; it does not by itself specify a duration in seconds. “No intermediate event” therefore does not mean zero elapsed time or infinite speed. A developed theory would have to explain how effective durations arise from comparisons among physical histories and physical clocks.

This restraint matters because finite constitution and uniform temporal spacing are different hypotheses. A locally finite history need not possess a universal smallest interval. The paper is about finite physical succession, not a cosmic metronome.

4. Representational surplus and effective motion

The continuum trajectory does more than summarize the order of a few distinguished events. It supplies indefinitely many coordinate positions and times, differentiable structure, and exact interpolating values. Some of that structure may be indispensable for prediction while still failing to correspond one-for-one with fundamental physical stages.

I use the phrase representational surplus in a deliberately restricted sense: mathematical structure present in an effective representation that is not intended to correspond one-to-one with fundamental physical events or relations. This usage should not be conflated with established debates over “surplus structure” in gauge theories, where the term concerns representational redundancy of a different kind (Dougherty 2025).

If H denotes a finite fundamental history, γ an effective continuous trajectory, and Θ the dynamical and reconstruction assumptions used to connect them, the relationship may be written schematically as

R_Θ(H) ≈ γ. (4)

Equation (4) is schematic, not a derived physical law. A finite H will generally not determine a unique smooth γ without additional dynamical, boundary, geometric, and coarse-graining information collected here under Θ. That non-uniqueness is important: representational surplus is not produced by a finite list of events alone, but by the richer continuum representation used to reconstruct and predict their effective behaviour.

4.1 A toy reconstruction

Consider, purely for illustration, a five-event fundamental history

H = {e₀, e₁, e₂, e₃, e₄}, e₀ → e₁ → e₂ → e₃ → e₄. (5)

Let Θ associate effective clock readings tᵢ = iT/4 and effective positions xᵢ = iD/4 with these events. One admissible reconstruction is the linear trajectory γ₀(t) = Dt/T. But the same five assignments are also fit by γₑ(t) = Dt/T + ε sin(4πt/T), provided |ε| < D/(4π). The sine term vanishes at t = 0, T/4, T/2, 3T/4, and T, while the stated bound keeps γₑ increasing. Thus the finite history and its assigned effective values do not select a unique smooth interpolation: at t = T/8, for example, γ₀(T/8) = D/8 whereas γₑ(T/8) = D/8 + ε. The interpolated value therefore depends on the reconstruction assumptions in Θ and does not by itself establish a sixth fundamental event. A continuous-state theory could instead treat a physical state at every t as fundamental.

Applied to the moving finger, this does not require a particle to teleport across a pre-existing continuous spatial gap. The Finite Constitution Hypothesis is neutral on whether spatial continuity is fundamental. It claims only that an interpolated spatial coordinate need not correspond to an additional fundamental event. A complete theory would still have to explain how its event structure relates to observed spatial geometry and local propagation.

5. The Zeno Reconstruction Test

The distinction above can be used as a diagnostic rather than merely as a defence of one ontology. “Test” is used here in the diagnostic rather than experimental sense: the first six questions expose a theory’s constitutive and reconstructive commitments, while empirical discrimination enters only in the final question. For any theory that claims to recover ordinary motion from more fundamental structure, ask:

1. What counts as a fundamental physical event, state, or other ontological element?

2. What counts as a physical transition or relation among those elements?

3. What reconstruction map and dynamical assumptions connect the fundamental structure to an effective continuous trajectory?

4. Under a nested sequence of finer subdivisions of the effective trajectory, do new subdivisions continue to resolve new fundamental physical distinctions, or does the mapping eventually saturate?

5. If saturation occurs, what exactly saturates—event constitution, spatial geometry, temporal structure, or some combination of them?

6. How does the theory reconstruct the smooth motion, spatial geometry, locality, and clock behaviour observed at larger scales?

7. What observation could distinguish its proposed constitution of motion from an empirically equivalent or appropriately constrained alternative?

The test is intended to make rival ontologies comparable rather than to privilege finite constitution in advance. Its specifically Zeno-like feature is question 4: repeated refinement is continued after the effective description is already dynamically adequate, and the theory is asked whether each new descriptive distinction must correspond to a new physical one.

5.1 Nested refinements and ontological saturation

Let P₁, P₂, … be a nested sequence of finite ordered sets of partition points on the parameter interval [0,T] of γ, with Pₙ properly contained in Pₙ₊₁. Let C(Pₙ) denote the set of cells induced by Pₙ, and let κ_Θ:H → [0,T] map each fundamental event represented in the effective history to its effective trajectory parameter. For each Pₙ, define cₙ:[0,T] → C(Pₙ) so that cₙ(t) is the cell containing t, using a fixed convention for boundary points. Define D_Θ(Pₙ,H) as the set of unordered pairs {eᵢ,eⱼ} of distinct mapped events for which cₙ(κ_Θ(eᵢ)) ≠ cₙ(κ_Θ(eⱼ)). Thus D_Θ(Pₙ,H) records which fundamental event distinctions are resolved at that granularity. Because Pₙ₊₁ refines Pₙ, D_Θ(Pₙ,H) ⊆ D_Θ(Pₙ₊₁,H). Call the refinement sequence separating for H if every pair of events represented at distinct effective parameters is eventually resolved.

If H contains only finitely many mapped fundamental events, there are only finitely many event-pair distinctions available to be resolved. Any separating nested refinement sequence must therefore eventually exhaust them. Thus there is some N such that further refinement adds descriptive structure without adding another distinction among members of H:

∃N ∀n ≥ N: D_Θ(Pₙ,H) = D_Θ(P_N,H). (6)

I call this ontological saturation with respect to the event history. The mathematical result is elementary and no mathematical novelty is claimed for it; its role is diagnostic, making explicit how finite physical constitution differs from indefinite descriptive refinement. Although the particular refinement level N at which saturation occurs depends on the reconstruction map and chosen refinement sequence, the existence of eventual event-level saturation for a finite mapped history does not, provided the refinement sequence is separating. The partitions themselves do not stop refining, and equation (6) does not define a smallest coordinate interval. It marks a difference between indefinite descriptive refinement and a finite stock of mapped fundamental event distinctions.

A continuous-state account, defined here as one in which a fundamental state-occurrence is associated with each time or spacetime location along the history, need not exhibit finite saturation: finer partitions can continue to resolve further members of the fundamental history. By contrast, any finitely constituted history with finitely many mapped fundamental events saturates at the event level regardless of whether its underlying geometry is continuous or non-continuum/emergent. Continuous spacetime with locally finite fundamental events is therefore one possible realization of finite constitution, not a separate rival to it. Ontological saturation diagnoses the constitution of the relevant physical history; it does not by itself determine the constitution of spacetime geometry.

5.2 Applying the test to the toy reconstruction

Return to the five-event history in equation (5). Let P₁ mark {0, T/2, T}; let P₂ refine this to {0, T/4, T/2, 3T/4, T}; and let P₃ add the eighth-time coordinates between them. Under the illustrative mapping Θ, P₂ is already fine enough to distinguish every member of H. P₃ contains additional coordinates and intervals on γ, but it distinguishes no additional event in H. Further bisections can continue without altering that fact. The finite account has reached event-level ontological saturation.

If, by contrast, the ontology assigns a fundamental physical state to every t in [0,T], the coordinates added at P₃ correspond to additional members of the physical history. The trajectory γ(t) may be mathematically identical in the two descriptions. What differs is the ontological interpretation of its refinement. This is the discrimination the Reconstruction Test is designed to expose; it is conceptual rather than, by itself, empirical.

5.3 Comparative application

| Physical constitution / geometry | Fundamentally continuous geometry | Non-continuum or emergent geometry |

| — | — | — |

| Continuum-many physical stages | Continuous-state realization: the physical history includes continuum-many fundamental states or stages. Finer refinement may continue to resolve additional fundamental distinctions. | Logically possible in principle: non-continuum fundamental geometry need not by itself imply a locally finite event history. Whether refinement saturates depends on the stage ontology. This case is not developed here. |

| Locally finite physical stages | A continuous-geometry realization of the Finite Constitution Hypothesis: event-level refinement saturates while geometric subdivision may continue indefinitely. | A causal-set-like realization of the Finite Constitution Hypothesis: event-level refinement saturates, while continuum geometry, if recovered, is effective or emergent. |

The matrix makes the two questions independent. The Finite Constitution Hypothesis occupies the locally finite row: it constrains the number of fundamental event occurrences in a causally delimited history but does not choose between fundamentally continuous and non-continuum/emergent geometry. A continuous-spacetime sparse-event model is therefore a continuous-geometry realization of finite constitution, not an alternative to it. Conversely, event-level saturation does not by itself imply a spacetime lattice, minimum length, or minimum duration.

6. Causal sets, relativity, and evidence

Causal-set theory supplies an existing mathematical setting in which local finiteness and continuum recovery can be studied. This matters because fundamental discreteness need not mean a regular lattice or a preferred sequence of simultaneous frames. Theorem-level results for Poisson sprinklings into Minkowski spacetime show that the sprinkling does not select a preferred direction in the way a regular lattice would (Bombelli, Henson, and Sorkin 2009). This does not establish Lorentz invariance for every discrete theory; it shows that discreteness alone need not introduce an observable preferred frame.

Evidence presents a separate difficulty. A continuously evolving system can be sampled at discrete times and produce the same list of records. Likewise, discrete quantum dynamics may admit continuous generators reproducing the same updates at comparison times, although the locality properties of those generators require separate analysis (Zimborás et al. 2022). Discrete-looking observations therefore do not, by themselves, identify finite physical constitution.

The converse caution is equally important. The empirical success of continuum mechanics, field theory, or relativistic models establishes the adequacy of continuum mathematics at the scales tested. It does not by itself establish that every interpolating coordinate or state variable in that successful representation has a one-to-one counterpart in the fundamental ontology. Evidence for an effective continuum description and evidence for continuum fundamentality are not automatically the same.

A successful finite-constitution theory would first have to reproduce the tested successes of ordinary continuous physics. It would then need a further consequence that follows from its transition structure and is not equally explained by an appropriately constrained continuous alternative. Christodoulou, Di Biagio, and Martin-Dussaud (2022), for example, derive an interferometric signature from a particular hypothesis of proper-time granularity. Their proposal does not test finite constitution in general, but it illustrates the right methodology: specify the underlying structure, derive an observable consequence, and let experiment constrain the model.

7. Objections and limits

7.1 Continuous motion already resolves Zeno

Yes. The hypothesis is not required to make motion mathematically possible. Convergence already establishes that an indefinitely subdivided continuous trajectory can have finite duration. The remaining claim concerns physical constitution: whether the continuum used by the successful description is fundamental or whether some of its indefinitely divisible structure is representational surplus.

7.2 Does the at-at theory already do the work?

An at-at account already rejects the idea that motion requires an additional act performed at every point: motion is represented by appropriate occupation of places at times. The Finite Constitution Hypothesis agrees that Zeno should not be read as an infinite checklist of tasks. But the at-at analysis alone is neutral on the further ontological question addressed here: whether the fundamental history includes continuum-many physical stages. For that reason the Reconstruction Test compares finite constitution with an explicitly continuous-state ontology rather than treating every at-at account as committed to one.

7.3 Does ontological saturation imply discrete spacetime?

No. Saturation in equation (6) is indexed to the fundamental event distinctions represented by H. A theory may posit continuous spacetime or temporal geometry while maintaining that only a locally finite set of event occurrences is fundamental. In that case it satisfies finite constitution at the event level while retaining continuous geometry. The comparison in section 5.3 is included precisely to prevent an inference from finite event constitution to a universal spacetime lattice, minimum length, or minimum duration.

7.4 Is finite constitution simply stipulated?

At this stage, yes: it is a hypothesis. Defining elementary succession or ontological saturation supplies no evidential advantage by itself. Their value is to state a physically intelligible alternative precisely enough to compare with continuous theories. Evidence would have to come from a developed dynamics and its consequences.

7.5 What about quantum correlations?

Quantum theory constrains how any elementary-change account may understand state information without providing evidence for finite succession. Entangled systems need not be representable as collections of independent pure states, so a transition law may have to act on a relevant joint physical state whose correlations are part of the dynamics (Horodecki et al. 2009). Finite constitution therefore should not be pictured as independently updated classical point-particle snapshots. This constraint does not imply temporal discreteness or a minimum time.

7.6 If a measurement occurs at an interpolated coordinate, does that make it a physical event?

If a measurement or other interaction is physically realised at an effective coordinate such as T/8, then the relevant interaction must be included in the realised history H′ according to whatever event ontology the theory adopts. Finite constitution is indexed to realised physical histories, not to every event that could counterfactually occur. The availability of the coordinate T/8 in the continuum representation therefore does not imply that an event already exists there in the original history H. An intervention can change the realised history and its event set without showing that every available continuum coordinate corresponds to a fundamental event in every history.

7.7 Is this causal-set theory renamed?

No novelty is claimed for local finiteness, causal links, or continuum recovery as research problems. The proposed novelty is the conceptual framework connecting Zeno to descriptive and physical divisibility, representational surplus, and the saturation-based Reconstruction Test. Causal-set theory is one natural realization to which that framework can be applied; the two-axis comparison in section 5.3 also shows that finite constitution is not restricted to theories with fundamentally non-continuous spacetime.

8. Conclusion

Convergence resolves the mathematical problem of completing an indefinitely subdivided continuous trajectory in finite time. The further question is whether the same indefinite divisibility belongs to the physical history itself.

This paper has separated two claims that the mathematical solution does not decide. Descriptive divisibility concerns the structure available in a continuum representation. Physical divisibility concerns whether further subdivision corresponds to additional fundamental stages. The Finite Constitution Hypothesis proposes that a causally delimited history segment may contain only finitely many such events, with elementary change understood as lawful physical succession rather than as a hidden smaller journey. Representational surplus names the possibility that an effective continuum contains useful interpolating structure without a one-to-one counterpart in the fundamental history.

The Zeno Reconstruction Test sharpens that distinction by following nested refinements of the effective trajectory. A finite event history exhibits ontological saturation once further descriptive refinement ceases to distinguish additional fundamental events. A continuous-state ontology need not saturate in this way. Finite constitution, however, is a thesis about event constitution rather than a complete theory of spacetime geometry: it is compatible both with fundamentally continuous geometry and with theories in which continuum geometry is recovered or emergent. The test therefore separates event constitution from spacetime continuity instead of treating them as the same question.

The proposal is not established by Zeno and is not new simply because it invokes local finiteness. Its contribution is a framework for asking which parts of a theory’s smooth description correspond to physical constitution and which arise in reconstruction. Whether nature has a finitely constituted event history remains a question for dynamics and evidence, not for convergence alone.

Acknowledgements and AI use

OpenAI ChatGPT (GPT-5.6 Sol, September 2026) was used during manuscript preparation through iterative prompts requesting critique, structural revision, drafting assistance, and language refinement. The author reviewed and approved all AI-assisted text and accepts full responsibility for the argument, accuracy, citations, and final manuscript.

References

Arntzenius, F. 2000. “Are There Really Instantaneous Velocities?” The Monist 83 (2): 187–208. doi:10.5840/monist20008328.

Bombelli, L., J. Lee, D. Meyer, and R. D. Sorkin. 1987. “Space-Time as a Causal Set.” Physical Review Letters 59: 521–524. doi:10.1103/PhysRevLett.59.521.

Bombelli, L., J. Henson, and R. D. Sorkin. 2009. “Discreteness without Symmetry Breaking: A Theorem.” Modern Physics Letters A 24: 2579–2587. doi:10.1142/S0217732309031958.

Christodoulou, M., A. Di Biagio, and P. Martin-Dussaud. 2022. “An Experiment to Test the Discreteness of Time.” Quantum 6: 826. doi:10.22331/q-2022-10-06-826.

Dougherty, J. 2025. “Gauge Theories in Physics.” The Stanford Encyclopedia of Philosophy, Fall 2025 edition, edited by E. N. Zalta and U. Nodelman. https://plato.stanford.edu/archives/fall2025/entries/gauge-theories/.

Hawkins, E., F. Markopoulou, and H. Sahlmann. 2003. “Evolution in Quantum Causal Histories.” Classical and Quantum Gravity 20: 3839–3854. doi:10.1088/0264-9381/20/16/320.

Horodecki, R., P. Horodecki, M. Horodecki, and K. Horodecki. 2009. “Quantum Entanglement.” Reviews of Modern Physics 81: 865–942. doi:10.1103/RevModPhys.81.865.

Huggett, N. 2024. “Zeno’s Paradoxes.” Stanford Encyclopedia of Philosophy. Substantive revision 6 March 2024. https://plato.stanford.edu/entries/paradox-zeno/.

Huggett, N., and C. Wüthrich. 2013. “Emergent Spacetime and Empirical (In)Coherence.” Studies in History and Philosophy of Modern Physics 44 (3): 276–285. doi:10.1016/j.shpsb.2012.11.003.

Okada, M. 2020. “A Small Remark on Hilbert’s Finitist View of Divisibility and Kanovich-Okada-Scedrov’s Logical Analysis of Real-Time Systems.” In Logic, Language, and Security, Lecture Notes in Computer Science 12300, 39–47. doi:10.1007/978-3-030-62077-6_3.

Pruss, A. R. 2018. Infinity, Causation, and Paradox. Oxford: Oxford University Press. See especially Chapter 8, “Discrete Time and Space.” doi:10.1093/oso/9780198810339.001.0001.

Surya, S. 2019. “The Causal Set Approach to Quantum Gravity.” Living Reviews in Relativity 22: 5. doi:10.1007/s41114-019-0023-1.

Zimborás, Z., T. Farrelly, S. Farkas, and L. Masanes. 2022. “Does Causal Dynamics Imply Local Interactions?” Quantum 6: 748. doi:10.22331/q-2022-06-29-748.