This note is a friendly technical check of Jeffrey Bub's There is No Quantum World (arXiv:2512.18400v2). I flag one unambiguous mathematical slip (a cardinality identity that implicitly assumes the Continuum Hypothesis) and then point out a few places where the discussion of infinite tensor products, "sectorization," and measurement updates would benefit from sharper wording. Nothing here is meant as a critique of Bub's interpretive goals; the aim is simply to separate what is mathematically forced from what depends on choices of algebra, representation, or philosophical stance. I end with a short remark on Philippe Grangier's reply (arXiv:2512.22965v1).
Bub's paper has two main threads [1]. First, it motivates a "neo-Bohrian" reading: the propositional structure is non-Boolean, contexts matter, and probabilities are taken to express indefiniteness rather than ignorance. Second, it discusses von Neumann infinite tensor products and the "sectorization" program (as used by Van Den Bossche & Grangier) as a proposed way to tame the measurement problem, and argues that it cannot deliver definite outcomes for any finite apparatus-hence an unavoidable classical "cut." My comments below are purely technical: what is plainly wrong, and what is correct but needs a qualifier.
In the discussion of an N -qubit measuring device, Bub uses the (standard) intuition that “more and more qubits” ultimately points toward an idealized infinite tensor product, and he links this to talk of “uncountable dimensionality” [1]. It is helpful to state the underlying set-theoretic facts explicitly.
For a finite register, the computational basis is labelled by the set of bitstrings of length N , {0, 1} N , so |{0, 1} N | = 2 N . In the usual infinite idealization, one instead meets the set of infinite bitstrings,
(Some authors compress this intuition into a heuristic notation like 2 N → 2 ℵ 0 as N → ∞, but the two cardinality statements above are the precise content.)
Problem. Bub then writes 2 ℵ 0 = ℵ 1 [1]. The equality 2 ℵ 0 = ℵ 1 is not a theorem of standard set theory (ZFC); it is equivalent to the Continuum Hypothesis (CH), which is independent of ZFC [9,10]. Without assuming CH, the correct statement is simply
and if one wants to identify 2 ℵ 0 with ℵ 1 one should say explicitly “assuming CH.”
This is the only unambiguous ZFC-level slip I noticed; everything else below is best read as a request for short qualifiers that prevent technical over-interpretation.
Several places in [1] would read more cleanly with short qualifiers. The points below are not meant as deep objections; they are mostly “precision patches.”
(i) Infinite tensor products are not just “limits of dimensions”
The paper can be read as if “the dimension becomes uncountable” is what drives the appearance of sectors. Even ignoring CH, that is not the best conceptual emphasis. What actually does the work in the operator-algebra picture is the choice of a physically relevant observable algebra (often quasi-local) and the choice of a representation/reference state used to build the infinite tensor product or GNS representation.
Sector/superselection structure is then tied to central decomposition and inequivalent representations, not to “uncountable dimension” by itself [8].
Bub writes that superpositions across sectors “represent mixtures,” and illustrates this with a block-diagonal mixture-like density operator [1]. As written, this is easy to misread.
A vector |Ψ⟩ defines a pure state on the full bounded-operator algebra B(H):
What becomes “mixture-like” is typically the restriction of that state to a physically relevant subalgebra (macroscopic/context/tail observables), where interference terms vanish or are operationally inaccessible [8,2]. In many algebraic discussions the physically relevant algebra is not B(H) at all, but a smaller quasi-local or macroscopic von Neumann algebra (often with a nontrivial center). In such settings, a vector state in a reducible representation need not be pure as a state on that smaller algebra. This is precisely why it helps to say explicitly which algebra is being used when one speaks of “mixtures” or “superselection.”
In the measurement postulates, the text moves from “the state jumps to one of the eigenstates” (selective, conditioned on an outcome) to writing a map that produces a mixture (non-selective, outcome ignored) [1]. This is a common shorthand, but here it matters because later discussion turns on whether one is conditioning on an actual outcome.
(iv) “Without decoherence” can be read too strongly Bub suggests the sector story yields classicality “without the usual appeal to decoherence” [1]. This may be intended as a presentational point, but it can be read as “without any dynamical/coarse-graining story.” In many concrete detector models, decoherence (environmental entanglement plus coarse-graining) is exactly what suppresses interference in the accessible observables and stabilizes macroscopic records. Sector language can repackage this structurally, but the wording would benefit from a narrower claim.
Grangier’s comment usefully pushes back on the slogan “infinity never comes” by reminding us that physics routinely relies on idealizations (thermodynamic/continuum limits) and by stressing that CSM treats “systems within contexts” as an explicit starting point rather than an “emergence proof” [2].
Two small technical notes about the appendix in [2]:
• The “tail observable commutes with locals in the sense that …” phrasing conflates an existence statement with a commutator statement. A standard tail-algebra definition avoids this ambiguity [8].
• The
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