A short technical comment on Bub's There is No Quantum World (arXiv:2512.18400v2) and a brief remark on related Grangier's reply (arXiv:2512.22965v1)

TL;DR

This note analyzes Bub's There is No Quantum World with a focus on mathematical precision in the infinite-tensor-product/sectorization discussion. It identifies a CH-dependent slip by noting that $2^{\aleph_0}=\aleph_1$ is CH-dependent and clarifies that sector structure arises from the choice of observable algebra and representation, not simply uncountable dimension. It clarifies measurement updates, the role of decoherence, and briefly comments on Grangier's reply, emphasizing careful language about mixtures and selective vs non-selective updates. The result is a practical patch-list to improve rigor in the interplay between algebraic quantum theories and interpretive claims.

Abstract

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).