contra scott aaronson on pen position cardinality

pens can take on $|\mathcal{P}(\mathbb{N})|$ many positions in a continuous universe

I am generally a fan of Scott Aaronson’s work, so when an acquaintance of mine said:

I also found Scott Aaronson's argument for a discrete universe plausible: otherwise, you'd have to either accept that you could observe the truth of weird set theory things like the continuem hypothesis, or the physical theory would have to additionally prevent you from observing it. The latter is an additional constraint, is kinda complicated and specific, and thus should get a complexity penalty

I was curious to know more. It was not at all obvious to me why one would expect a continuous universe to permit the observation of the continuum hypothesis, but perhaps Scott Aaronson had a clever scheme of some sort? So my friend tracked down the actual quote they were remembering for me:

What I've tried to impress on you is that there are profound difficulties if we want to assume the world is continuous. Take a pen, for example: how many different positions can I put it on the surface of a table? $\aleph_1$? More than $\aleph_1$? Less than $\aleph_1$? We don't want the answers to "physics" questions to depend on the axioms of set theory! Ah, but you say my question is physically meaningless, since the pen's position could never actually be measured to infinite precision. Sure - but the point is you need a physical theory to tell you that.

— Scott Aaronson, Quantum Computing Since Democritus.

I’m not sure that this line of argument makes very much sense. If I think that a pen’s position can be modeled with a real-valued parameter (or a tuple of real-valued parameters), then I know precisely how many different positions I can put the pen on the surface of a table: $\mathfrak{c}$, the cardinality of the continuum, $|\mathcal{P}(\mathbb{N})|$, $2^{\aleph_0}$. To my eyes, that part is pretty settled.

The weird cardinality here isn’t $\mathfrak{c}$, it’s $\aleph_1$. Asking whether $\mathfrak{c}$ is more than $\aleph_1$¹¹ Note that to even establish that “more, equal, or less” exhausts the possibilities you need the axiom of choice — or rather, the axiom of choice is equivalent to the statement that for any two sets $A$ and $B$, either $A$ injects into $B$, $B$ into $A$, or both. Concretely, if $\mathbb{R}$ isn’t well-orderable, then it can have incomparable cardinality to $\aleph_1$ (which, when the axiom of choice doesn’t hold, is defined as the cardinality of the least uncountable ordinal, as there isn’t necessarily a least uncountable cardinality). is equivalent to asking whether or not there exists some set $X$ satisfying:

1. $\mathbb{N}$ injects into $X$.²² One could also compare cardinalities by saying that $\mathbb{N} \le^* X$ if $X$ surjects onto $\mathbb{N}$; this can differ from the injection-based notion in the absence of the axiom of choice. This notion is generally worse behaved — $\mathsf{ZF}$ can’t disprove the existence of a case where either $A$ or $B$ can surject onto the other, but there’s no bijection between them, so $\le^*$ isn’t even necessarily antisymmetric. On the other hand, the Shröder-Bernstein theorem establishes (without choice) that if either $A$ or $B$ can inject into the other, then there’s a bijection between them.

2. $X$ does not inject into $\mathbb{N}$.

3. $X$ injects into $\mathbb{R}$.

4. $\mathbb{R}$ does not inject into $X$.

It seems very natural to me that the existence of such a set would depend on the axioms of set theory, and I don’t see any reason to consider whether such a set exists to be a “physics” question.

I don’t have any confident opinion on whether the universe is ultimately discrete or not, nor how we could be certain we wouldn’t ever encounter weird set theory demons were it continuous. But I don’t think the “how many positions” example in particular is compelling, and wanted a less ephemeral home for the counterpoint than Discord. My overall opinion is mostly that it would be premature to rule out either the discrete or the continuous possibility without presenting a better argument than any I recall.