They said "prove" but really it's about trust. Google has lost many peoples' trust and it's on Google to restore that trust.

Sure you can. Apple does not run its image classification on your images using its cloud servers. You can test this by stepping inside a microwave or other cage and seeing that image classification and search still works on the iPhone.

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On the other hand, what Apple does with your photos that you allow to be exfiltrated through iCloud... that's your own stupid fault.

We're not talking about mathematical or scientific levels of proof, but assurance and trust.

The usual methods for achieving this are government regulation and oversight (free of capture), and independent third-party audits (likewise).

The good news is that there seems to be ... some, slight ... progress in this direction.

You definitely can [0], but this one would probably be hard for google without significantly modifying the architecture of gmail in ways that would remove its revenue model. For example, they could open source a client that had audit-able end-to-end encryption, but then they couldn't optimize ad revenue by aggregating and mining large email datasets.

[0]: https://en.wikipedia.org/wiki/Proof_of_impossibility

> a proof demonstrating that a particular problem cannot be solved as described in the claim, or that a particular set of problems cannot be solved in general

did you even read the article you linked

Apologies, I thought you were saying that you can't prove a negative... that negative proofs (like the examples linked) do not exist.