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I feel like it's a lot easier to think about in the My Grandfather's Axe form. The axe only has identity because you give it one and therefore it retains that identity through your possession of it.

If everyone who knew Theseus's name died, neither ship would ever be the Ship of Theseus again.

For the same reason, I drew this Snail of Theseus as the logo of my digital garden/lab notes: https://untested.sonnet.io/notes/the-snail-of-theseus/

I heard someone describe it as "The System of Theseus" that it's not an object but a set of things that lead to usefulness.