Is the distinction between comparison and counting as accumulation? Like, comparing 0 and 1 is a direct comparison between the two things and does not require any grasp of higher numbers? It seems like sets and numbers encoded in unary form here are equivalent, for instance like Church encoding of numbers in the lambda calculus. Certainly it's computationally equivalent; so the only thing I can think of is that counting requires history, a stack, an accumulator, something; whereas that approach of equivalence allows you to throw out information at each step, the final state is independent of the previous states. This is easy to demonstrate in lambda calculus, but I'm not sure there's a meaningful distinction over counting since, like, the set itself represents all of the information that would be encoded in an accumulator, or on a stack, or whatever. It's the same information in different forms.
(I am wildly out of my depth on these things, but the linkage between numbers in Church encoding and sets is so clear that the faultiness seems evident to me at least in that space. Like, this is no different to a trivial equivalence function which does no math, and a naive one which checks if either is zero, and then decrements each and recurses.)
![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
no subject
(I am wildly out of my depth on these things, but the linkage between numbers in Church encoding and sets is so clear that the faultiness seems evident to me at least in that space. Like, this is no different to a trivial equivalence function which does no math, and a naive one which checks if either is zero, and then decrements each and recurses.)