# Finiteness assumption in explorable media

**Video games work with finite and discrete objects**. Explorable and interactive media (especially explorable explanations) seem to bake in an assumption of finiteness, which makes it challenging to interact with the infinite and arbitrary objects that appear in mathematics. An extremely common move in a math proof is to say "Let x be an arbitrary element of X" where X is some class of objects of interest, typically infinite. There is no obvious way to allow "moves" like this in a game.

Examples:

- the explorables in https://explorabl.es/math/
- The Witness builds up complex puzzles starting from simple ones. But each puzzle is finite (finite board size, finite state space)

The kind of explorable medium that I'm interested in would somehow encode the proof-writing/proof-generating process so the user can do real math. But how do you prove things about e.g. an arbitrary compact metric space? How would you "explore" something that requires a proof by contradiction (which would require representing impossible situations)?

The tricky thing about math is that you regularly have to deal with infinity, i.e. you need to figure out how to draw objects (such as sequences) which are not finitely representable. You could retreat to finite objects in fields like combinatorics i think, but it feels more satisfying to come up with a general solution that would work in any field.