Finiteness assumption in explorable media

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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.

Examples:

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)?

See also