Insisting on finiteness us the wrong approach, even though finite approximations are all that can ever be practically used.
even though your computer is a finite automaton, it is not useful to think of it in these terms (?)
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it seems most useful for resolving things like the Halting Problem (not a problem if you restrict # of program steps)
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but I guess there's an analogy to physics where it's best to think of solids as infinite lattices unless edge conditions matter
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also you miss out on neat power series tricks in combinatorics, and other surprises like in enumerative geometry
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