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Cake day: July 2nd, 2023

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  • But the vector space of (all) real functions is a completely different beast from the space of computable functions on finite-precision numbers. If you restrict the equality of these functions to their extension,

    defined as f = g iff forall x\in R: f(x)=g(x),

    then that vector space appears to be not only finite dimensional, but in fact finite. Otherwise you probably get a countably infinite dimensional vector space indexed by lambda terms (or whatever formalism you prefer.) But nothing like the space which contains vectors like

    F_{x_0}(x) := (1 if x = x_0; 0 otherwise)

    where x_0 is uncomputable.