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Cake day: June 20th, 2023

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  • I have two questions.

    1. After the revolution, how to you prevent the people that were influential during the revolution from seizing power for themselves, becoming the new bourgeoisie. This happened time and time again in practice.

    2. Even in the best case scenario, the decisions on what to produce become centralized in the hands of politicians. Political systems that we tried so far don’t work that well in practice. Is this really the best solution?



















  • As you already figured out the types are sets with a certain number of elements.

    Two types are isomorphic if you can write a function that converts all elements of the first one into the elements of the second one and a function which does the reverse. You can then use this as the equality.

    The types with the same number of elements are isomorphic, i.e True | False = Left | Right. For example, you can write a function that converts True to Left, False to Right, and a function that does the reverse.

    Therefore you essentially only need types 0, 1, 2, 3, …, where type 0 has 0 elements, type 1 has 1 element, etc. and all others are isomorphic to one of these.

    Let’s use (*) for the product and (+) for the sum, and letters for generic types. Then you can essentially manipulate types as natural numbers (the same laws hold, associativity, commutativity, identity elements, distributivity).

    For example:

    2 = 1 + 1 can be interpreted as Bool = True | False

    2 * 1 = 2 can be interpreted as (Bool, Unit) = Bool

    2 * x = x + x can be interpreted as (Bool, x) = This of x | That of x

    o(x) = x + 1 can be interpreted as Option x = Some of x | None

    l(x) = o(x * l(x)) = x * l(x) + 1 can be interpreted as List x = Option (x, List x)

    l(x) = x * l(x) + 1 = x * (x * l(x) + 1) + 1 = x * x * l(x) + x + 1 = x * x * (l(x) + 1) + x + 1 = x * x * l(x) + x * x + x + 1 so a list is either empty, has 1 element or 2 elements, … (if you keep substituting)

    For the expression problem, read this paper: doi:10.1007/BFb0019443