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> one to populate it and one to query it I'm not a math expert, but "populating" a set strikes me as an imperative programming mindset. In math, I understand

> one to populate it and one to query it I'm not a math expert, but "populating" a set strikes me as an imperative programming mindset. In math, I understand a set more like a function returning a boolean that indicates membership. If the contents of the set change, then it's just a new set.

lisper 4y ago
Yeah, this. A set does not have "two interfaces", it only has one: a membership function that takes an object and returns a boolean indicating if the object is a member of the set or not. Note that this function need not be computable, only well-defined.
- cgio 4y ago
  Is the membership function not equivalent to what I describe above on the query side? My difference is that I am making explicit that the set needs to be populated. The uttering of a set of “indescribable “ objects does not a set give you. That membership function works both inwards and outwards.
  - recursive 4y ago
    This concept of "population" doesn't really seem to be consistent with my understanding of a set. Set A is the set of all widgets that have a frozz. The existence of Set A doesn't imply that there exists a widget that has a frozz. Perhaps the existence of such a widget is an open question. But the set is perfectly well defined.
    - lisper 4y ago
      Right. The set of all integers which are both even and odd is a well-defined set. It just happens to be empty. The set of all counter-examples to the Goldbach conjecture is a well-defined set. We don't know if it's empty or not.
    - cgio 4y ago
      If I don’t know what frozz is, is the set well defined? Maybe in form, but not in essences. Lisper’s example in sibling comment is indeed well formed, but I know that if I get a counter-example to the conjecture it will pass the population interface and become part of the set. Therefore, I don’t see how we differ comparing simple membership with population and query. My feeling is we disagree over an ex nihilo vs procedural emergence of sets.

lisper 4y ago

Yeah, this. A set does not have "two interfaces", it only has one: a membership function that takes an object and returns a boolean indicating if the object is a member of the set or not. Note that this function need not be computable, only well-defined.

cgio 4y ago
Is the membership function not equivalent to what I describe above on the query side? My difference is that I am making explicit that the set needs to be populated. The uttering of a set of “indescribable “ objects does not a set give you. That membership function works both inwards and outwards.
- recursive 4y ago
  This concept of "population" doesn't really seem to be consistent with my understanding of a set. Set A is the set of all widgets that have a frozz. The existence of Set A doesn't imply that there exists a widget that has a frozz. Perhaps the existence of such a widget is an open question. But the set is perfectly well defined.
  - lisper 4y ago
    Right. The set of all integers which are both even and odd is a well-defined set. It just happens to be empty. The set of all counter-examples to the Goldbach conjecture is a well-defined set. We don't know if it's empty or not.
  - cgio 4y ago
    If I don’t know what frozz is, is the set well defined? Maybe in form, but not in essences. Lisper’s example in sibling comment is indeed well formed, but I know that if I get a counter-example to the conjecture it will pass the population interface and become part of the set. Therefore, I don’t see how we differ comparing simple membership with population and query. My feeling is we disagree over an ex nihilo vs procedural emergence of sets.