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In your example with marbles, the superset has the property of every element being comprised of a set of colors, let's say {black, red, blue}. The subset is {r

In your example with marbles, the superset has the property of every element being comprised of a set of colors, let's say {black, red, blue}. The subset is {red}. Can't every possible property we can conceive of a subset be constructed in a similar fashion such that the superset contains at least the same number of elements and never less?

lmm 4y ago
> Can't every possible property we can conceive of a subset be constructed in a similar fashion such that the superset contains at least the same number of elements and never less? What do you mean? A superset always contains at least the same number of elements as a subset, by definition. A superset generally has fewer or at least weaker properties; "colour is one of A, B or C" is a weaker property than "colour is A", and there's no real difference between that and saying the superset doesn't have the property at all (I could say "my subset consists of shiny marbles; my subset has the property that they're shiny and my superset does not have that property", or I could say "my subset has the property that shininess is true, and my superset has the property that shininess is true or false", and there isn't really any difference once you get down to the set-theoretic level - a property is just a boolean-valued function). The only superset that has exactly the same properties as the set itself is the set itself (since "is a member of the subset" is a valid property).
- antihipocrat 4y ago
  Ok, how about I define a new property that my superset contains a 'rainbow' of colors. The subset has only one of the colors, it doesn't contain the rainbow property. The original post claimed that the superset always has the same or fewer properties. I was questioning whether this was true. You've mentioned that it is generally true, which seems like a better way to describe the concept. -- edit: I understand now, in my example above the subset must contain a property by which it could be identified. The original rainbow property is lost but a new one is gained.
  - lmm 4y ago
    > Ok, how about I define a new property that my superset contains a 'rainbow' of colors. The subset has only one of the colors, it doesn't contain the rainbow property. You're mixing levels. The property is something satisfied by members of the set, not by the set as a whole.
Karrot_Kream 4y ago
The way "subset of properties" is being used here is loose and that's where the confusion is arising from. If we want to be a bit more rigorous: * Define O as a set of possible (finite) objects * Define properties as functions P: O -> {0, 1} * A given property then is a function of the form p(o) = 1 for all o in O that satisfy the property p. You can use property functions to generate subsets of O ({ o for o in O where p(o) = 1 }). The intersection of these generated subsets of multiple property functions will be strict subsets (strictly "smaller" as O is finite) of O as long as at least one property function is not of the form P(o) = 1 for * in O. Generally assuming that each property function generates strict subsets, applying more properties derives a smaller intersection set. That's all.
- antihipocrat 4y ago
  Thanks for that, it makes sense using your definition. I'm curious as to whether you agree with the parent comment that the subset could contain 'more' properties than the superset. My understanding from your stricter definition is that (by definition) all potential properties (o for o in O where p(o) = 1) must be contained within O. If the answer is that the superset O has fewer properties where every p(o) = 1 then I agree, and I was overthinking it
  - Karrot_Kream 4y ago
    > If the answer is that the superset O has fewer properties where every p(o) = 1 then I agree, and I was overthinking it It's just this honestly. A superset will generally have fewer properties which apply across the whole set.

lmm 4y ago

> Can't every possible property we can conceive of a subset be constructed in a similar fashion such that the superset contains at least the same number of elements and never less? What do you mean? A superset always contains at least the same number of elements as a subset, by definition. A superset generally has fewer or at least weaker properties; "colour is one of A, B or C" is a weaker property than "colour is A", and there's no real difference between that and saying the superset doesn't have the property at all (I could say "my subset consists of shiny marbles; my subset has the property that they're shiny and my superset does not have that property", or I could say "my subset has the property that shininess is true, and my superset has the property that shininess is true or false", and there isn't really any difference once you get down to the set-theoretic level - a property is just a boolean-valued function). The only superset that has exactly the same properties as the set itself is the set itself (since "is a member of the subset" is a valid property).

antihipocrat 4y ago
Ok, how about I define a new property that my superset contains a 'rainbow' of colors. The subset has only one of the colors, it doesn't contain the rainbow property. The original post claimed that the superset always has the same or fewer properties. I was questioning whether this was true. You've mentioned that it is generally true, which seems like a better way to describe the concept. -- edit: I understand now, in my example above the subset must contain a property by which it could be identified. The original rainbow property is lost but a new one is gained.
- lmm 4y ago
  > Ok, how about I define a new property that my superset contains a 'rainbow' of colors. The subset has only one of the colors, it doesn't contain the rainbow property. You're mixing levels. The property is something satisfied by members of the set, not by the set as a whole.

Karrot_Kream 4y ago

The way "subset of properties" is being used here is loose and that's where the confusion is arising from. If we want to be a bit more rigorous: * Define O as a set of possible (finite) objects * Define properties as functions P: O -> {0, 1} * A given property then is a function of the form p(o) = 1 for all o in O that satisfy the property p. You can use property functions to generate subsets of O ({ o for o in O where p(o) = 1 }). The intersection of these generated subsets of multiple property functions will be strict subsets (strictly "smaller" as O is finite) of O as long as at least one property function is not of the form P(o) = 1 for * in O. Generally assuming that each property function generates strict subsets, applying more properties derives a smaller intersection set. That's all.

antihipocrat 4y ago
Thanks for that, it makes sense using your definition. I'm curious as to whether you agree with the parent comment that the subset could contain 'more' properties than the superset. My understanding from your stricter definition is that (by definition) all potential properties (o for o in O where p(o) = 1) must be contained within O. If the answer is that the superset O has fewer properties where every p(o) = 1 then I agree, and I was overthinking it
- Karrot_Kream 4y ago
  > If the answer is that the superset O has fewer properties where every p(o) = 1 then I agree, and I was overthinking it It's just this honestly. A superset will generally have fewer properties which apply across the whole set.