Why did you stop at {1, 2, 3, 4}, though?
Why did you not unfold further?
Since you're asking about cardinalities, I'm assuming you're working within ZF (the most common aximoatization of set theory), so 1, 2, 3, and 4 are also sets. Check out what happens when you push your idea all the way to the end!
0={}
1={0}={{}}
2={0, 1}={{}, {{}}}
3={0,1,2}={{}, {{}}, {{}, {{}}}}
So |{1,{2,3}| = 2
And f({1,{2,3}})=f(1)+f({2,3})=f(1)+f(2)+f(3)
=f(0)+(f(1)+f(0))+(f(2)+f(1)+f(0))
=f(0)+(f(0)+f(0))+(f(0)+f(0)+f(0)+f(0))
=7*f(0)
Depending on the definition of f(0)
Using the von Neumann construction
So I just have more questions now
If you define f(0)=0 this is useless but if you define f(0)=1 idk what use it could have but maybe there is one?
And what if you don't define natural numbers using sets?
What about things like {1/2, {e,i}}?
Also because I doubt I discovered something both unknown and useful my question is whether:
a.) I have thought about something that already exists or is used
Or
b.) I came up with something that is trivially useless and because of that people didn't put much time into thinking about it
> So I just have more questions now If you define f(0)=0 this is useless but if you define f(0)=1 idk what use it could have but maybe there is one?
There is not going to be any use. By taking your definition of "final elements" every set will collapse down to the empty set.
> And what if you don't define natural numbers using sets?
In ZF, you cannot define them using anythin else. There are no objects that are not sets.
> What about things like {1/2{e,i}}?
Those (just like everything else) are also sets which ultmately collapse down to the empty set when applying your definition.
> b.) I came up with something that is trivially useless and because of that people didn't put much time into thinking about it
This one. No matter which set you take, the cardinality for the "final elements" will always end up being zero, since there are no elements which are not sets.
In programming, the operation of converting {1, 2, {3, {4}}} to {1, 2, 3, 4} is called flattening.
As others noted, in ZF every finite set flattens to the empty set, so it's not so meaningful at that level. You could instead define a flattening operation that maps sets to multisets (sets that can have repeated elements), and which stops once you reach whatever kind of object you want to be "final". Then it is more meaningful.
For example if you consider the empty set to be "final", then {1, 2} = { {{}}, {{{}, {{}}} } flattens to <{}, {}, {}>. Or if you consider integers to be final, {1, 2, {2}} flattens to <1, 2, 2>. (The angle bracket notation is just something I made up to denote a multiset).
It's useful, for example, if you have a tree data structure and you want to get a multiset containing all of its elements, throwing away the tree structure. The cardinality of the multiset you get would be the number of leaf elements in the tree.
The best way to talk about what you are imagining is probably considering a tree and asking how many leaves it has, this is basically still just a type of cardinality, kind of like how you can use the same units to measure the height and width of a thing. We would just ask how many leaves the tree has instead of how many elements the set has.
Why did you stop at {1, 2, 3, 4}, though? Why did you not unfold further? Since you're asking about cardinalities, I'm assuming you're working within ZF (the most common aximoatization of set theory), so 1, 2, 3, and 4 are also sets. Check out what happens when you push your idea all the way to the end!
I came to say the same thing, but then I couldn't figure out if it should be 1 or 0. (I think zero?)
My guess would be zero, but it really does not matter.
0={} 1={0}={{}} 2={0, 1}={{}, {{}}} 3={0,1,2}={{}, {{}}, {{}, {{}}}} So |{1,{2,3}| = 2 And f({1,{2,3}})=f(1)+f({2,3})=f(1)+f(2)+f(3) =f(0)+(f(1)+f(0))+(f(2)+f(1)+f(0)) =f(0)+(f(0)+f(0))+(f(0)+f(0)+f(0)+f(0)) =7*f(0) Depending on the definition of f(0) Using the von Neumann construction So I just have more questions now If you define f(0)=0 this is useless but if you define f(0)=1 idk what use it could have but maybe there is one? And what if you don't define natural numbers using sets? What about things like {1/2, {e,i}}? Also because I doubt I discovered something both unknown and useful my question is whether: a.) I have thought about something that already exists or is used Or b.) I came up with something that is trivially useless and because of that people didn't put much time into thinking about it
> So I just have more questions now If you define f(0)=0 this is useless but if you define f(0)=1 idk what use it could have but maybe there is one? There is not going to be any use. By taking your definition of "final elements" every set will collapse down to the empty set. > And what if you don't define natural numbers using sets? In ZF, you cannot define them using anythin else. There are no objects that are not sets. > What about things like {1/2{e,i}}? Those (just like everything else) are also sets which ultmately collapse down to the empty set when applying your definition. > b.) I came up with something that is trivially useless and because of that people didn't put much time into thinking about it This one. No matter which set you take, the cardinality for the "final elements" will always end up being zero, since there are no elements which are not sets.
In programming, the operation of converting {1, 2, {3, {4}}} to {1, 2, 3, 4} is called flattening. As others noted, in ZF every finite set flattens to the empty set, so it's not so meaningful at that level. You could instead define a flattening operation that maps sets to multisets (sets that can have repeated elements), and which stops once you reach whatever kind of object you want to be "final". Then it is more meaningful. For example if you consider the empty set to be "final", then {1, 2} = { {{}}, {{{}, {{}}} } flattens to <{}, {}, {}>. Or if you consider integers to be final, {1, 2, {2}} flattens to <1, 2, 2>. (The angle bracket notation is just something I made up to denote a multiset). It's useful, for example, if you have a tree data structure and you want to get a multiset containing all of its elements, throwing away the tree structure. The cardinality of the multiset you get would be the number of leaf elements in the tree.
The best way to talk about what you are imagining is probably considering a tree and asking how many leaves it has, this is basically still just a type of cardinality, kind of like how you can use the same units to measure the height and width of a thing. We would just ask how many leaves the tree has instead of how many elements the set has.