So I need to find .
I start with .
(After , i'm just taking the union of the sets before it. Therefore (writing the elements as a list)):
To me this makes sense since i'm constructing the natural numbers:
etc
However, my tutor wrote that I should have 16 elements in . How is he getting all these extra elements?
Since I have already started writing...
If , then . Therefore, indeed, .
You need one more element: .
If we denote the elements of as follows: , , , , then your list can be written as follows:
, , , .
It is missing 12 sets, including , , , , as well sets that contain .
Though you can encode natural numbers in this way, a standard construction (von Neumann ordinals) defines the successor of as , so the th set contains elements.To me this makes sense since i'm constructing the natural numbers