SW VandeCarr said:
With the exception of my informal use of the term "series", you've not challenged anything I've stated in post 4. I'm not saying the positive integers are primitive constructions and it's not my formal system. I'm citing the Peano construction of zero and the positive integers from set theory. In this construction {}=0. If so, the ordered pair (0,0) can then be written as ({},{}). I'm saying that this is suspect since it uses a unique set to express two non-interchangable components of the null or zero vector in R^2. How can we have countably many representations of the empty set in a single mathematical object? What does two empty sets mean when there is only one empty set? I agree that a set is not a number. That is just my point.
Regarding {0}, what is the cardinality of this set? What is the cardinality of {}? If they are both of same cardinality and {}=0, why can't we say that {0}=0?
-Regarding your problem with ({}(0), {}(1),...,{}(n),...) - {}(0), {}(1), etc are not different empty sets. They are all {}. Th integers assigned to them only determine the position. It's ok to do this with sets, in the same way that it is ok to have n-tuple (0,0,0,...) in a ring where 0 in a unique element. In ({}(0), {}(1),...,{}(n),...), {}(0) and {}(1) are different elements of the tuple, but not because they are different sets, merely because they are in different positions. Now if you swap the position of a few of these {}(i) you get the same thing since:
({}(0), {}(1),...,{}(n)) = (0(0), 0(1),...,0(n)) = (0(1), 0(0),...,0(n)) = ({}(1), {}(0),...,{}(n))
as you know two n-tuples (a(0), a(1),...), (b(0), b(1),...) are the same if a(i) = b(i) for all i. So if you continue permuting the elements of ({}(0), {}(1),...,{}(n)) as above you get that {}(0) = {}(1) = {}(2) =...= {}(n).
-Regarding your original question about {} and {0}. The cardinality of finite sets is defined to be the number of elements in them. And if {0} confuses you, then think of the function
f: {0} -> {p} by f(0) = p . Here p is any object you. The function f creates a bijection between {0} and {p}. You can now treat {0} and {p} the same. Their cardinalities for example are the same.