I'm trying to solve one puzzle, which deals with infinite set, so I wonder what operations on infinite sets are allowed in mathematics and what operations have no sense?
Let me explain my problem in more details.
For example, consider an infinite subsets of natural numbers. Like odd numbers: {1,3,5,7,...}. It is clear that you can find Unions, Intersections, Complements and Cartesian product.
But can you add all elements of this set by modulo? Or, at least say, for example, that if $x=(...((3+5) mod 2)+7) mod 2)+9)...$ and $y=(...((5+7) mod 2)+9) mod 2)+11)...$ then for sure $x$ and $y$ has different, though unknown values? Since $x=(3+y) mod 2$.
Can you compare sets one to another? For example if you define comparison of two different well-ordered sets like:
1) take the first (smallest) elements of the sets: $a1$ and $b1$. if $a1 < b1$, then $set A < set B$.
2) if $a1 = b1$, then take second elements of the sets: $a2$ and $b2$. if $a2 < b2$, then $set A < set B$.
3) continue, until you find different elements.
You can clearly do this with finite sets, but can you do this with infinite sets? Can you say that set of all possible subsets of natural numbers can be ordered with this comparison? If not, then why?
What if we take not natural numbers, but some sets with higher cardinality, like real numbers?