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?


closed as too broad by Zain Patel, Andrés E. Caicedo, Ramiro, 6005, hardmath Jul 25 '16 at 0:07

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    $\begingroup$ Your ordering is basically lexicographic order, though you may want shortlex order instead, in order to accommodate the finite subsets. $\endgroup$ – Brian M. Scott Jul 24 '16 at 13:26
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    $\begingroup$ If this were to be closed, IMO "too broad" would be more appropriate than "unclear" or "off-topic". $\endgroup$ – Hurkyl Jul 24 '16 at 13:27
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    $\begingroup$ I don't think it makes much sense to talk about infinite sums modulo $2$. Infinite sums require a concept of convergence so you need a metric and the only metric on a finite set is the discrete metric which means your sums have to really be finite sums to make sense. As for comparing sets the way you have, that seems reasonable to me. $\endgroup$ – Gregory Grant Jul 24 '16 at 13:29
  • $\begingroup$ @BrianM.Scott I googled and: "In mathematics, and particularly in the theory of formal languages, shortlex is a total ordering for finite sequences of objects that can themselves be totally ordered.". I need order for infinite sequences. $\endgroup$ – klm123 Jul 24 '16 at 13:33
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    $\begingroup$ @klm123: You get a linear order; it's not a well-order, but you didn't specify that you wanted one. $\endgroup$ – Brian M. Scott Jul 24 '16 at 13:55