My discovery about mathematics

Let C={c₁}∪{c₂}∪… (c_i≠c_k for i≠k) be a countable set.
∑_i∈C x_i=x_c1+x_c2+…(infinitely)
Set up that F={0",1",2",3"}≅Z/4Z, O=2F={0",2"}.
Define A+B={a+b | a∈A,b∈B} for A,B∈F/O. F/O is a module then. O is the zero of the module.
∑_i∈C O =O+O+…(infinitely)={n₁+n₂+…(infinitely) |n₁∈O, n₂∈O,…}
Put b_i=0 for ∀i∈N. ∑_{i∈{b1}∪{b2}} O=∑_i∈{0} O=O=O+O=∑_{i∈{c1}∪{c2}} O,…
∑_{i∈{b1∪…∪{bm}} O=∑_i∈{0} O=O=O+…+O=∑_{i∈{c1}∪…∪{cm}} O,…
Repeating this infinitely,you get O=∑_i∈{0} O=∑_{i∈{b1}∪{b2}∪…(infinitely)} O=∑_{i∈{c1∪{c2}∪…(infinitely)} O
=O+O+…(infinitely). But O+O+…∋ e=2"+2"+…(infinitely). e=2"+e So e∉O
This is contradiction.