A Finite Set of Numbers Is Incomplete Unless Infinity Is Assumed

In this post we explore how infinity shapes the number system. Without infinity the number system would be incomplete. I would like to explain this with examples. Consider a set of two integers {1,2}. I want to multiply them element by element, we get
A= [1, 2]
B= [1, 2]
N= A × B= [1, 2, 3, 4]
C= [1, 2, 4]
Elements not on the set= [3]
I was expecting to get set N but I got set C which is missing 3. But I want to get 3 on the list. So enlarge the set I get
A= [1, 2, 3]
B= [1, 2, 3]
N= [1, 2, 3, 4, 5, 6, 7, 8, 9]
C= [1, 2, 3, 4, 6, 9]
Elements not on the set= [5, 7, 8]
You observe set C contains 3 but now it missing 5, 7, 8. I think you got the main idea. To get 5, 7, 8 on the list, I to increase the set size but I would be missing.
Similarly in case of addition, it always the first number as in the example below
A= [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
B= [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
C= [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]
diff [1]