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Number Theory/Cardinality of sets


If we manufacture fours sets of numbers from scratch (each set synchronized by) sharing the same index value i (i being a number in N) such that:
Ai={1,2,3...i} having n(Ai)=i
Bi={-1,-2,-3...-i} having n(Bi)=i
Ci={2,4...2*i} having n(Ci)=i
Di={1,3,5...2*i-1} having n(Di)=i

for i=1: A1={1}, B1={-1}, C1={2}, D1={1}, n(A1)=n(B1)=n(C1)=n(D1)=1
for i=2: A2={1,2}, B2={-1,-2}, C2={2,4}, D2={1,3}, n(A2)=n(B2)=n(C2)=n(D2)=2
for i=k-1: A(k-1)={1,2,...k-1}, B(k-1)={-1,-2...-(k-1)}, C(k-1)={2,4...2(k-1)}, D(k-1)={1,3...2(k-1)-1}, n(A(k-1))=n(B(k-1))=n(C(k-1))=n(D(k-1))=k-1
for i=k: Ak=A(k-1) U {k}, Bn=B(k-1) U {-k}, Cn=C(k-1) U {2k}, Dn=D(k-1) U {2k-1), n(Ak)=n(Bk)=n(Ck)=n(Dk)=(k-1) + 1 = k

By induction the sizes of these four sets will obey the relationship n(Ak)=n(Bk)=n(Ck)=n(Dk)=k for any k in N

As such
n(Ak)= k, and n(Ck U Dk) = n(Ck) + n(Dk) = k+k = 2k
This size relationship is shown to hold for any k

If we allow the value 2k to go to infinity such that Ck U Dk U {0} becomes the set of natural numbers (N), then the set Ak U Bk U {0} cannot possibly be the set of integers (Z), because Bk is shown to be a pure subset (also by pigeonhole principle) of Ck U Dk U {0} for all eternity.

As such when we pay careful attention to the **relative** sizes of sets Ck U Dk U {0} and Bk U {0} then the set of positive integers cannot possibly manufacture (element by element) a set larger than itself.

If we are to obey this relationship, then if the set of natural numbers is considered a unit infinity (say U_N), the set of integers is 2*U_N)

Question: Why do mathematicians readily dismiss (neglect, or violate) the above relative relationship (even though it holds eternal) when making the assertion that N and Z have the same cardinality ?

P.S. I cannot seem to be able to read any response from this site (The emails come in clipped with no links to read the full text response)

Hello ThomasAn
Like you I didn't receive the full text of your question.  I suggest you come to the site and look at past questions if you want to see the full response to a previous question.

Now, cardinality.  This cannot be treated like an algebraic number.  Any infinite set whose elements can be counted with the natural numbers, has the same cardinality by definition.  Clearly, each of your sets can be counted with the natural numbers.  The rationals can also be counted by putting them in a square array and counting diagonally starting in the top left corner.
But the real numbers cannot.
I suggest you study's_paradox_of_the_Grand_Hotel
This gives an insight into cardinality.



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