Warning: Please ignore the following.
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Should 0 be included in the set of natural numbers?
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No, \(0\notin \mathbb N\)
The numbers 1, 1+1=2, 2+1=3, ... are said to be natural.
Given two elements N and E, we can struct a field by the rules
(i) N+N=N
(ii) N+E=E
(iii) E+E=N
(iv) N\(\cdot\)N = N
(v) N\(\cdot\)E = N
(vi) E\(\cdot\)E = E
Then in keeping with our notation, we should be N=0, E=1.
From (iii), E+E = N, so, we have 1+1 = 0
However from the first statement, we have 1+1=2 \(\ne\) 0, so we need to exclude 0.
Problems:
1) Why would we construct such a field with rules (i) to (vi), particularly with rule (iii)?
2) What would happen if we picked some other values for N and E?