Integers are not Densely Ordered
Jump to navigation Jump to search
The integers $\Z$ are not densely ordered.
- $\forall n \in \Z: \not \exists m \in \Z: n < m < n + 1$
By definition of immediate successor element, this is equivalent to the statement:
- $\forall n \in \Z: n + 1$ is the immediate successor to $n$
We have that Integers form Ordered Integral Domain.
From One Succeeds Zero in Well-Ordered Integral Domain:
- $\not \exists r \in \Z: 0 < r < 1$
From Properties of Ordered Ring:
- $a < b \implies n + a < n + b$
Putting $a = 0, b = 1, m = n + r$:
- $\not \exists m \in \Z: n + 0 < m < n + 1$
Hence the result.
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers