An infinite series of permutations, in the context of a countable set X (like the natural numbers ℕ or the integers ℤ), can be understood as a linear ordering ≤π of X that may differ from the usual ordering. This is analogous to how a finite permutation rearranges elements, but extended to infinite sets.
Here's a breakdown of what that means:
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Countable Set (X): A set that can be put into a one-to-one correspondence with the natural numbers. This means you can, in principle, list all the elements of the set, even though the list might be infinitely long. Examples include the set of natural numbers (1, 2, 3, ...) and the set of integers (... -2, -1, 0, 1, 2, ...).
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Permutation (in the finite case): For a finite set, a permutation is simply a rearrangement of its elements. It's a bijective (one-to-one and onto) mapping from the set to itself. For example, a permutation of {1, 2, 3} could be {2, 3, 1}.
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Linear Ordering (≤π): This is the key concept for extending the idea of a permutation to infinite sets. A linear ordering defines how elements are compared to each other. For any two elements a and b in the set X, either a ≤π b or b ≤π a, and this relationship is transitive (if a ≤π b and b ≤π c, then a ≤π c).
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Infinite Permutation (π): An infinite permutation establishes a specific linear ordering (≤π) on the infinite countable set X. This ordering might be the standard ordering (e.g., 1 < 2 < 3 < ... for natural numbers), or it might be a completely different ordering. The crucial aspect is that it defines a new way to arrange the elements of the infinite set.
Example:
Consider the set of natural numbers ℕ = {1, 2, 3, ...}.
- The usual ordering is 1 < 2 < 3 < ...
- An infinite permutation could define a new ordering like this: 2 < 4 < 6 < ... < 1 < 3 < 5 < ... (all the even numbers come before all the odd numbers, and within each group, the ordering is standard).
In essence, an infinite permutation reorders the elements of an infinite countable set by defining a new way to compare and arrange them. This new arrangement, defined by the linear ordering ≤π, is what constitutes the infinite permutation.
