Bourbaki–Witt theorem

In mathematics, the Bourbaki–Witt theorem in order theory, named after Nicolas Bourbaki and Ernst Witt, is a basic fixed point theorem for partially ordered sets. It states that if X is a non-empty chain complete poset, and

such that

then f has a fixed point. Such a function f is called inflationary or progressive.

If the poset X is finite then the statement of the theorem has a clear interpretation that leads to the proof. The sequence of successive iterates,

where x₀ is any element of X, is monotone increasing. By the finiteness of X, it stabilizes:

It follows that x_∞ is a fixed point of f.

Pick some ${\displaystyle y\in X}$. Define a function K recursively on the ordinals as follows:

If ${\displaystyle \beta }$ is a limit ordinal, then by construction

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