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The Bourbaki-Witt theorem states that, in a chain-complete poset, the subset $X$ generated by an inflationary monotone function $s$ from the least element and joins of chains satisfies $$ \forall x,y\in X.\quad s y\leq x \quad\lor\quad x\leq y. $$ From this one may deduce that $X$ is well ordered (in the original sense of Cantor) and has a top element, which is the unique fixed point of $s$ in $X$ and the least one in the larger poset.

Since we have a well-ordering, there is an induction principle: any property of the least element that is preserved by $s$ and joins of chains also holds for the least fixed point.

In fact Zermelo's second (1908) proof of the well ordering principle is exactly this. so the Bourbaki-Witt theorem should be attributed to Zermelo.

However, the Bourbaki and Witt papers appeared in 1949 and 1950.

Serge Lang's Algebra textbook contains it in an appendix. Or rather, the $n$th printing does, for some $1< n\leq 9$.

This fixed point theorem (or its constructive replacement due to Dito Pataraia, 1997) should be in the core of the pure mathematics curriculum.

Is there any textbook that proves it in Chapter 1 and then uses it throughout its development of its subject?

I am interested in books in subjects such as some branch of algebra or topology, but not logic, set theory, lattice theory or order theory. Lattice theory considered as a form of algebra or topology would be reasonable.

I asked this question in connection with a seminar that I gave on 9 Dec 2022 about the history of order-theoretic fixed-point theorems, including my modification of Pataraia's constructive result. I did that work because I needed it for my paper on Well founded coalgebras and recursion. The slides for the seminar and the draft paper are on my "ordinals" web page and include the full bibliographical details of the papers mentioned above.

The Wikipedia page supposedly about the Bourbaki-Witt theorem gives the correct citations but wrongly states that the proof was by transfinite recursion.

This question discusses the constructive result that should perform that same role as I am suggesting that the Bourbaki--Witt theorem ought to have done.

For further background please see my webpage. If you would like to take this up as a historical investigation, please contact me by email to get my trove of material.

My new translations page includes the relevant one by Bourbaki and another by Walter Felscher comparing versions of the proof.

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    $\begingroup$ Can we give you lectures about the axiom of replacement though? (Joke explained here.) $\endgroup$
    – Burak
    Commented Dec 10, 2022 at 18:13
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    $\begingroup$ Why do you think this should be in the core of the undergraduate curriculum? That seems like a pretty bold claim, and frankly I don’t understand your reasoning at all. $\endgroup$ Commented Mar 3, 2023 at 16:31
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    $\begingroup$ @PaulTaylor Regarding this being in the core of the undergraduate curriculum: not all undergraduate curricula in pure mathematics include even a course on set theory. That's worth keeping in mind here. $\endgroup$ Commented Mar 3, 2023 at 18:25
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    $\begingroup$ @PaulTaylor If it is not relevant to the question, why do you include a discussion (well, an assertion) about undergraduate curricula in your question? $\endgroup$ Commented Mar 3, 2023 at 21:23
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    $\begingroup$ @PaulTaylor: I think it's poor form to make a wild statement like that and then get annoyed and tell someone to read your papers (which I'm not going to do since I don't work in set theory) if they want to understand what you are saying. $\endgroup$ Commented Mar 3, 2023 at 22:32

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