The Bourbaki-Witt theorem (Theorem 2.3.1) is a very natural result: take a chain-complete poset *X*, an inflationary map *f* (namely, *f*(*x*) is always above *x*), and a point *x*_{0} in *X*, then *f* has a fixed point above *x*_{0}. This is obtained by the following process: start from* x=x_{0}*, and repetitively replace the current value of

*x*by

*f*(

*x*). If you ever build infinitely many values of x without reaching a fixed point, then take the supremum of all values obtained so far. Repeat these two operations (replacing

*f*(

*x*) by

*x*, taking sups) for as long as needed.

This is the intuitive idea, but making it a formal proof is more challenging. Notably, the proof I’m giving of Theorem 2.3.1 seems to progress rather laboriously, going through a series of auxiliary results that should be mostly obvious, but nonetheless require some ingenuity.

Another proof would go through so-called ordinal iteration. In set theory, one can show that there are objects called *ordinals*, which generalize the natural numbers by including infinite ordinals. The sleight-of-hand definition is: the class of ordinals is the smallest class containing 0, such that *a*+1 is an ordinal for every ordinal *a*, and the sup of every chain of ordinals is an ordinal. Actually defining it, which means in particular defining the ordering between ordinals, requires some set-theoretic cunning. (See the wikipedia page on ordinals.) One can then obtain the fixed point that the Bourbaki-Witt theorem claims exists by ordinal induction. That is, we define a collection *x _{a}* of points of

*X*by:

*x*is already defined,

_{0 }*x*=f(

_{a+1}*x*), and if

_{a}*a*is the sup of a chain of ordinals

*b*, then

*x*is the sup in X of the chain of elements f(

_{a}*x*). The class of all ordinals is not a set, i.e., it is not small. But

_{b}*X*is small, so the elements

*x*cannot be all distinct. One then shows that if

_{a }*x*is equal to

_{a}*x*, then it is a fixed point of

_{b}*f*. If you include all the needed set theory, this is far from elementary.

Dito Pataraia once found an elegant, and rather elementary of something very close to the Bourbaki-Witt theorem [1]. Apparently, he would only very rarely publish his results, so his proof first appeared in a paper by Martín Escardó [2]. My attention was drawn to this result by an anonymous referee, whom I am thanking here.

What Dito Pataraia proved was a version of the Bourbaki-Witt theorem that requires a bit more assumptions, and produces a vastly stronger result.

The main change in assumptions is that we now require our inflationary maps to be *monotonic* as well. This is the case in many applications. One exception is the Caristi-Waszkiewicz theorem (Exercise 7.4.42). The other change is that he requires *X* not to be an inductive poset, rather to be a *dcpo*; that does not make much of a change, since Markowsky’s Theorem (p.61) tells us that inductive posets and dcpos are one and the same thing, but Markowsky’s Theorem is itself non-trivial.

The big change is in the result: we can find a fixed point *x* above *x*_{0}, not of one inflationary map, but of an arbitrary family (*f _{i}*)

_{i}_{ in }

*of inflationary (monotonic) maps*

_{I}*simultaneously*: we shall have

*f*(x)=x with the

_{i}*same*x for every

*i*in

*I*.

Here is how Pataraia does it. In a bold move, he goes to a higher-order concept right away. Given a dcpo *X*, he considers the set Infl(*X*) of all inflationary monotonic maps from *X* to *X*. With the pointwise ordering, Infl(*X*) is a dcpo, and sups are computed pointwise, as usual. More: Infl(*X*) is itself directed. The key is that for any two inflationary monotonic maps *f*, *g*, their composition *f* o *g* is inflationary, monotonic, and above both *f* (since *g* is inflationary and *f* monotonic) and *g* (since *f* is inflationary). It follows that Infl(*X*) has a supremum ⊤. This is an inflationary, monotonic map. For every *f* in Infl(*X*), *f* o ⊤ is below ⊤ because ⊤ is the top element, and *f* o ⊤ is above ⊤ because *f* is inflationary: so *f* o ⊤ = ⊤. It follows that, for every *x* in *X*, *f*(⊤(*x*)) = ⊤(*x*). In other words, ⊤(*x*) is a fixed point of *f*, and one that is above *x* as well since ⊤ is inflationary. Hence:

**Theorem** ([2], improving on [1]): On a dcpo *X*, every family (*f _{i}*)

_{i}_{ in }

*of inflationary (monotonic) maps has a common fixed point*

_{I}*x*above any given point

*x*. We can take x=⊤(

_{0}*x*), where ⊤ is an inflationary monotonic map that is independent of

_{0}*x*and of the family (

_{0 }*f*)

_{i}

_{i}_{ in }

*.*

_{I}We can also require the common fixed point *x* to be least. In this case, we must accept that *x* will depend on the family (*f _{i}*)

_{i}_{ in }

*.*

_{I}**Theorem** ([2]): On a dcpo *X*, every family (*f _{i}*)

_{i}_{ in }

*of inflationary (monotonic) maps has a least common fixed point*

_{I}*x*above any given point

*x*.

_{0}The proof goes as follows. Let *F* be smallest subset of *X* that contains *x _{0}*, is closed under applications of every

*f*, and is closed under taking sups of directed families. This is a dcpo, and the restrictions

_{i}*f’*of

_{i}*f*to

_{i }*F*define inflationary maps from

*F*to

*F*again, which therefore have a common fixed point

*x*=⊤'(

*x*) in

_{0}*F*, as we have just seen.

To show that it is least, let *y* be another common fixed point of the maps *f _{i}* above

*x*The downward closure

_{0.}*Y*of y contains

*x*, is closed under directed sups, and we claim that

_{0}*Y*is closed under applications of every

*f*: for every

_{i}*z*in

*Y*,

*f*(

_{i}*z*) is below

*f*(

_{i}*y*)=

*y*since

*f*is monotonic and y is a fixed point of

_{i }*f*. It follows that

_{i}*Y*contains

*F*, which is the smallest subset with these properties. Therefore

*x*=⊤'(

*x*) is also in

_{0}*Y*, which means that

*x*is below

*y*, hence

*y*is smallest.

We then obtain the following fixed point induction principle, akin to Fact 2.3.3:

**Trick:** To show that a property *P* holds of the least common fixed point *x*, it is enough to show that *P*(*x _{0}*) holds, that

*P*(

*y*) implies

*P*(

*f*(

_{i}*y*)) for every

*y*in

*X*and every

*i*in

*I*, and that for every directed family

*D*of elements

*y*of

*X*such that

*P*(

*y*) holds,

*P*(sup

*D*) holds.

Indeed, the set of points that satisfy *P* is one that contains *x _{0}*, is closed under applications of every

*f*, and is closed under taking sups of directed families, hence contains the set

_{i}*F*mentioned in the proof of the previous theorem.

— Jean Goubault-Larrecq(June 13th, 2013)

- Dito Pataraia. A constructive proof of Tarski’s fixed-point theorem for dcpo’s. Presented in the 65th Peripatetic Seminar on Sheaves and Logic, in Aarhus, Denmark, November 1997.
- Martín H. Escardó. Joins in the frame of nuclei. Applied Categorical Structures, April 2003, Volume 11, Issue 2, pp 117-124.