Section: New Results

Bourbaki, Sets and Ordinals

Participant : José Grimm [correspondant] .

In previous years, we developped a formal library describing the part of the Bourbaki books on set theory, cardinals and ordinals, [18] . Here are ome additions to the library.

Since addition of ordinals is non-commutative, the sum of $n$ ordinals $x_1$ to $x_n$ depends on their ordering; the maximum number $f\left(n\right)$ is a priori bounded by $n!$, and we have shown that it satisfies a recurrence relation (R), Bourbaki asks, in an exercise, to show that $f\left(n\right)=81f(n-5)$ for $n\ge 20$. This is an easy consequence of an explicit formula (F) for $f$. That (R) implies (F) can be expressed in pure Coq (with binary integers), but we have no idea how to prove it.

We proved some facts of the theory of models: the set $V_{\omega }$ of hereditarily finite sets satisfies ZF (but not the axiom of infinity); the von Neumann universe satisfies ZF and AF, there is a subset of the universe satisfiying ZF containing no inaccessible cardinal. We have also studied the set of formulas and show the theorem of Lövenheim-Skolem.

The main contribution this year is the study of some families of ordinals. If the family is internally closed and too big to be a set, then it is the image of a normal (continuous and strictly increasing) function, called the enumeration function of the family. The family of fix-points of a normal function satisfies this property, and the enumeration of this family is called the first derived function. There is a derivation at every order. For instance, the first derivation of $x\mapsto 1+x$ is $x\mapsto \omega x$, and the derivation of order $n$ is $x\mapsto \phi (n,x)$. The least $x$ such that $x={\omega }^x$ is known as ${ϵ}_0$; the least $x$ such that $x=\phi (x,0)$ is known as ${\Gamma }_0$.

We have shown that the inductive type $T$ defined by zero and a constructor of type $T\to N\to T\to T$, without the terms that are not in “normal form” , is isomorphic to the set of ordinals less than ${ϵ}_0$; in the case of $T\to T\to N\to T\to T$, we get all ordinals less than ${\Gamma }_0$; we have also studied the case with one more $T$ (the first two types were first implemented by Castéran, the last was suggested by Ackermann) [19]