English translation of Bloch’s paper

Question

I'd like to know if there is an English translation of Bloch's "Sur les ensembles stationnaires de nombres ordinaux et les suites distinguees de fonctions regressives". Also I'd like to know where to find a pdf version with text selection of Fodor's "Eine Bemerkung zur Theorie der regressiven Funktionen".

Answer 1

About C. R. Acad. Sci. Paris 236 (1953), 265–268.

I obtained it by asking "Gallica" and then the full title.

I have translated (see below) a part of the document.

Beginning of the document's translation:

SET THEORY : About stationary sets of ordinal numbers and distinguished sequences of regressive functions. Note by Mr. Gérard Bloch, presented by Mr. Arnaud Denjoy.

On the set of ordinal numbers of class I and II, a family of subsets, said “stationary” is defined and studied ; the results are then applied to distinguished sequences associated with ordinal numbers of second kind.

O denotes the set of cardinals of classes I and II with order topology. $\Omega$ is the first number of class III. Please note that the intersection of a denumerable family of non denumerable closed sets is a non denumerable closed set.

A function $f(x)$, defined on a subset $A$ of $O$ taking its values in $O$, will be said regressive if $f(x)\leq x$ for all x in A with equality excluded but for $x=1$, if $1 \in A$. A set $A$ of points belonging to $\Omega$ is stationary if it is non denumerable and if for any regressive function $f(x)$ defined on $A$, one has $\lim \cdots \Omega$; otherwise said if there exists at least a point a such that $f^{-1}(a)$ is non denumerable. In all other cases, A is non stationary.

Theorem 1. _ A necessary and sufficient condition of stationarity for a given set is that any closed subset of its complement set is at most denumerable.

In particular, any set containing a closed non denumerable subset (in particular the whole set) is stationary. Moreover, any set which is the union of a denumerable family of non-stationary sets is itself non-stationary; and the intersection of a non denumerable closed set and a stationary set is a stationary set.

Let $O’$ denote the set of numbers of the second type, a distinguished sequence over a subset $A$ of $ O'$ will be by definition a sequence of regressive functions $f_n(x) \ n \in \mathbb{N}$ defined on $A$, such that, for all $x \in A$

$f_1(x)\leq f_2(x)\leq\cdots$ and $\lim_{n} f_n(x)=x$.

Theorem II:- A distinguished sequence being given over a stationary subset $A$ of $O’$, there exists an integer $n_0$ such that for any $n>n_0$, and for any partition of $A$ into two subsets $B_n$ and $C_n$ such that $f_n(x)$ is upper bounded on $B_n$, the set $C_n$ is stationary.

The proof is omitted by lack of place.

By suppressing then, if needed, a finite number of terms in the distinguished sequence we may assume that $n_0=i$. Consider then function $f(x)$ and the set of points of $O$ whose reciprocal image by $f_i(x)$ is non denumerable. $H$ is itself non denumerable. Indeed if $K=f_i^{-1}(H)$ and $L=A-K$, set $L$ is not stationary; were $H$ be denumerable, $f_i(x)$ would be bounded from above on $K$ and theorem II would be invalidated. Therefore, one can number the elements of $h_i$ of $H$ by increasing order of magnitude, index $i$ describing the whole space $O$.

[...]

Corollary : any stationary set is the union of two stationary disjoint sets ; any non denumerable closed set (in particular the whole space) is the union of two disjoint sets none of which contains a non denumerable closed set.

By the way this corollary is a particular case of the following theorem which can be proved directly : if, on a set E whose cardinality is at most the cardinality of continuum, a family $F$ of sets, closed for operation “denumerable intersection”, is given, there exists a partition of $E$ into two disjoints sets, none of which contains a set of $F$.

[...]

Theorem III : On any closed non denumerable set $F$ (especially on the set $O’$ of numbers of the second category), there exists a distinguished sequence $f_1(x) \cdots f_n(x)…$ with the following properties :

1) The set F is the union of $\aleph_1$ stationary sets pairwise without common points, on each of which $f_i(x)$ takes a constant value, all of these values being different.

2) Each set of the $p$th order is the union of $\aleph_1$ stationary set pairwise without common points, on which each $f_{p+1}(x)$ takes a constant values, all of them different. These sets will be called “sets of $(p+1)$st order”.

3) The intersection of the sequence of sets $E_1,E_2\cdots$ such that $E_p$ is a set of $p$th order” with $E_1 \supset E_2\cdots \supset \cdots$ is void, or reduced to a point $x$. In the latter case, $x$ is the limit of increasing sequence $x_1,x_2, \cdots$, where $x_p$ denotes the value of $f_p(x)$ on $E_p$.

In a reverse way, any point $x \in F$ is the intersection of such a sequence of sets $E_p$, and it it possible to prove that the set of $x$ elements which are not equal to their rank in at least one of the sets $E_p$ containing it is a non stationary set.

Note : Mr. Neumer Walter has presented, in the case of the set of ordinal numbers strictly less than an initial given number of classes, a part of the previous results dealing with stationary sets, but for results dealing with “distinguished sequences"(Cf. Math. Zeitung, 54, p. 254-261). This article was unknown to me when I wrote this Note.

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If you want to upload this text as a pdf file, here is how you should proceed

• be on the first page (efficiently by typing 265 in the downright place)

• In the icon menu on the left, click on the sixth one (with a down arrow). A popup window appears, select options like on this figure :

(the "4" here is the total number of pages). A pdf file will be created. If you open it, you will be able to Ctrl-Save it.