Where did Nachman Aronszajn prove the existence of Aronszajn trees?

Question

I understand that Nachman Aronszajn is credited for proving the existence of Aronszajn trees. But I've not had any success finding where this was published. Does anyone know of a specific paper or book where Aronszajn published his proof?

Answer 1

I believe that these first appeared in the paper

Georges Kurepa, Ensembles linéaires et une classe de tableaux ramifiés (tableaux ramifiés de M.Aronszajn), Publ. Math. Univ. Belgrade, 6 (1936) 129-160.

The author attributes them to a letter that Aronszajn wrote to him (see page 132). This paper is available here.

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Edit by A. Caicedo: The following is from Stevo Todorcevic's essay introducing the papers on "Theory of partially ordered sets", part A of Selected papers of Ðuro Kurepa, Edited and with commentaries by Aleksandar Ivić, Zlatko Mamuzić, Žarko Mijajlović and Stevo Todorčević, Matematički institut SANU, Belgrade, 1996. MR1429393 (97m:01106).

The papers it refers to are

• A[35] Ensembles ordonnés et ramifiés, Publ. Math. Univ. Belgrade 4 (1935), 1-138.
• A[37] Ensembles lineaires et une classe de tableaux ramifies (Tableaux ramifies de M. Aronszajn), Publ. Math. Univ. Belgrade 6/7 (1937/38), 129-160.
• [1] J. E. Baumgartner, Decomposition and embedding of trees, Notices Amer. Math. Soc. 17 (1970), 967.
• [6] K. J. Devlin, Note on a theorem of J. Baumgartner, Fund. Math. 76 (1972), 255-260. MR0540759 (58 #27476).
• [12] W. P. Hanf, Incompactness in languages with infinitely long expressions, Fund. Math. 53 (1964), 309-324. MR0160732 (28 #3943).

The most important publication in this group is Kurepa's thesis A[35] written under the direction of M. Fréchet. It was the first systematic study of trees and ramified partially ordered sets and of their close relationship to linear orderings. It was the source of many crucial notions and problems in this area such as, for example, the notions of Aronszajn and Souslin tree. It is the source of the problem whether inaccessible cardinals have the tree property i.e., whether they satisfy the analogue of König's infinity lemma, which was later proved by Hanf, Tarski and others ([12]) to be equivalent to the large cardinal property of weak compactness. Trees are classified in §8 A11 as "large", "étroit" and "ambigu" according to their heights and widths. The very thin and tall trees ("étroit") always have cofinal branches i.e., chains intersecting every level (Theorem $5^{bis}$). This is a fine result representing a recurring theme in applications of trees, especially in the partition calculus. This result was also a source of the problem whether the same fact is true about the class of slightly wider trees ("ambigu") i.e., the trees of height equal to some cardinal $\theta$ and whose levels are now only assumed to be of size $\lt \theta$ (rather than $\lt\lambda$ for some cardinal $\lambda<\theta$ as it was the case with the trees in Theorem $5^{bis}$). This is the problem known today as the problem whether $\theta$ has the tree property. For $\theta= \omega_1$ the problem was solved in June 1934 by N. Aronszajn and appeared (with an acknowledgement) as Theorem 6 of A[35]. According to the footnote on the same page, Aronszajn constructed his tree as a subtree of the tree of all $1-1$ sequences from $\mathbb Q^{\lt\omega_1}$, while Kurepa's version of the proof presented in A[35] (and also in A[37]) was to build such a tree inside the tree $\sigma\mathbb Q$ (denoted by $\sigma_0$) of all (nonempty bounded) well-ordered subsets of rationals. This is the construction most frequently used in subsequent expositions of this result. In the sequel A[37;§27] he modified his construction in order to produce an Aronszajn tree with the surprising property that it is a union of countably many antichains, thus introducing yet another remarkable notion, the notion of a special Aronszajn tree. (It is known today that some care is needed to get such a tree, as it is possible to have nonspecial Aronszajn subtrees of both $\sigma\mathbb Q$ and the set of all $1-1$ sequences from $\mathbb Q^{\lt\omega_1}$; see [1] and [6].)

The essay goes on to describe additional work (by Kurepa and others) on the tree property.