In set theory, an Aronszajn tree is an uncountable tree with no uncountable branches and no uncountable levels. For example, every Suslin tree is an Aronszajn tree. More generally, for a cardinal κ, a κ-Aronszajn tree is a tree of height κ in which all levels have size less than κ and all branches have height less than κ. They are named for Nachman Aronszajn, who constructed an Aronszajn tree in 1934; his construction was described by. A cardinal κ for which no κ-Aronszajn trees exist is said to have the tree property .
Existence of κ-Aronszajn trees
states that -Aronszajn trees do not exist. The existence of Aronszajn trees was proven by Nachman Aronszajn, and implies that the analogue of Kőnig's lemma does not hold for uncountable trees. The existence of -Aronszajn trees is undecidable : more precisely, the continuum hypothesis implies the existence of an -Aronszajn tree, and Mitchell and Silver showed that it is consistent that no -Aronszajn trees exist. Jensen proved that V = L implies that there is a κ-Aronszajn tree for every infinite successor cardinalκ. showed that it is consistent that no -Aronszajn trees exist for any finite n other than 1. If κ is weakly compact then no κ-Aronszajn trees exist. Conversely if κ is inaccessible and no κ-Aronszajn trees exist then κ is weakly compact.
Special Aronszajn trees
An Aronszajn tree is called special if there is a function f from the tree to the rationals so that f < f whenever x < y. Martin's axiom MA implies that all Aronszajn trees are special. The stronger proper forcing axiom implies the stronger statement that for any two Aronszajn trees there is a club set of levels such that the restrictions of the trees to this set of levels are isomorphic, which says that in some sense any two Aronszajn trees are essentially isomorphic . On the other hand, it is consistent that non-special Aronszajn trees exist, and this is also consistent with the generalized continuum hypothesis plus Suslin's hypothesis.
Construction of a special Aronszajn tree
A special Aronszajn tree can be constructed as follows. The elements of the tree are certain well-ordered sets of rational numbers with supremum that is rational or −∞. If x and y are two of these sets then we define x ≤ y to mean that x is an initial segment of the ordered sety. For each countable ordinal α we write Uα for the elements of the tree of level α, so that the elements of Uα are certain sets of rationals with order type α. The special Aronszajn tree T is the union of the sets Uα for all countable α. We construct the countable levels Uα by transfinite induction on α as follows starting with the empty set as U0:
If α + 1 is a successor then Uα+1 consists of all extensions of a sequence x in Uα by a rational greater than sup x. Uα + 1 is countable as it consists of countably many extensions of each of the countably many elements in Uα.
If α is a limit then letTα be the tree of all points of level less than α. For each x in Tα and for each rational numberq greater than sup x, choose a levelα branch of Tα containing x with supremum q. Then Uα consists of these branches. Uα is countable as it consists of countably many branches for each of the countably many elements in Tα.
The function f = sup x is rational or −∞, and has the property that if x < y then f < f. Any branch in T is countable as f maps branches injectively to −∞ and the rationals. T is uncountable as it has a non-empty level Uα for each countable ordinal α which make up the first uncountable ordinal. This proves that T is a special Aronszajn tree. This construction can be used to construct κ-Aronszajn trees whenever κ is a successor of a regular cardinal and the generalized continuum hypothesis holds, by replacing the rational numbers by a more general η set.
