# Suslin Trees

**Definition 1.** A **tree** is a strict poset such that set of predecessors of is well-ordered for all .

This motivates the notion of a root. We can define the **root of ** as . In particular we can say an element is a **root** if it is the root of some element. We define the **height** of as , where is the order-type (or ordinality) of . The ordinality of a toset (totally ordered set) is the unique ordinal number to which is isomorphic (as tosets).

Note that . For example . But since in the bijection defined by and for finite ordinals, we have , but .

Correspondingly we define the **height** of to be . An –**tree** is a tree of height . We also define the **th level** of a tree as the set .

**Definition 2.** A **branch** in is a maximal toset/chain in . That is, it is not properly contained within any chain in .

**Definition 3.** A tree is **normal** if

- has a unique root;
- each level of is countable;
- if is not maximal, then it has infinitely many successors in ;
- all branches have the same height;
- if is a limit ordinal and such that , then .

Note that tosets can be endowed with a topology called an **order topology**. It is generated by predecessor and successor sets. A poset is **dense** if for any comparable , there is a such that . A toset is **complete** if every bounded subset has an infimum and supremum. A toset satisfies the **countable chain condition** if every collection of disjoint open rays (predecessor/successor sets) is countable.

**Suslin’s Problem.** Let be a dense, complete, and unbounded toset that satisfies the countable chain condition. Is isomorphic to ?

Note that is the unique toset which is dense, complete, unbounded and separable (has a countable dense subset). The separability of implies that it satisfies the countable chain condition. Since any interval in has a rational number (from denseness), then any collection of disjoint open rays must be countable. Hence Suslin’s Problem asks if the converse is true on tosets that are dense, complete, and unbounded. This is not provable in ZFC, ZFC+CH, ZFC+GCH, or ZFC+CH.

**Definition 4.** A **Suslin line** is a dense, complete, and unbounded toset that satisfies the countable chain condition but is not separable.

Hence Suslin’s Problem asks whether or not a Suslin line exists. This turns out to be equivalent to the existence of Suslin trees. While a **chain** of a poset is any sub toset, an** antichain** is a subset in which no elements are comparable.

**Definition 5.** A **Suslin tree** is a tree such that , every branch is countable, and every antichain is countable.

Note these assumptions are not contradictory, since the height of is not equivalent to the supremum of heights of branches (where height of a branch is defined by supremum of heights of its elements).

**Lemma 6.** If there exists a Suslin tree, then there exists a normal Suslin tree.

**Proposition 7**. There exists a Suslin line if and only if there exists a Suslin tree.

If SH (Suslin’s Hypothesis) is the nonexistence of a Suslin line and MA is Martin’s Axiom, then the proof of independence follows from and where is the Axiom of Constructibility.

[1] Jech, Thomas. *Set Theory*. Third Edition. Springer Monographs in Mathematics. Springer-Verlag. 2003.