We first start with a tree structure on the set of real coefficient polynomials with domain and codomain , and call it the polynomial tree. We will say iff for some
Proposition 1. This gives us a partial order on .
Proof. since If and then and thus so And lastly if and then and Thus which implies and thus so
The fact that differentiation yields a unique function gives us the tree structure: the chain of derivatives (predecessors) is well ordered with minimal element By the Weierstrass approximation theorem we can attempt to extend this to the closure of the class of smooth functions with compact support. We extend our indexing set to the ordinal and define the th (and th) derivative linearly by the rule
Recall integration is a set valued operation–sending a function to its set of antiderivatives which all differ by some constant.
Example 2. Let We have
Hence for finite since the “last term” is and If we begin to integrate, we will add polynomials trailing after Continuing this denumerably we then obtain
where is the conumber of in the sense that it satisfies in the th term of the th partial sum. The denumerable integration converges if the sequence of constants chosen satisfies
Suppose the sequence satisfies this condition and let us assume the sum converges to some then we can write
In particular we thus have since
We thus have and We also define the degree of the functions which are not polynomials as Let be a polynomial with degree and have infinite degree. Then we clearly have that and have degree
Definition 3. A function is cyclic if for some nonzero finite number The number is called the differential order of and is denoted If a function is not cyclic we say (note in this definition ).
Example 4. The functions and are cyclic with orders and respectively. The only function with finite degree which is cyclic is with order
In fact being cyclic occurs iff
for at least some where and Thus we have
With this presentation it is clear that the order of the th term in the above term is if Otherwise the order is
Let us now define the equivalence relation where iff and are cyclic with same order and where such that Reflexivity is clear, and symmetry/transitivity involve simple arithmetic.
Example 5. Hence some equivalence classes are and for any polynomial
Proposition 6. gives a tree structure on which we call the calculus tree.
Proof. This is trivial considering the relation we modded out by was precisely loops in the poset–together with the fact that the poset has a unique minimal element.
Also see update.
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.
 Jech, Thomas. Set Theory. Third Edition. Springer Monographs in Mathematics. Springer-Verlag. 2003.