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.
Los Angeles Unified School District put in place a new policy that limits the impact of homework on a students’ grade to 10%. This appears to be resultant from complaints from parents and students regarding either a time issue or a hindering home environment issue. “According to the new policy, ‘Varying degrees of access to academic support at home, for whatever reason, should not penalize a student so severely that it prevents the student from passing a class, nor should it inflate the grade’. It was distributed to schools last month”. Well, as a hypothetical teacher, this policy for my class would just mean upping the weight of examinations to 90%. And if the premise is based on possible deficiencies of academic support at home, I’m worried this would make the situation even worse when it came time for tests. Then again, I may be an anomaly in that my grades would be based solely on tests and homework–in general no projects or make-ups on exams. I mean, I’d be all for this policy since it would separate those who really know the material from those who don’t, but their basis for implementing it seems to be at odds with this ideology.
God is a common yet unclear term. One definition is “creator of universe”. We first must clarify what “universe” means. As far as I can tell, the traditional interpretation of universe is everything (i.e. class of all sets). In this case, the creator/cause of it (i.e. a larger class, which would hence make the universe a set) makes no sense.
Other common interpretations naively personify the God object as human-like, with notions of benevolence/morality. Others include concepts of omnipresence, omnipotence, and omniscience. I’m not really sure how to make use of terms like benevolence or omniscience, but I’d probably go with utilitarianism and omnipresence as respective equivalences. Now, any object with omnipotence and omnipresence=omniscience is at least as “big” (in the containment/causal sense) as the universe. But again, since the universe contains everything, by definition, then any object with these qualities is equal to the universe.
So it makes more to sense to “start” with the universe. Where one can possibly get by using this term is asking what caused variation/lack of uniformity in a hyperfluid with an initially uniform density and 0 velocity gradient. The universe could then be defined as the interval of nonuniform density/velocity of this fluids’ existence–spurred on by nothing more than some random perturbation.
As a theory, God is simply defined as an initial object in the category of events. As an initial object it can explain any arbitrary thing, and this is precisely the problem. Science strives to find a necessary and sufficient theory such that the universe (the set of observable events) models that theory. While God is clearly a sufficient theory, there is no reason to suggest it is necessary.