Osborne's Rules

|sin(z)| = (sin^2(x) + sinh^2(y))^(1/2)

|cos(z)| = (cos^2(x) + sinh^2(y))^(1/2)

Axiom of Foundation (Axiom of Regularity)

∀A ≠ ø, ∃x∈A such that x∩A = ø

Cardiinals and Ordinals

Both cardinals and ordinals are generalizations of natural numbers. But they are different in many ways. What follows is some of my observations:

1. Commutativity: Cardinals are commutative, while ordinals are not. Here's an simple example:

In ordinal arithmetic, 1+ω = ω ≠ ω+1, and it is because what I call the "order stuff". 1+ω = ord(1Uω) and it is easy to see that ω has one-to-one correspondance with ord(1Uω) with a proper order, while in the case ω+1 = ord(ωU1), we don't have such correspondance. Why? It is because we can't map 1 to any natural numbers, or we will mess up with order. By definition, 1 has to be mapped "after" all the elements of ω, and this is the reason why we don't have ordinal commutativity in general.

In cardinal arithmetic, 1+ω = ω+1 = ω, because this time we can map 1 in ω+1 to some natural number without to be worried about the order stuff.

A very important thing to remember is, the cardinality of a set A is the least ordinal that has one-to-one correspondance with A.

Chain

A chain C is a subset of a partial order subset P which is linearly ordered.

Example:

Let P = {1, 2, 3, 4, 4.5, 5, 5.8, 6}, and (P, ≤) is a partial order, where ≤ stands for the usual order of natural numbers.

Then: A = {1, 2, 3}, B = {1, 2, 4, 5, 6} are chains.