<The Detail of This Chapter.pdf>


0.1.1.the prologue

In any situation, we have to take some concepts about our objects, and we have to express them as clearer as possible, no matter how complicate are them. Which means we need settle down all the coarse concepts that inspired by our practice, then pick some simplest and self-explanatory concepts as littler as possible, OK, let's begin to describe our image from them!

The simplest and self-explanatory concepts: class; element; equality.

The first-order predicate: and; or; non; implicate; exist.

The basic sentences:

--element a belong to class A;

--element a is equal to element b;

--class A is equal to class B.

Then we have some axioms to express meaning of these concepts:

(1)For element a and class A, we can always judge that if a belong to A. <the fundamental relation between element and class, meanwhile, express the meaning of element and class.>

(2)["a belong to A" imply "a belong to B"; and "a belong to B" imply "a belong to A"] imply A is equal to B. <the meaning of two classes are equal>

(3)Equality has transitivity. <the fundamental property of equality>

(4)For any statement that only use the first-order predicate, exist a corresponding class A that cause the statement is true for all the element of A. <another definition of class>


From the concepts of class and element, we can get a finer concept:

If a class itself can became a element in certain situation, then we call such a class a set.

If a class itself can not became a element in any situation, then we call such a class an essential class.


the class A=: {x | x is a set, and x don't belong to x.} is an essential class.


From now on, we can express any object clearly.


power set of set X: its element is any subset of X.

a filter F on X, a subset of power set of X:

if k is a element of F, then any subset b of X that contain a as subset is a element of F;

if a and b are element of F, then their intersection is element of F;

null set is not element of F.


ultrafilter, or max filter