more rigorous. In this course, you should learn how to formalize a problem and make your thinking rigorous 4 Order Logic for application adopts a special set of its own notations. For convenience to read text book we just follow the terminology in text book. Then, we need change the definition of partial order, which are slightly different with the one learned in last semester. And tree is also a very important tool to our class. And it is totally represented in an order approach, as a partial order In Logic for application, partial order is defined as following: Definition 1(Partial order). A partial order is a set S with a binary relation on S, which transitive and irreflexive You should mind that partial order is irreflexive other than reflexive defined in previous semester and the most of text book on set theory Whatever the change of partial order, linear order is always the same. Definition 2(Linear order). A partial order is a linear order, if it satisfies the trichotomy law: <y or a=y or y <i As we know, there are many partially ordered sets. For further investigation, we can divide them into two categories, something bad and something good. Here, we define what is good as followin Definition 3(Well ordering). A linear order is well ordered if every nonempty set A of S has a least element The set N is a reference of countable set. countable is defined base on set of natural number Definition 4(Countable). A set A is countable if there is a one-to-one mapping from A to M As a rational number can be represent as a pair of two natural number. Q is a set of countable elements, which also contains natural number as a very small part of it Specially, when we can count a set clearly, here we mean a exact number, we define the following Definition 5(Finite). A set A is finite if there is a one-to-one mapping from A to (0,1,.,n-1 or some n∈ In this semester, finite is our favor and unfortunately evil too. As a student who study computer area, here we just mean building machine or coining software, infinite number of objects is always out of our control With these two, we can define its negative side Definition 6. 1. If a is not countable, it is uncountablemore rigorous. In this course, you should learn how to formalize a problem and make your thinking rigorous. 4 Order Logic for application adopts a special set of its own notations. For convenience to read text book, we just follow the terminology in text book. Then, we need change the definition of partial order, which are slightly different with the one learned in last semester. And tree is also a very important tool to our class. And it is totally represented in an order approach, as a partial order. In Logic for application, partial order is defined as following: Definition 1 (Partial order). A partial order is a set S with a binary relation < on S, which is transitive and irreflexive. You should mind that partial order is irreflexive other than reflexive defined in previous semester and the most of text book on set theory. Whatever the change of partial order, linear order is always the same. Definition 2 (Linear order). A partial order < is a linear order, if it satisfies the trichotomy law: x < y or x = y or y < x. As we know, there are many partially ordered sets. For further investigation, we can divide them into two categories, something bad and something good. Here, we define what is good as following. Definition 3 (Well ordering). A linear order is well ordered if every nonempty set A of S has a least element. The set N is a reference of countable set. countable is defined base on set of natural number. Definition 4 (Countable). A set A is countable if there is a one-to-one mapping from A to N . As a rational number can be represent as a pair of two natural number. Q is a set of countable elements, which also contains natural number as a very small part of it. Specially, when we can count a set clearly, here we mean a exact number, we define the following concept. Definition 5 (Finite). A set A is finite if there is a one-to-one mapping from A to {0, 1, . . . , n−1} for some n ∈ N . In this semester, finite is our favor and unfortunately evil too. As a student who study computer area, here we just mean building machine or coining software, infinite number of objects is always out of our control. With these two, we can define its negative side. Definition 6. 1. If A is not countable, it is uncountable. 3