用户: Dforsign/分析学讲义/数理逻辑与集合论基础

大纲:

基礎邏輯符號

公理化集合論與樸素集合論

第一個例子: 集合論中的自然數集

In order to state mathematical ideas rigorously, people abstracted a “new" language from our daily natural language. It is now called logic.

Logic is a displine that studies reasoning. We consider a reasoning consists of a premise or a set of premises, a conclusion and a set of inference rules.

For premises and conclusions, we call them prositions: statements that are either true or false.

From now on, we will use $\top$ to denote “true”, and $\perp$ to denote “false”. And we will use $⊢$ to denote “deduce”. For a complete reasoning, we use a line to seperate the premises and conclusion. Above lie the premises and beneath lies the conclusion.

例 0.1. $$\frac{\text{Cats are cute}\text{He is a cat}}{\text{He is cute}}\frac{\text{He is not cute},\{\{\text{Cats are cute},\text{He is a cat}\}⊢\text{He is cute}\}}{\text{He is not a cat}}$$

Obviously, it is the inference rules that play an important role in logic. Different rules result in different logical systems.

For this course, which is also valid for most mathematical research, we constrain ourselves to a minimal knowledge of logic, upon which the fundamental concept of mathematics, set theory, is founded. This logic is called first-order logic.

We consider the classic first-order logic the "rigorous language" for mathematics. Therefore, before the definition of it, there is no such thing as rigour. So, the classic first-order logic is written in loose natural language or "metalanguage" if one prefers logical jargon.

In short, the classic first-order logic consists of

1.	variable
2.	predicate of $n$ free variables
3.	function sign (a term coined to distinguish from function in mathematics) of $n$ variables
4.	logic signs
5.	inference rules of classic propositional logic
6.	inference rules of classic predicate logic

More precisely, a variable is just a variable. It is a term to represent a varying object. It is just like what we learned in high school.

A predicate is a proposition with $n$ free variables. And a free variable means there is no qualifier before the variable mentioned above.

A qualifier is a certain type of logic signs: $\forall$ (for all) and $\exists$ (exist).

There are two more types of logic signs: "logical connective" and "truth value".

Logical connectives are $\neg$, $\wedge$, $\vee$, representing "negtive", "and", "or".

Truth values are as mentioned above.

Inference rules of classic propositional logic are straightforward yet cumbersome postulates. Including

Inference rules of classic propositional logic.

1.	$$\frac{}{\phi ⊢\phi }\frac{\Gamma ⊢\chi }{\Gamma ,\phi ⊢\chi }(\text{structural rule})$$
2.	$$\frac{\Gamma ⊢\phi \Gamma ⊢\neg \phi }{\Gamma ⊢\perp }(\text{Reductio ad absurdum})\frac{\Gamma ⊢\perp }{\Gamma ⊢\chi }(\text{Law of Explosion})$$
3.	$$\frac{\Gamma ⊢\neg (\neg \chi )}{\Gamma ⊢\chi }(\text{Law of Excluded Middle})$$
4.	$$\frac{\Gamma ,\phi ⊢\perp }{\Gamma ⊢\neg \phi }\frac{\Gamma ⊢\neg \phi }{\Gamma ,\phi ⊢\perp }(\text{Law of Negation})$$
5.	$$\frac{\Gamma ⊢{\chi }_1\Gamma ⊢{\chi }_2}{\Gamma ⊢{\chi }_1\wedge {\chi }_2}\frac{\Gamma ⊢{\chi }_1\wedge {\chi }_2}{\Gamma ⊢{\chi }_1}\frac{\Gamma ⊢{\chi }_1\wedge {\chi }_2}{\Gamma ⊢{\chi }_2}(\text{Law of And})$$
6.	$$\frac{\Gamma ⊢{\chi }_1}{\Gamma ⊢{\chi }_1\vee {\chi }_2}\frac{\Gamma ⊢{\chi }_2}{\Gamma ⊢{\chi }_1\vee {\chi }_2}$$$$\frac{\Gamma ⊢{\chi }_1\vee {\chi }_2\Gamma ,{\chi }_1⊢\phi \Gamma ,{\chi }_2⊢\phi }{\Gamma ⊢\phi }(\text{Law of Or})$$
7.	$$\frac{\Gamma ,\chi ⊢\phi }{\Gamma ⊢\chi \to \phi }(\text{Deduction Theorem})$$
8.	$$\frac{\Gamma ⊢\chi \to \phi \Gamma ⊢\chi }{\Gamma ⊢\phi }(\text{Modus ponens})$$