Now let's discuss some related topics of interest. People have different views on these topics. For me, it is the exclamation point in the table below! The number represents the degree to which it promotes my work.

Topic a: the source of interest in set theory

Basic Mathematics/Applied to Philosophy!

The application of mathematics! ! !

Historical reasons! ! !

Internal development! ! ! !

Beauty! ! ! ! ! ! ! ! !

The fun of proof! ! ! ! !

Generalization! ! ! ! ! !

Game entertainment [plus hot rules]! ! !

We can also use these topics to classify the current work and scholars' evaluation of set theory, so we will focus on their differences below. To a great extent, I was attracted by mathematics and then mathematical logic because of their generalization, and I think my generalization is correct; It seems that I was wrong. I feel that examples often confuse you: special nature is just a trap, because it doesn't hold water in general. Pay attention to "generalization" I mean that I would rather take the general first-order completeness theory as the research object than the limited Morley rank simple group as the research object, but my creed is not "don't just look at the tree, don't look at the forest". To deal with every problem, we should deal with it according to its characteristics. Finding the application of your own field to other fields is to show what others will be interested in; But give you a question, why not do your best to popularize it to the greatest extent? Of course, if the theorem has been proved that the additional generalization is common, it is also boring.

On the other hand, many of my colleagues, including some of the best brains in the field of set theory, surprised me with their inferiority complex. Facing mathematicians, many of them feel inferior. It seems that there are mathematicians here and logicians here, both of which are irrelevant fields. They think that mathematicians really work in deeper, more difficult, richer and more meaningful fields, so we mathematical logicians must find "mathematical logic", which leads to the application of mathematics, and logicians have done a lot of work, just like Abraham Robinson School did. Now I like to prove theorems in many fields of mathematics as long as I can, but I don't like this servile attitude in the field of mathematical logic.

Many other people have done a lot of work in giving play to the role of set theory in mathematics foundation and philosophy. I have no objection to this, but I have questions. My feeling is similar to that of many writers: they understand the role of critics in cultural life, but they think that sticking to critics' thoughts will only produce boring works, and these thoughts themselves will shine forever because of their inner beauty. Some people also complain that set theory has lost its "good old days". At that time, it was proved to be composed of ideas, not as technical as it is now. Generally speaking, I am not a supporter of Old Times, because I neglected your technical ability at that time, but technology is my banner. In many cases, technology is not a routine matter to realize ideas, but plays a role in all aspects such as organization and ideas. These technologies are quite difficult and usually contain important new ideas. My feeling, exaggerated, is that the aesthetic feeling of set theory is eternal, but its philosophical value is guided by the trend. And I feel that the words of these complainants are contradictory. For example, some of them say that mathematical logic is now more mathematical than before, while others say that what it deals with is meaningful. By the way, these contradictory views are not contradictory in practice, and many people support more than one.

About the aesthetic feeling of set theory, I mean the aesthetic feeling of defining, theorem and proving the harmonious position in a structure. But I'm not afraid of complicated proofs. When I was an undergraduate, I found that Galois' theory was beautiful in boekhoff McLay's book, and later I found that Molly's theory and its proof were beautiful. Tired readers may be furious: "Beauty?" You can find traces of beauty in your mess. ",I can only say that each has its own hobbies, and mine is like this.

Topic b: the framework of set theory

ZFC (Eight Axioms of Zemello-frankl+axiom of choice)! ! ! ! ! ! !

Force method! ! ! !

Internal model! ! !

Big base! ! !

ZF+ Dependent axiom of choice (DC)+ Some forms of decisive axioms!

This is a reasonable but overlapping division. In any case, we have proved this theorem within the framework of ZFC. From the point of view of the supporters of ZFC framework, the proof theorem is proved within the framework of ZFC, and other frameworks are auxiliary, which I quite agree with. The compulsory method tells us that when a theorem cannot be proved, the consistency can be proved with a large radix, and when we are lucky, the large radix can be arranged in linear order to compare the sizes. Finally, it is necessary to use the internal model to explain that a large cardinality is necessary, or a better equivalent result can be obtained. My feeling is that apart from the results of coordination, the ZFC framework has covered our intuitive scope, so a proof refers to a proof under the ZFC framework, which is of course a powerful proof that the ZFC framework is reasonable. In essence, the method of strengthening coercion tells us that all set domains are equally legal, so it is not representative to study all set domains with special representativeness, such as the set L that can be formed. The coercive method shows that it is not important to prove the theorem or assume that the generalized continuum hypothesis is established under the ZFC framework. This is a strong conclusion of the coercive method, but I doubt that the viewpoint of this coercive method will be supported. From a compromise point of view, the mandatory method framework and the ZFC framework are complementary, and one framework negates the results in the other framework, so if you are interested in one framework, you will also be interested in the other framework. In fact, I was forced to take the forced law seriously. I want to prove that it is correct for me to use the diamond theorem on each stable subset of Alef 1 potential set when solving the Whitehead problem of Abel group cardinality, because it is continuous. The coercive method in [BD] is too weak).