How is the modernization system of Infinitely small calculus developed?

In the historical period of modern times, the calculus of an infinitely small number is expanded on the super real number * R, and the calculus of the traditional real number system R will gradually fall behind the scenes. Why?

Calculus in the traditional real number R is represented by a set of statements. Generally, two expressions containing function symbols are connected by equations or inequalities to form a declarative sentence. Our goal is to transfer the declarative sentence on R to the super real number * r so that it retains the Original Meaning (true or false, and no meaning). Then, in the * r system, a simple and intuitive mathematical proof is provided using an infinitely small method, which indirectly proves the original proposition on R.

The idea of "turning" the modern calculus system is very clear, and the advantages and disadvantages can be seen through actual comparison. In 1973, an infinitely small Calculus Teaching Experiment in Chicago proved this point completely. China is a big country, and tens of millions of students need to learn calculus every year. The advantages and disadvantages of teaching methods are very important.

J. Keisler
The following transfer principles are introduced in the calculus textbook ":

Transfer Principle

Every real statement that holds for one or more special real functions holds for the hyperreal natural extensions of these functions.

This "transfer principle" refers to the meaning mentioned above. So, let's ask: is it possible that in the super real number system, an infinitely small method is used to prove a theorem, but it is not proved in the traditional real number system? Undoubtedly, this is certainly possible. Why not? What are the doubts about this?

How is an infinitely small method equivalent to (ε, Delta) Limit Theory? J. Keisler is clearly stated in the tutorial book. After clerk Xue Lili transcribe the book, we will tell you in the first place. The cloud around the infinitely small method will go with the wind. What a clerk does is black and white.

Laveniz (1646-1716) invented the "infinitely small" ideal number at the age of 29 and has been favored by people so far. According to the modern model theory, an infinitely small number is a super real number that is closer to zero than a real number. Therefore, we need to use the thinking logic "microscope" to observe their existence. Is there a gap between zero and non-zero real numbers? Of course, there are infinite super real numbers in this gap. To put it bluntly, there are many real number system vulnerabilities. 1948, 28-Year-Old Edwin

How is the modernization system of Infinitely small calculus developed?

Hewitt was surprised to find these vulnerabilities. Some people like to say bad things about infinity, but none of them say it is incorrect in mathematics theory. If I had a daughter, I would have taught her an infinite number of ways to do things.