A Tower of The Theory of Mathematics
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There are several ways of constructing elementary mathematics. Some well known types of them are , philosophic method, intuition method, and formalized method. They are all used to solve the problem ---- the lack of a taugh basis for nowaday mathematics. In fact, people always "learn" the usage of natural numbers from their everyday lives, and take for granted the truth of those properties of natural numbers.
This phenomenon can also be found in understanding of real numbers . We always agrees the communitivity and associativity of standard additive and multiplicative operations on real numbers, and then the theory of Calculus and mathematical analysis was built up.
We shall now focus on the last one ----- The Formalized method, ane the thinking with the perference of formalized method is called "formalism". In this way people give a strict rule for specialize language for mathematical, which contains the following parts: several symbols (e.g. alphebats, characters), a grammar to construct a "complete sentence", a list of chosen sentences to start deduction, and a list of rules to operate sentences (this process is called deduction).
Formalists (those who perfer formalism) deals with all basic mathematics by (axiomatic) set theory, which is a powerful tool to support all other subjects in mathematics. There are some useful types of axiomatic set theories, the Zermelo-Fraenkel-Choice Axiom (ZFC) Set Theory, the Von Neumann-Bernays-Goedel (NBG) Set Theory, and the Morse”VKelley (MK) Set Theory, with the same goal, similar thoughts, but different expressions.
The following is a discussion of the first type of set theory. Note that formalistic thoughts emphasize the structure the sentences. Only with a "complete enought" grammatic, deduction convention, can mathematical concepts precisely develop.
However, we omit the illustration of grammatic and deduction parts. We believe the familiarity of formal logic for most people.
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