Analysis I (Volume 1) by Terence Tao

By Terence Tao

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5 are true. We will make this assumption a bit more precise once we have laid down our notation for sets and functions in the next chapter. 12. We will refer to this number system N as the natural number system. , we could have the Hindu-Arabic number system {0, 1, 2, 3, ... } and the Roman number system {O,I,II,III,IV,V,VI, ... }, and if we really wanted to be annoying we could view these number systems as different. But these number systems are clearly equivalent 24 2. The natural numbers (the technical term is isomorphic), because one can create a oneto-one correspondence 0 +-+ 0, 1 +-+ I, 2 +-+ I I, etc.

How are we to define what the natural numbers are? 1. (Informal) A natural number is any element of the set N := {0,1,2,3,4, ... }, which is the set of all the numbers created by starting with 0 and then counting forward indefinitely. We call N the set of natural numbers. 2. In some texts the natural numbers start at 1 instead of 0,. but this is a matter of notational convention more than anything else. In this text we shall refer to the set {1, 2, 3, ... } as the positive integers z+ rather than the natural numbers.

But these number systems are clearly equivalent 24 2. The natural numbers (the technical term is isomorphic), because one can create a oneto-one correspondence 0 +-+ 0, 1 +-+ I, 2 +-+ I I, etc. , if 2 corresponds to I I, then 2++ will correspond to I I++). 13. Since all versions of the natural number system are equivalent, there is no point in having distinct natural number systems, and. we will just use a single natural number system to do mathematics. 7), and it will be the only assumption we will ever make about our numbers.

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