I am beginning real analysis and got stuck on the first page (Peano Postulates). It reads as follows

DIAMMIBENVERMk1 2022-06-29 Answered
I am beginning real analysis and got stuck on the first page (Peano Postulates). It reads as follows, at least in my textbook.

Axiom 1.2.1 (Peano Postulates). There exists a set N with an element 1 N and a function s : N N that satisfies the following three properties.
a. There is no n N such that s ( n ) = 1.
b. The function s is injective.
c. Let G N be a set. Suppose that 1 G, and that g G s ( g ) G. Then G = N .

Definition 1.2.2. The set of natural numbers, denoted N , is the set the existence of which is given in the Peano Postulates.

My question is: From my understanding of the postulates, we could construct an infinite set which satisfies the three properties. For example, the odd numbers { 1 , 3 , 5 , 7 , }, or the powers of 5 { 1 , 5 , 25 , 625 }, could be constructed (with a different s ( n ), of course, since s ( n ) is not defined in the postulates anyway). How do these properties uniquely identify the set of the natural numbers?
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Answered 2022-06-30 Author has 13 answers
Yes, you can find other sets on which a successor function is defined that satisfies all the Peano axioms.

What makes the natural numbers unique is that you can use the Peano postulates to prove that when you have two such sets you can build a bijection between them that maps one successor function to the other. That means the sets are really "the same" - the elements just have different names.

So you might as well use the traditional names 1,2,3,….
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When I read this question I am like it is obvious, so I got kind of confused. But I took a crack at it anyways.

If you read the postulate. This is not obvious because you dont know if the right angle is a right angle. What if a triangle was drawn differently but with one line perpendicular to another. You need to know if the line is perpendicular or not, and that is why it is not obvious.

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