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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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?

vrtuljakwb · Accepted Answer

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 &quot;the same&quot; - the elements just have different names.So you might as well use the traditional names 1,2,3,….

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

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