In Apostol's book we start by defining a set called the set of real numbers which satisfies the field and order axioms. Then we define the set of positive integers as being the subset of every inductive set(i.e. sets which contain 1 and contain x+1 whenever it contains x) of real numbers. From the field and order axioms and the the definition of positive integers how can one prove that the sum of two positive integers is also a positive integer? Also their product? I attempted this problem by restating it as follows: Assume x,y belong to the set of positive integers. then we may reach x+y by successive operation of adding 1 to x ( i.e x+1, (x+1)+1, ((x+1)+1)+1, etc are all positive integers and x+y =((...(x+1)+1)+1)...)+1 where the plus one(+1) operation is repeated y times. How can one prove that x+y is that number which is obtained from x by this repeated operation?
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$\begingroup$ Your method will work. You want to argue that each of the numbers $x, x+1, x+2...,x+y$ is in each inductive set and, therefore, is in the set of integers. $\endgroup$– John DoumaCommented Jan 2, 2022 at 5:24
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1$\begingroup$ In order to do such a proof you would also need an inductive type definition of a+b. One I saw once was (roughly): a+1=successor of a, and if a+b has been defined, then a+(b+1) is defined as (a+b)+1. $\endgroup$– coffeemathCommented Jan 2, 2022 at 5:25
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