Two Questions about the Inductive Set

[The Student]

My notes have,

Definition: A set of real number is called an inductive set if

a) The number 1 is in the set.
b) For every x in the set, x + 1 is in the set also.

The set R⁺ of positive real numbers is an example of an inductive set. [The number 1 is in R⁺ because 1 > 0. And if x is in R⁺ (so that x > 0), then x + 1 is in R⁺ (since x + 1 > 1 > 0).]

Question 1: Does "every x" (in part b) mean every element in the set?

Definition: A real number that belongs to every inductive set is called a positive integer; such a number is necessarily positive because R⁺ is an inductive set.

Question 2: What kind of set do you think it's referring to? I ask this because I can find a set that has a 1 but no integers in it, such as {1/n: n ∈ ℕ}.

 

[HallsofIvy]

My notes have,




Question 1: Does "every x" (in part b) mean every element in the set?

Yes!

Question 2: What kind of set do you think it's referring to? I ask this because I can find a set that has a 1 but no integers in it, such as {1/n: n ∈ ℕ}.

Yes, your set satisfies (1) above but that is NOT an "inductive set" because it does not satisfy (2). It does not contain "1+ 1"= 2. The only inductive set that is a subset of the positive integers is the set of positive integers itself. If we allow subsets of the rational numbers then there are many "inductive sets": {1/2, 1, 3/2, 2, 5/2, 3, ...} or {1/3, 1, 4/3, 2, 7/3, ...} each of which contains the positive integers as a subset. But we normally restrict "inductive sets" to the positive integers in order to make use of the fact that the only inductive subset of the positive integers is the entire set itself.

 

[The Student]

Yes!






Yes, your set satisfies (1) above but that is NOT an "inductive set" because it does not satisfy (2). It does not contain "1+ 1"= 2. The only inductive set that is a subset of the positive integers is the set of positive integers itself. If we allow subsets of the rational numbers then there are many "inductive sets": {1/2, 1, 3/2, 2, 5/2, 3, ...} or {1/3, 1, 4/3, 2, 7/3, ...} each of which contains the positive integers as a subset. But we normally restrict "inductive sets" to the positive integers in order to make use of the fact that the only inductive subset of the positive integers is the entire set itself.

Oh yeah, thank-you very much!