Inductive sets

Inductive sets are sets such that

1. \emptyset\in A, where A is the inductive set under consideration.
2. x\in A\implies x^+\in A, where x^+=x\cup \{x\}

Natural numbers (here, natural numbers include 0) are those sets that are present in every inductive set. Let us explore this strand of thought more.

Clearly, \emptyset belongs to every inductive set. Hence, it also belongs to the set of natural numbers (or \Bbb{N}). Also, seeing as 0^+,0^{++},\dots belong to every inductive set, all these elements make up the set of natural numbers.

An inductive set that is a proper superset of \Bbb{N} would be \{0,0.5,1,1.5,2,2.5,\dots\}. The successor of every element is there in this set, and it also contains 0. Hence, it is an inductive set.

EDIT: I forgot to mention the motivation behind writing this post. To prove a property for all elements in an inductive set, we have to prove it for all elements without predecessors, and then show that if $a$ satisfies the property, then so does a^+. For example, in the inductive set \{0,0.5,1,1.5,2,2.5,\dots\}, we have to prove the property for 0 and 0.5, and then also prove that a satisfies the property \implies a^+ satisfies the property. Because the only starting point of \Bbb{N} (element without predecessor) is 0, now you  know how and why induction works in natural numbers.

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Graduate student

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