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Below is taken from page 41 of the book Classic Set Theory.

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The set of natural numbers is defined as the smallest inductive set.

How do I show that each natural number is the set of all its predecessors, for example $10=\{0,1,2,3,4,5,6,7,8,9\}$ and why does this observation "the set $n$ contains intuitively, $n$ elements" involves a circularity?

Little Rookie
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  • Well, you know $\forall m<n,;m\subseteq n$ – Rushabh Mehta Oct 19 '18 at 13:24
  • Your question "How do I show that each natural number is the set of all its predecessors" will depend on how natural numbers are defined in the book you are quoting from. But you did not include the definition in your question, making it impossible to answer. – Lee Mosher Oct 19 '18 at 13:29
  • @RushabhMehta it should be $m$ is a proper subset of $n$ – Little Rookie Oct 19 '18 at 13:33
  • Prove that the set ${n\in\mathbb N\mid n\subseteq\mathbb N}\subseteq\mathbb N$ is inductive. That implies ${n\in\mathbb N\mid n\subseteq\mathbb N}=\mathbb N$ or equivalently "every element of a natural number is again a natural number". – drhab Oct 19 '18 at 14:34
  • @drhab I have done the proof. – Little Rookie Oct 19 '18 at 14:38
  • Good. Does this result bring satisfaction concerning your question? – drhab Oct 19 '18 at 14:39
  • @drhab No. I still don't see a reason why $10 = {0,1,2,3,4,5,6,7,8,9}$ – Little Rookie Oct 19 '18 at 14:42
  • $10$ is defined as $9\cup{9}$ so if $9={0,1,2,3,4,5,6,7,8}$ then $10={0,1,2,3,4,5,6,7,8,9}$. This shifts the question to: how to prove that $9={0,1,2,3,4,5,6,7,8}$? Again this question can be shifted and within a finite number of shiftings it is enough to prove that $0=\varnothing$ which is true by definition. – drhab Oct 19 '18 at 14:48
  • You could prove that the set ${n\in\mathbb N\mid n\text{ is the set of all its predecessors }}$ is an inductive set. Tricky however is the question: "what exactly is a predecessor"? – drhab Oct 19 '18 at 14:51
  • @drhab So it all just because of the design of symbols for numbers? Because we name the empty set as 0, its successor as 1, the successor of 1 to be 2, and so for. – Little Rookie Oct 20 '18 at 01:31
  • Essential is that every element of a natural is again a natural number (see my first comment). Then if you start naming them by the familiar symbols $0,1,2,\dots$ as done in your question then this leads to things like $10={0,1,2,3,4,5,6,7,8,9}$. – drhab Oct 20 '18 at 06:47

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