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I have heard that back in 1908 Zermelo proposed to use $\emptyset,\{\emptyset\},\{\{\emptyset\}\},\ldots$ as the natural numbers. However, later von Neumann proposed the alternative approach of defining natural numbers in the following way,

$$0=\emptyset\\1=\{0\}\\2=\{1,0\}\\\vdots$$My question is,

Where can I find the detailed construction of natural numbers in the way Zermelo proposed? It will also be very good if there be a comparative analysis of the von Neumann construction and the Zermelo construction.

I have searched internet for sometime but couldn't find anything relevant. Can anyone give me some reference regarding this?

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  • $\begingroup$ What kind of details are you looking for? Zermelo's construction can be rigorously defined by recursion as: $0 = \emptyset$, and $n + 1 = \{ n\}$, where the singleton operation is easily deduced from set theory (in ZFC, it's the pair operation applied to $n$ and $n$). Zermelo's and Von Neumann's constructions are isomorphic in any reasonable sense - you can easily verify that either one satisfies the Peano axioms. $\endgroup$
    – Dorebell
    Jun 24 '15 at 5:13
  • $\begingroup$ I actually have never read the detailed construction of natural numbers by Zermelo's premises. I am looking just for the details of that construction. By the way, I don't understand why did you say that both of the constructions are isomorphic. In von Neumann's construction, for example, we have the very interesting consequence that every predecessor of a natural number ($\ne 0$, of course) belongs to it. Whereas in Zermelo's construction the conclusion holds for the immediate predecessor and I don't see exactly how we can extend this to all predecessors. $\endgroup$
    – user170039
    Jun 24 '15 at 5:23
  • $\begingroup$ @user170039 They are isomorphic in the sense that the two sets can be placed in bijection with each other in such a way that the successor (and by extension all other natural number operations) is preserved. So the only thing that is different is the truth value of statements like $1 \in 2$ which are not used in number theory. $\endgroup$ Jun 24 '15 at 5:29
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The reason you are having difficulty finding such a reference is probably due to the fact that presentations of the natural numbers rarely deal with these construction directly. Instead we simply require an element $0 \in \mathbb{N}$ and a funciton $S : \mathbb{N} \to \mathbb{N}$ that gives the "successor" of each number. For the Zermelo naturals, $0 = \emptyset$ (as you mentioned) and $S(x) = \{ x \}$.

The typical construction proceeds by proving the Peano axioms for the given $\mathbb{N}$. The only part which will be significantly different in the case of Zermelo's naturals is proving that there exists a set $\mathbb{N}$ s.t. $\forall x \in \mathbb{N} \; S(x) \in \mathbb{N}$ (i.e. that there is a set which contains "all" the natural numbers). This requires invocation of the axiom of infinity. If you look at the definition of that axiom, you can see that the axiom in fact directly asserts that there exists a set which contains the von Neumann naturals, which is probably part of the reason presentations use this construction.

To prove the existence of such a set for Zermelo's naturals, the easiest way I can think of is establishing a bijection between the von Neumann naturals and Zermelo's, and then using the axiom schema of replacement.

The other interesting part of the Peano axioms is the induction axiom. The same proof that works for the von Neumann naturals (detailed here) will also work if we use the bijection between the Zermelo and von Neumann naturals again.

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  • $\begingroup$ Thank you for your answer. But my question explicitly asks for the details of the construction of natural numbers from "Zermelo Naturals". Its answer is supposed to be some kind of book, article, research papers and similar things of literature. I don't think your answer fulfills this criteria exactly. $\endgroup$
    – user170039
    Jun 24 '15 at 6:08
  • $\begingroup$ @user170039 I have detailed the parts of the construction that vary from the von Neumann naturals, for which you can turn up many presentations of the proofs of the Peano axioms. I suggest you look at those if you have difficulty understanding the construction, because as I said only the very beginning of the development (i.e. proving Peano's axioms) has any differences. I would be surprised if you could find a reference on this, because even number theorists and mathematical logicians don't particularly care since nobody writes out the natural numbers as sets. $\endgroup$ Jun 24 '15 at 6:18
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    $\begingroup$ As an added point, ZFC is equivalent to ZFC', where the usual axiom of infinity is replaced by one which states that there is a set containing all Zermelo naturals; we could prove that the Von Neumann ones exist from this using the same bijection/replacement argument given here. $\endgroup$
    – Dorebell
    Jun 24 '15 at 8:22

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