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What representations of the natural numbers have been used, historically, and who invented them? Are there any notable advantages or disadvantages?

I read about Frege's definition not long ago, which involves iterating over all other members of the universe; clearly not possible in set theories without a universal set. The usual one today is

  • $0 = \{\}\text{ (sometimes }\{\{\}\}\text{?)}$
  • $S(n)=n\cup\{n\}$

but I know that another early definition was

  • $0=\{\{\}\}$
  • $S(n)=\{n\}$

Unfortunately I don't know who first used or popularized these last two, nor whether there were other early contenders.

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The second definition doesn't generalize at all to ordinals whereas the first generalizes very well; you get the von Neumann ordinals (en.wikipedia.org/wiki/…) this way and perhaps von Neumann popularized this definition. –  Qiaochu Yuan Nov 25 '11 at 23:49
    
@Qiaochu: It was indeed von Neumann who came up with it, and I would suspect popularized it as well, perhaps Bernays which worked with von Neumann had his part in that too. The second notation is by Zermelo. –  Asaf Karagila Nov 26 '11 at 0:00
    
@AsafKaragila: I suspected it was Zermelo in the second case, thanks for confirming that. :) –  Charles Nov 26 '11 at 6:33
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2 Answers

up vote 8 down vote accepted

Cantor defined the ordinals in his early work. Zermelo later proved that under the axiom of choice every set can be well ordered.

Well orders are very rigid in the sense that if $A\cong B$ are two well ordered sets then the isomorphism is unique. This allows us to construct explicit well orders for each order type.

Zermelo's ordinals were $\varnothing=\{\}$ for $0$, and $n+1=\{n\}$ for successors. The set of natural numbers is simply all the finite ordinals, however this could not have been defined in his original theory.

However in his set theoretic work, von Neumann popularized the axioms added by Fraenkel and Skolem (as well by himself) to Zermelo's early work on axiomatic set theory. He added the axiom of foundations and the schema of replacement. He then continued to define ordinals as transitive sets which are well ordered by $\in$. This work was perhaps popularized even further by Bernays and Goedel when they developed the extension of ZF which allows proper classes.

von Neumann's definition was that $0=\varnothing$ and $n+1=n\cup\{n\}$. This is a terribly convenient definition, since it allows us to say with clarify that $n< m\Leftrightarrow n\in m$, furthermore it allows us to define the natural numbers as the least inductive set.

von Neumann's idea easily carried over to the transfinite ordinals as well, I am not sure about Zermelo's convention. I do recall reading that Zermelo was less worried about ordinals, though.

(I also suggest the Introduction section of Akihiro Kanamori's The Higher Infinite in which he writes about the history of set theory. It might not be exactly about ordinals or the natural numbers, but it is an interesting read and it gives a few insights to add on what I have written above.)

Added: When searching more on the topic, I ran into Kanamori's recent paper cited below. There he says that Bernays worked with Zermelo, and it is seems that he was the one popularizing the Zermeloian definitions, as von Neumann's work relied on well foundedness of $\in$, and the replacement schema.

  1. Akihiro Kanamori, Bernays and Set Theory. The Bulletin of Symbolic Logic, Vol. 15, No. 1 (Mar., 2009), pp. 43-69
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There are also some brief but interesting historical notes in both Enderton's classic ELEMENTS OF SET THEORY and Yiannis Moschovakis's equally wonderful NOTES ON SET THEORY. But I don't think he's going to find a better short answer then the one you gave him-well done. : ) –  Mathemagician1234 Nov 26 '11 at 1:29
    
Fantastic answer. @Mathemagician1234: Actually I've read Enderton; that's probably where much of my hazy recollection in the question came from. Maybe I'll look up Moschovakis as well, thanks for the recommendation. –  Charles Nov 26 '11 at 6:27
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Maybe this book might be useful for you, too. I'll include a short quote from §1.2 Natural numbers.

Ebbinghaus et al.: Numbers, p.14

Counting with the help of number symbols marks the beginning of arithmetic. Computation counting. Until well into the nineteenth century, efforts were made to trace the idea of number back to its origins in the psychological process of counting. The psychological and philosophical terminology used for this purpose met with criticism, however, after FREGE's logic and CANTOR'S set theory had provided the logicomathematical foundations for a critical assessment of the number concept. DEDEKIND, who had been in correspondence with CANTOR since the early 1870's, proposed in his book Was sind und was sollen die Zahlen? [9] (published in 1888, but for the most part written in the years 1872—1878) a "set-theoretical" definition of the natural numbers, which other proposed definitions by FREGE and CANTOR and finally PEANO'S axiomatization were to follow. That the numbers, axiomatized in this way, are uniquely defined, (up to isomorphism) follows from DEDEKIND'S recursion theorem.

Dedekind's book Was sind und was sollen die Zahlen? seems to be available online: Wikipedia article on Richard Dedekind gives two links, one at ECHO and one at GDZ.

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