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I have been trying to get my head round ZF set theory and Peano's axioms, but I have hit some confusion over Peano's definition of the successor function, or more accurately von Neumann's model.

Why did von Neumann use $S(x) := x \bigcup \{x\} $ and not just plain old $S(x) := \{x\} $? The latter seems a lot more simple and easy to work with, so am I missing some major advantage of the former, or is my function incompatible in some way?

Edit: is the only advantage just that the cardinality of the set is equal to the value it represents?

Cyclic3
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    We'd like that the cardinality of set $n$ will be the natural number $n$, via bijections, and also that sets will be transitive. Your idea doesn't do that. – Henno Brandsma Jun 27 '19 at 22:08
  • When you say the "only" advantage, you perhaps minimise the convenience of the advantage you have. Often in mathematics there are inferior definitions which require more work - why not choose the easiest and most natural? Sometimes the easies definition is hard won (read the history of the ideas) and the simplicity of the idea belies the effort behind it. When I read GH Hardy or other masters, for example, I want to do better, but discover by personal sweat how efficient their exposition is. – Mark Bennet Jun 27 '19 at 22:42
  • @MarkBennet I suppose. I am coming to understand that a concise way of accessing the properties and "values" of these sets is a critical part of interacting with these systems, and not just a nice thing to have. – Cyclic3 Jun 27 '19 at 23:03

2 Answers2

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It is harder to work with. For example, we can "check" whether an arbitrary set $x$ is a von Neumann natural:

$x$ is finite and transitive and well-ordered by $\in$.

where finite can be formalized as, e.g., "every injective map $x\to x$ is onto" and transitive is defined as "every element is also a subset". Try to find a similar characterization for the alternative (without using "$\ldots$" anywhere). Or try to find a simple way to express the order relation between natural numbers in a simple way for the sets representing the numbers (for von Neumann we have $x<y$ iff $x\in y$ iff $x \subsetneq y$).

Henno Brandsma
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Hagen von Eitzen
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    Another huge convenience is that von Neumann's definition easily extends to define the whole class of ordinal numbers -- the other definition can't possibly do that. – Bob Krueger Jun 27 '19 at 22:10
  • Ah, that makes a lot of sense! Formalising the comparison in that way is very elegant. – Cyclic3 Jun 27 '19 at 22:18
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The most immediately obvious benefit of the Von Neumann representation is that the set that represents the number $n$ has exactly $n$ elements. This makes it technically easy to use the representation to reason about counting.

It also provides a connection to a "naive" definition of numbers where, for example, the number two is regarded as "the property that a set may have that it has one more than one element", and represented by the set of all sets that have exactly two elements. Unfortunately such a set can't actually exist in standard set theory, but the Von Neumann definition points to a particular representative of this class that we can use to represent it instead.

A more advanced -- but technically quite important -- benefit of the Von Neumann naturals is that they generalize directly to a representation of transfinite ordinal numbers. Representing finite numbers as towers of singletons has no natural continuation beyond the finite ones, but the Von Neymann representation does.

hmakholm left over Monica
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