motivation for von Neumann ordinals

The idea of the von Neumann ordinal can be traced back to the followingwell-known fact: for any natural number $n$, there are exactly $n$natural numbers which are less than $n$. For instance, the set

$$\{0,1,2,3,4\}$$

has $5$ elements, the set

$$\{0,1,2,3,4,5,6\}$$

has $7$ elements, etc.

To obtain von Neumann ordinals, we turn this idea around. Instead oftaking it for granted that numbers exist (and have certain properties),we want to start with the more primitive notion of set and definenumbers (and derive their properties). The way to define a number isas a set of objects which have that number of elements. For instance,consider counting on fingers — in that case, a set of fingers standsfor a number. We will apply the same idea here in a more sophisticatedform — the counters we will use are not going to be fingers or beadson an abacus, but abstract elements of an abstract set.

To do this, we turn the observation made earlier around anddefine a natural number to be the set of all natural numbersless than it. At first sight, this definition appears circular, butupon closer examination, we see that it is legitimate. The reasonis that to define a particular number, we only need to make useof the numbers smaller than it as counters, so cn use our definitionrepeatedly to express numbers as sets.

To begin, we notice that, since there are no natural numbers smallerthan zero, we represent zero by the empty set. Next, since the onlynumber smaller than $1$, is zero, which corresponds to the empty set,we see that $1$ corresponds to the set whose only element is theempty set, i.e. $1=\{0\}=\{\mathrm{\varnothing }\}$. Then we can go on toexpress all other numbers in terms of the empty set in a manner whichmay be explained with a typical example:

$4$$=\{0,1,2,3\}$

$=\{\mathrm{\varnothing },\{0\},\{0,1\},\{0,1,2\}\}$

$=\{\mathrm{\varnothing },\{\mathrm{\varnothing }\},\{\mathrm{\varnothing },\{\mathrm{\varnothing }\}\},\{0,1,\{0,1\}\}$

$=\{\mathrm{\varnothing },\{\mathrm{\varnothing }\},\{\mathrm{\varnothing },\{\mathrm{\varnothing }\}\},\{\mathrm{\varnothing },\{\mathrm{\varnothing }\},\{\mathrm{\varnothing },\{\mathrm{\varnothing }\}\}\}.$

As we already see in this example, this representation of integersin terms of the empty set is extremely clumsy. While it is of littleuse in practical application (even tally marks or Roman numerals aremore concise) it is of use theoretically because it is easy todefine the basic operations on numbers in terms of set-theoreticaloperations.

As an example of such a definition, we note that the ordering relation —given two numbers $m$ and $n$, we have exactly when $m\subset n$as sets. As it turns out, our numbers are totally ordered, in factwell-ordered under this relation, so our numbers are ordinal numbers,hence the name “von Neumann ordinals”.

Another important example is the successor function. Thinking for aminute about how we define a number as the set of numbers smallerthan itself, we see that the next number is gotten by adding theset denoting the previous element to itself as an element, in symbols,$n+1=n\cup \{n\}$. For example, we have

$$4+1=4\cup \{4\}=\{0,1,2,3\}\cup \{4\}=\{0,1,2,3,4\}=5.$$

Using this definition, one may do things like derive the Peanoaxioms from the axioms of set theory. From a foundational pointof view, that derivation is important because it shows that it isnot necessary to separately postulate natural numbers, but thatthey arise naturally from set theory.

Finally, this definition applies equally well to transfinite numbers.For instance, consider the first transfinite ordinal $\omega$. Bydefinition, this is the ordinal number of the ordered set of naturalnumbers. In our scheme, we simply define $\omega$ to be the set ofall natural numbers. Furthermore, given any well-ordered set,one can show by transfinite induction that it is isomorphic to somevon Neumann ordinal, so all ordinal numbers are represented.