motivation for von Neumann ordinals

motivation for von Neumann ordinals

The idea of the

von Neumann ordinal

can be traced back to the following
well-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 of
taking it for granted that numbers exist (and have certain

properties

),
we want to start with the more primitive notion of set and define
numbers (and derive their properties). The way to define a number is
as a set of objects which have that number of elements. For instance,
consider counting on fingers — in that case, a set of fingers stands
for a number. We will apply the same idea here in a more sophisticated
form — the counters we will use are not going to be fingers or beads
on an abacus, but abstract elements of an abstract set.

To do this, we turn the observation made earlier around and

define

a natural number to be the set of all natural numbers
less than it. At first sight, this

definition

appears circular, but
upon closer examination, we see that it is legitimate. The reason
is that to define a particular number, we only need to make use
of the numbers smaller than it as counters, so cn use our definition
repeatedly to express numbers as sets.

To begin, we notice that, since there are no natural numbers smaller
than zero, we

represent

zero by the

empty set

. Next, since the only
number smaller than

1

, is zero, which corresponds to the empty set,
we see that

1

corresponds to the set whose only element is the
empty set, i.e.

1

=

{

0

}

=

{

∅

}

. Then we can go on to
express all other numbers in terms of the empty set in a manner which
may be explained with a typical example:

4

=

{

0

,

1

,

2

,

3

}

=

{

∅

,

{

0

}

,

{

0

,

1

}

,

{

0

,

1

,

2

}

}

=

{

∅

,

{

∅

}

,

{

∅

,

{

∅

}

}

,

{

0

,

1

,

{

0

,

1

}

}

=

{

∅

,

{

∅

}

,

{

∅

,

{

∅

}

}

,

{

∅

,

{

∅

}

,

{

∅

,

{

∅

}

}

}

.

As we already see in this example, this

representation

of integers
in terms of the empty set is extremely clumsy. While it is of little
use in practical application (even tally marks or Roman numerals are
more concise) it is of use theoretically because it is easy to
define the basic

operations

on numbers in terms of set-theoretical
operations.

As an example of such a definition, we note that the

ordering relation

—
given two numbers

m

and

n

, we have

m

<

n

exactly when

m

⊂

n

as sets. As it turns out, our numbers are

totally ordered

, in fact

well-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 a
minute about how we define a number as the set of numbers smaller
than itself, we see that the next number is gotten by adding the
set denoting the previous element to itself as an element, in symbols,

n

+

1

=

n

∪

{

n

}

. For example, we have

4

+

1

=

4

∪

{

4

}

=

{

0

,

1

,

2

,

3

}

∪

{

4

}

=

{

0

,

1

,

2

,

3

,

4

}

=

5

.

Using this definition, one may do things like derive the

Peano
axioms

from the axioms of

set theory

. From a foundational point
of view, that

derivation

is important because it shows that it is
not necessary to separately

postulate

natural numbers, but that
they arise naturally from set theory.

Finally, this definition applies equally well to

transfinite numbers

.
For instance, consider the first transfinite ordinal

ω

. By
definition, this is the ordinal number of the ordered set of natural
numbers. In our scheme, we simply define

ω

to be the set of
all natural numbers. Furthermore, given

any

well-ordered set,
one can show by

transfinite induction

that it is

isomorphic

to some
von Neumann ordinal, so all ordinal numbers are represented.

Title

motivation for von Neumann ordinals

Canonical

name

MotivationForVonNeumannOrdinals

Date of creation

2013-03-22 16:50:40

Last modified on

2013-03-22 16:50:40

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

13

Author

yark (2760)

Entry type

Derivation

Classification

msc 03E10

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