# Ordinal arithmetic

(Redirected from Cantor normal form)

In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set which represents the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. The so-called "natural" arithmetical operations retain commutativity at the expense of continuity.

The union of two disjoint well-ordered sets S and T can be well-ordered. The order-type of that union is the ordinal which results from adding the order-types of S and T. If two well-ordered sets are not already disjoint, then they can be replaced by order-isomorphic disjoint sets, e.g. replace S by {0} × S and T by {1} × T. This way, the well-ordered set S is written "to the left" of the well-ordered set T, meaning one defines an order on S $\cup$ T in which every element of S is smaller than every element of T. The sets S and T themselves keep the ordering they already have. This addition of the order-types is associative and generalizes the addition of natural numbers.

The first transfinite ordinal is ω, the set of all natural numbers. For example, the ordinal ω + ω is obtained by two copies of the natural numbers ordered in the usual fashion and the second copy completely to the right of the first. Writing 0' < 1' < 2' < ... for the second copy, ω + ω looks like

0 < 1 < 2 < 3 < ... < 0' < 1' < 2' < ...

This is different from ω because in ω only 0 does not have a direct predecessor while in ω + ω the two elements 0 and 0' do not have direct predecessors. As another example, here are 3 + ω and ω + 3:

0 < 1 < 2 < 0' < 1' < 2' < ...
0 < 1 < 2 < ... < 0' < 1' < 2'

After relabeling, the former just looks like ω itself, i.e. 3 + ω = ω, while the latter does not: ω + 3 is not equal to ω since ω + 3 has a largest element (namely, 2') and ω does not. Hence, this addition is not commutative.

However, addition is still associative; one can see for example that (ω + 4) + ω = ω + (4 + ω) = ω + ω.

The definition of addition can also be given inductively (the following induction is on β):

• α + 0 = α,
• α + (β + 1) = (α + β) + 1 (here, "+ 1" denotes the successor of an ordinal),
• and if β is a limit ordinal then α + β is the limit of the α + δ for all δ < β.

Using this definition, ω + 3 can be seen to be a successor ordinal (it is the successor of ω + 2), whereas 3 + ω is a limit ordinal, namely, the limit of 3 + 0 = 3, 3 + 1 = 4, 3 + 2 = 5, etc., which is just ω.

Zero is an additive identity α + 0 = 0 + α = α.

Addition is associative (α + β) + γ = α + (β + γ).

Addition is strictly increasing and continuous in the right argument:

$\alpha < \beta \Rightarrow \gamma + \alpha < \gamma + \beta$

but the analogous relation does not hold for the left argument; instead we only have:

$\alpha < \beta \Rightarrow \alpha+\gamma \le \beta+\gamma$

Ordinal addition is left-cancellative: if α + β = α + γ, then β = γ. Furthermore, one can define left subtraction for ordinals βα: there is a unique γ such that α = β + γ. On the other hand, right cancellation does not work:

$3+\omega = 0+\omega = \omega$ but $3 \neq 0$

Nor does right subtraction, even when βα: for example, there does not exist any γ such that γ + 42 = ω.

## Multiplication

The Cartesian product, S×T, of two well-ordered sets S and T can be well-ordered by a variant of lexicographical order that puts the least significant position first. Effectively, each element of T is replaced by a disjoint copy of S. The order-type of the Cartesian product is the ordinal which results from multiplying the order-types of S and T. Again, this operation is associative and generalizes the multiplication of natural numbers.

Here is ω·2:

00 < 10 < 20 < 30 < ... < 01 < 11 < 21 < 31 < ...

which has the same order type as ω + ω. In contrast, 2·ω looks like this:

00 < 10 < 01 < 11 < 02 < 12 < 03 < 13 < ...

and after relabeling, this looks just like ω. Thus, ω·2 = ω+ω ≠ ω = 2·ω, showing that multiplication of ordinals is not commutative.

Distributivity partially holds for ordinal arithmetic: R(S+T) = RS+RT. However, the other distributive law (T+U)R = TR+UR is not generally true: (1+1)·ω = 2·ω = ω while 1·ω+1·ω = ω+ω which is different. Therefore, the ordinal numbers form a left near-semiring, but do not form a ring.

The definition of multiplication can also be given inductively (the following induction is on β):

• α·0 = 0,
• α·(β+1) = (α·β)+α,
• and if β is a limit ordinal then α·β is the limit of the α·δ for δ < β.

The main properties of the product are:

• α·0 = 0·α = 0.
• One (1) is a multiplicative identity α·1 = 1·α = α.
• Multiplication is associative (α·βγ = α·(β·γ).
• Multiplication is strictly increasing and continuous in the right argument: (α < β and γ > 0) $\Rightarrow$ γ·α < γ·β
• Multiplication is not strictly increasing in the left argument, for example, 1 < 2 but 1·ω = 2·ω = ω. However, it is (non-strictly) increasing, i.e. αβ $\Rightarrow$ α·γβ·γ.
• Right cancellation does not work, e.g. 1·ω = 2·ω = ω, but 1 and 2 are different.
• α·β = 0 $\Rightarrow$ α = 0 or β = 0.
• Distributive law on the left: α·(β+γ) = α·β+α·γ
• No distributive law on the right: e.g. (ω+1)·2 = ω+1+ω+1 = ω+ω+1 = ω·2+1 which is not ω·2+2.
• Left division with remainder: for all α and β, if β > 0, then there are unique γ and δ such that α = β·γ+δ and δ < β. (This does not however mean the ordinals are a Euclidean domain, since they are not even a ring, and the Euclidean "norm" is ordinal-valued.)
• Right division does not work: there is no α such that α·ω ≤ ωω ≤ (α+1)·ω.

## Exponentiation

The definition of ordinal exponentiation for finite exponents is straightforward. If the exponent is a finite number, the power is the result of iterated multiplication. For instance, ω2 = ω·ω using the operation of ordinal multiplication. Note that ω·ω can be defined using the set of functions from 2 = {0,1} to ω = {0,1,2,...}, ordered lexicographically with the least significant position first:

(0,0) < (1,0) < (2,0) < (3,0) < ... < (0,1) < (1,1) < (2,1) < (3,1) < ... < (0,2) < (1,2) < (2,2) < ...

Here for brevity, we have replaced the function {(0,k), (1,m)} by the ordered pair (k, m).

Similarly, for any finite exponent n, $\omega^n$ can be defined using the set of functions from n (the domain) to the natural numbers (the range). These functions can be abbreviated as n-tuples of natural numbers.

But for infinite exponents, the definition may not be obvious. A limit ordinal, such as ωω, is the supremum of all smaller ordinals. It might seem natural to define ωω using the set of all infinite sequences of natural numbers. However, we find that any absolutely defined ordering on this set is not well-ordered. To deal with this issue we can use the variant lexicographical ordering again. We restrict the set to sequences which are nonzero for only a finite number of arguments. This is naturally motivated as the limit of the finite powers of the base (similar to the concept of coproduct in algebra). This can also be thought of as the infinite union $\bigcup_{n<\omega}\omega^n$.

Each of those sequences corresponds to an ordinal less than $\omega^\omega$ such as $\omega^{n_1} c_1 + \omega^{n_2} c_2 + \cdots + \omega^{n_k} c_k$ and $\omega^\omega$ is the supremum of all those smaller ordinals.

The lexicographical order on this set is a well ordering that resembles the ordering of natural numbers written in decimal notation, except with digit positions reversed, and with arbitrary natural numbers instead of just the digits 0-9:

(0,0,0,...) < (1,0,0,0,...) < (2,0,0,0,...) < ... <
(0,1,0,0,0,...) < (1,1,0,0,0,...) < (2,1,0,0,0,...) < ... <
(0,2,0,0,0,...) < (1,2,0,0,0,...) < (2,2,0,0,0,...)
< ... <
(0,0,1,0,0,0,...) < (1,0,1,0,0,0,...) < (2,0,1,0,0,0,...)
< ...

In general, any ordinal α can be raised to the power of another ordinal β in the same way to get αβ.

It is easiest to explain this using Von Neumann's definition of an ordinal as the set of all smaller ordinals. Then, to construct a set of order type αβ consider all functions from β to α such that only a finite number of elements of the domain β map to a non zero element of α (essentially, we consider the functions with finite support). The order is lexicographic with the least significant position first. We find

• 1ω = 1,
• 2ω = ω,
• 2ω+1 = ω·2 = ω+ω.

The definition of exponentiation can also be given inductively (the following induction is on β, the exponent):

• α0 = 1,
• αβ+1 = (αβα, and
• if δ is limit, then αδ is the limit of the αβ for all β < δ.

Properties of ordinal exponentiation:

• α0 = 1.
• If 0 < α, then 0α = 0.
• 1α = 1.
• α1 = α.
• αβ·αγ = αβ + γ.
• (αβ)γ = αβ·γ.
• There are α, β, and γ for which (α·β)γαγ·βγ. For instance, (ω·2)2 = ω·2·ω·2 = ω2·2 ≠ ω2·4.
• Ordinal exponentiation is strictly increasing and continuous in the right argument: If γ > 1 and α < β, then γα < γβ.
• If α < β, then αγβγ. Note, for instance, that 2 < 3 and yet 2ω = 3ω = ω.
• If α > 1 and αβ = αγ, then β = γ. If α = 1 or α = 0 this is not the case.
• For all α and β, if β > 1 and α > 0 then there exist unique γ, δ, and ρ such that α = βγ·δ + ρ such that 0 < δ < β and ρ < βγ.

Warning: Ordinal exponentiation is quite different from cardinal exponentiation. For example, the ordinal exponentiation 2ω = ω, but the cardinal exponentiation $2^{\aleph_0}$ is the cardinality of the continuum which is larger than $\aleph_0$. To avoid confusing ordinal exponentiation with cardinal exponentiation, one can use symbols for ordinals (e.g. ω) in the former and symbols for cardinals (e.g. $\aleph_0$) in the latter.

## Cantor normal form

Ordinal numbers present a rich arithmetic. Every ordinal number α can be uniquely written as $\omega^{\beta_1} c_1 + \omega^{\beta_2}c_2 + \cdots + \omega^{\beta_k}c_k$, where k is a natural number, $c_1, c_2, \ldots, c_k$ are positive integers, and $\beta_1 > \beta_2 > \ldots > \beta_k \geq 0$ are ordinal numbers. This decomposition of α is called the Cantor normal form of α, and can be considered the base-ω positional numeral system. The highest exponent $\beta_1$ is called the degree of $\alpha$, and satisfies $\beta_1\le\alpha$. The equality $\beta_1=\alpha$ applies if and only if $\alpha=\omega^\alpha$. In that case Cantor normal form does not express the ordinal in terms of smaller ones; this can happen as explained below.

A minor variation of Cantor normal form, which is usually slightly easier to work with, is to set all the numbers ci equal to 1 and allow the exponents to be equal. In other words, every ordinal number α can be uniquely written as $\omega^{\beta_1} + \omega^{\beta_2} + \cdots + \omega^{\beta_k}$, where k is a natural number, and $\beta_1 \ge \beta_2 \ge \ldots \ge \beta_k \ge 0$ are ordinal numbers.

The Cantor normal form allows us to uniquely express—and order—the ordinals α which are built from the natural numbers by a finite number of arithmetical operations of addition, multiplication and exponentiation base-$\omega$: in other words, assuming $\beta_1<\alpha$ in the Cantor normal form, we can also express the exponents $\beta_i$ in Cantor normal form, and making the same assumption for the $\beta_i$ as for α and so on recursively, we get a system of notation for these ordinals (for example,

$\omega^{\left( \omega^{\left( \omega^7\cdot6+\omega+42 \right)}\cdot1729+\omega^9+88 \right)}\cdot3+\omega^{\left( \omega^\omega \right)}\cdot5+65537$

denotes an ordinal).

The ordinal ε0 (epsilon nought) is the set[1] of ordinal values $\alpha$ of the finite[2] arithmetical expressions of this[3] form. It is the smallest ordinal that does not have a finite arithmetical expression, and the smallest ordinal such that $\varepsilon_0 = \omega^{\varepsilon_0}$, i.e. in Cantor normal form the exponent is not smaller than the ordinal itself. It is the limit of the sequence

$0, \, 1=\omega^0, \, \omega=\omega^1, \, \omega^\omega, \, \omega^{\omega^\omega}, \, \ldots \,.$

The ordinal ε0 is important for various reasons in arithmetic (essentially because it measures the proof-theoretic strength of the first-order Peano arithmetic: that is, Peano's axioms can show transfinite induction up to any ordinal less than ε0 but not up to ε0 itself).

The Cantor normal form also allows us to compute sums and products of ordinals: to compute the sum, for example, one need merely know that

$\omega^{\beta} c+\omega^{\beta'} c' = \omega^{\beta'}c' \,,$

if $\beta'>\beta$ (if $\beta'=\beta$ one can obviously rewrite this as $\omega^{\beta} (c+c')$, and if $\beta'<\beta$ the expression is already in Cantor normal form); and to compute products, the essential facts are that when $\alpha = \omega^{\beta_1} c_1 + \cdots + \omega^{\beta_k}c_k$ is in Cantor normal form (and α>0) then

$\alpha\omega^{\beta'} = \omega^{\beta_1 + \beta'} \,$

and

$\alpha n = \omega^{\beta_1} c_1 n + \omega^{\beta_2} c_2 + \cdots + \omega^{\beta_k}c_k \,,$

if n is a non-zero natural number.

To compare two ordinals written in Cantor normal form, first compare $\beta_1$, then $c_1$, then $\beta_2$, then $c_2$, etc.. At the first difference, the ordinal which has the larger component is the larger ordinal. If they are the same until one terminates before the other, then the one which terminates first is smaller.

## Large countable ordinals

As discussed above, the Cantor Normal Form of ordinals below $\epsilon_0$ can be expressed in an alphabet containing only the function symbols for addition, multiplication and exponentiation, as well as constant symbols for each natural number and for $\omega$. We can do away with the infinitely many numerals by using just the constant symbol 0 and the operation of successor, $S$ (for example, the integer 4 may be expressed as $S(S(S(S(0))))$). This describes an ordinal notation: a system for naming ordinals over a finite alphabet. This particular system of ordinal notation is called the collection of arithmetical ordinal expressions, and can express all ordinals below $\epsilon_0$, but cannot express $\epsilon_0$. There are other ordinal notations capable of capturing ordinals well past $\epsilon_0$, but because there are only countably many strings over any finite alphabet, for any given ordinal notation there will be ordinals below $\omega_1$ that are not expressible. Such ordinals are known as large countable ordinals.

## Natural operations

The natural sum and natural product operations on ordinals were defined in 1906 by Gerhard Hessenberg, and are sometimes called the Hessenberg sum (or product). They are also sometimes called the Conway operations, as they are just the addition and multiplication (restricted to ordinals) of Conway's field of surreal numbers. They have the advantage that they are associative and commutative, and natural product distributes over natural sum. The cost of making these operations commutative is that they lose the continuity in the right argument which is a property of the ordinary sum and product. The natural sum of α and β is sometimes denoted by α # β, and the natural product by a sort of doubled × sign: α ⨳ β. To define the natural sum of two ordinals, consider once again the disjoint union $S\cup T$ of two well-ordered sets having these order types. Start by putting a partial order on this disjoint union by taking the orders on S and T separately but imposing no relation between S and T. Now consider the order types of all well-orders on $S\cup T$ which extend this partial order: the least upper bound of all these ordinals (which is, actually, not merely a least upper bound but actually a greatest element) is the natural sum.[4] Alternatively, we can define the natural sum of α and β inductively (by simultaneous induction on α and β) as the smallest ordinal greater than the natural sum of α and γ for all γ < β and of γ and β for all γ < α.

The natural sum is associative and commutative: it is always greater or equal to the usual sum, but it may be greater. For example, the natural sum of ω and 1 is ω+1 (the usual sum), but this is also the natural sum of 1 and ω.

To define the natural product of two ordinals, consider once again the cartesian product S × T of two well-ordered sets having these order types. Start by putting a partial order on this cartesian product by using just the product order (compare two pairs if and only if each of the two coordinates is comparable). Now consider the order types of all well-orders on S × T which extend this partial order: the least upper bound of all these ordinals (which is, actually, not merely a least upper bound but actually a greatest element) is the natural product. There is also an inductive definition of the natural product (by mutual induction), but it is somewhat tedious to write down and we shall not do so (see the article on surreal numbers for the definition in that context, which, however, uses Conway subtraction, something which obviously cannot be defined on ordinals).

The natural product is associative and commutative and distributes over the natural sum: it is always greater or equal to the usual product, but it may be greater. For example, the natural product of ω and 2 is ω·2 (the usual product), but this is also the natural product of 2 and ω.

Yet another way to define the natural sum and product of two ordinals α and β is to use the Cantor normal form: one can find a sequence of ordinals γ1 > … > γn and two sequences (k1, …, kn) and (j1, …, jn) of natural numbers (including zero, but satisfying ki + ji > 0 for all i) such that

$\alpha = \omega^{\gamma_1}\cdot k_1 + \cdots +\omega^{\gamma_n}\cdot k_n$
$\beta = \omega^{\gamma_1}\cdot j_1 + \cdots +\omega^{\gamma_n}\cdot j_n$

and defines

$\alpha \#\beta = \omega^{\gamma_1}\cdot (k_1+j_1) + \cdots +\omega^{\gamma_n}\cdot (k_n+j_n).$

## Notes

1. ^ assuming that an ordinal number is the set of (smaller) ordinal numbers, cf. Von Neumann's definition of ordinals
2. ^ "finite" here means the expressions are of finite length, but not necessarily of finite value
3. ^ i.e. Cantor normal form with $\beta_1<\alpha$
4. ^ Philip W. Carruth, Arithmetic of ordinals with applications to the theory of ordered Abelian groups, Bull. Amer. Math. Soc. 48 (1942), 262-271. See Theorem 1. Available here

## References

• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
• Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.