Ordinal Addition by Zero

Theorem

Let $x$ be an ordinal.

Let $\O$ be the zero ordinal.

Then:

$x + \O = x = \O + x$

where $+$ denotes ordinal addition.

Proof

By definition of ordinal addition, it is immediate that:

$x + \O = x$

$\Box$

We shall use Transfinite Induction on $x$ to prove $\O + x = x$

Base Case

The induction basis $x = \O$ comes down to:

$\O + \O = \O$

This follows by the above.

$\Box$

Inductive Case

For the induction step, suppose that $\O + x = x$.

Then, also:

\(\ds x^+\)

\(=\)

\(\ds \paren {\O + x}^+\)

Substitutivity of Equality

\(\ds \)

\(=\)

\(\ds \O + x^+\)

Definition of Ordinal Addition

$\Box$

Limit Case

Finally, the limit case.

So let $x$ be a limit ordinal, and suppose that:

$\forall y \in x: \O + y = y$

Now we have:

\(\ds x\)

\(=\)

\(\ds \bigcup_{y \mathop \in x} y\)

Limit Ordinal Equals its Union

\(\ds \)

\(=\)

\(\ds \bigcup_{y \mathop \in x} \paren {\O + y}\)

Indexed Union Equality

\(\ds \)

\(=\)

\(\ds \O + x\)

Definition of Ordinal Addition

$\Box$

Hence the result, by Transfinite Induction.

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 8.3$