Ordinal Addition by Zero
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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: $\S 8.3$