Ordinal Addition/Examples/Ordinal Addition by One

Theorem

Let $x$ be an ordinal.

Let $x^+$ denote the successor of $x$.

Let $1$ denote (ordinal) one, the successor of the zero ordinal $\O$.

Then:

$x + 1 = x^+$

where $+$ denotes ordinal addition.

Proof

\(\ds x + 1\)

\(=\)

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

Definition of (ordinal) $1$

\(\ds \)

\(=\)

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

Definition of Ordinal Addition

\(\ds \)

\(=\)

\(\ds x^+\)

Ordinal Addition by Zero

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 3$ Some ordinals