Successor of Omega

Theorem

$\omega + 1 = \set {0, 1, 2, \ldots; \omega}$

where $\omega$ is the minimally inductive set and $\omega + 1$ is the successor of $\omega$.

Note the use of the semicolon; this is the notation for multipart infinite sets.

Proof

\(\ds \omega + 1\)

\(=\)

\(\ds \omega \cup \set {\omega}\)

Definition of Successor Set

\(\ds \)

\(=\)

\(\ds \set {0, 1, 2, \ldots} \cup \set \omega\)

Definition of Von Neumann Construction of Natural Numbers

\(\ds \)

\(=\)

\(\ds \set {0, 1, 2, \ldots; \omega}\)

Definition of Set Union

$\blacksquare$

Comment

It is customary to use $\omega + 1$ rather than $\omega^+$ for transfinite arithmetic.

However, it needs to be borne in mind that this is not conventional natural number addition.

For example, $\omega + 1 \ne 1 + \omega$.

This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code.

Sources

• 1979: Irving M. Copi: Symbolic Logic (5th ed.) p. $205$

This article, or a section of it, needs explaining. In particular: Telling us what page it's on is less than adequate because different editions may have the relevant work on different pages. So please, instead, place a result by its chapter, section, etc. Please see what is done on other pages and go and do thou likewise. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

• Weisstein, Eric W. "Ordinal Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OrdinalNumber.html