Successor of Omega
Contents |
Theorem
- $\omega + 1 = \left\{{0, 1, 2, ...; \omega}\right\}$
where $\omega$ is the minimal infinite successor set and $\omega + 1$ is the successor of $\omega$.
Note the use of the semicolon; this is the notation for multipart infinite sets.
Proof
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \omega + 1\) | \(=\) | \(\displaystyle \omega \cup \left\{ {\omega}\right\}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | definition of successor set | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \left\{ {0, 1, 2, \ldots}\right\} \cup \left\{ {\omega}\right\}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Natural Numbers are Elements of the Minimal Infinite Successor Set | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \left\{ {0, 1, 2, ...; \omega}\right\}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | 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, $\omega + 1 \ne 1 + \omega$.
If you are fairly certain that none exist, please remove {{proofread|both}} from the code.
Sources
- Weisstein, Eric W. "Ordinal Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/OrdinalNumber.html
- Irving M. Copi: Symbolic Logic: 5th Edition (1979) p 205
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.
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