In the following symbolic mathematical statement $n \in \omega $, what does $\omega$ stand for? Does it have something to do with the continuum, or is it just another way to denote the set of natural numbers?
Confusion about a specific notation
2 Answers
According to Wikipedia the character $\omega$ represents "the set of natural numbers in set theory".
It goes on to say "...though $\mathbb{N}$ or $\mathbf{N}$ are more common in other areas of mathematics."
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2So, outside set theory, use one of those other designations! – 2011-10-01
The notation of $\omega$ is coming from ordinals, and it denotes the least ordinal number which is not finite.
The von Neumann ordinals are transitive sets which are well ordered by $\in$. We can define these sets by induction:
- $0=\varnothing$;
- $\alpha+1 = \alpha\cup\{\alpha\}$;
- If $\beta$ is limit and all $\alpha<\beta$ were defined, then $\displaystyle\beta=\bigcup_{\alpha<\beta}\alpha$.
That is to say that after we have defined all the natural numbers, we define $\omega=\{0,1,2,3,\ldots\}$, then we can continue if so and define $\omega+1 = \omega\cup\{\omega\}$ and so on.
In set theory it is usual to use $\omega$ to denote the least infinite ordinal, as well the set of finite ordinals. It still relates to the continuum since $\mathcal P(\omega)$ is of cardinality continuum, since $\omega$ is countable.