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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?

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up vote 5 down vote accepted

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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So, outside set theory, use one of those other designations! – GEdgar Oct 1 '11 at 17:14

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.

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