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# Help: Infinite Ordinals

So, recently I learned of the existence of numbers known as infinite ordinals and while I understand them to a degree I still have a few questions about them.

Keep in mind I have already tried to find answers to these questions myself so either very few people know or I'm really bad at finding answers to maths questions.

First I should probably state what I know already to prevent any confusion or repetition of facts.

• $$\omega = \{1, 2, 3, 4, 5, \cdots \}$$

• $$n \cdot \omega = \{n, 2n, 3n, 4n, 5n, \cdots \} = \omega$$

• $$\omega \cdot n = \omega + \omega + \overbrace{\cdots \cdots}^{n \text{ times}} + \omega$$

Now, onto the questions;

1. Does the equation $$n \cdot \omega = \omega$$ hold true for $$n \leq 0$$? If it doesn't what is the result?

2. Concerning the nature of $$\omega$$; is $$\frac{1}{\omega}$$ defined? If not, why?

3. If $$\frac{1}{\omega}$$ is defined how would you find the result of $$n \cdot \omega$$ with $$n=\frac{1}{\omega}$$?

Any answer to any of the above questions would be greatly appreciated.

Note by Jack Rawlin
5 months, 1 week ago