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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

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