Definition. A cardinal is an ordinal which it is not in bijection with any smaller ordinal.

Notation. $|X|$ means that cardinal of $X$.

Proposition 1. Let $w$ be an ordinal. Then $|w+1|=w.$

Proposition 2. $|w^2|=w$.

I couldn't understand proposition 1 and 2, can you explain that why $|w+1|=w$ and $|w^2|=w$.

  • 1
    $\begingroup$ It should assume that $w$ is an infinite ordinal. Then you can find a bijection between $w+1$ and $w$ and another between $w^2$ and $w$. $\endgroup$ – Ross Millikan Apr 17 at 23:40
  • $\begingroup$ @RossMillikan Ahh yess!! Thanks for comment! $\endgroup$ – PozcuKushimotoStreet Apr 17 at 23:40
  • 2
    $\begingroup$ As standalone statements, they are not true. They should be modified in either of the following ways: a) add "infinite" next to "ordinal" and substitute all "$=w$" with "$=\lvert w\rvert$"; b) substitute all "ordinal" with "inifinite cardinal". $\endgroup$ – Saucy O'Path Apr 17 at 23:41

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