# Basic Proposition of Cardinals

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

• 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 $w$. – Ross Millikan Apr 17 at 23:40
• @RossMillikan Ahh yess!! Thanks for comment! – PozcuKushimotoStreet Apr 17 at 23:40
• 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". – Saucy O'Path Apr 17 at 23:41