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May 2nd, 2015, 02:14 AM   #1
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monotonic laws for ordinal subtraction

I have to prove some monotonic laws for ordinals. It's quite comfortable for me to show monotonic laws of ordinal addition (e.g. $\beta\leq\gamma\Rightarrow\alpha+\beta\leq\alpha+ \gamma$). But when it comes to laws with subtraction, then I'm not sure where to start.

Maybe it's because of definition of subtraction for ordinals
$\alpha-\beta=\gamma$, if $\alpha=\beta+\gamma$, which is not constructive.

So, maybe someone can give me a hint on how to prove those:

$\alpha,\beta,\gamma$ - ordinals.

(i) $\alpha>\beta\Rightarrow \gamma(\alpha-\beta)=\gamma\alpha-\gamma\beta$

(ii)$\alpha>\beta>\gamma \Rightarrow \alpha-\gamma>\beta-\gamma$
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May 2nd, 2015, 05:40 AM   #2
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Well you can always use $p \gt q \implies p \lt (-q)$ and $p - q = p + (-q)$.
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May 4th, 2015, 10:17 AM   #3
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I haven't seen $\displaystyle -q$ defined for an ordinal $\displaystyle q$...
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