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得る（うる）
Express possibility
Examples:
- 地球の温暖化が進むと、気候の変化、海面の上昇などの問題が生じうる。As global warming progresses, problems such as climate changes and rising sea levels can occur.
- 今から必死になって頑張れば、志望の大学に入ることも十分にありうる。If you work hard from now on, it is quite possible that you will enter the university of your choice.
- 今度のサッカーの決勝戦では、どんなことでも起こりうる。Anything can happen in the upcoming football finals.
うる is a written language. The negative form is ～得（え）ない
得ない（えない）
Means no possibility
Examples:
- 他人の夢の中に潜入して、アイディアを盗むなんてありえない話だ。It's impossible to infiltrate someone else's dream and steal an idea.
- 根性がないと、この目標を達しえないでしょう。Without guts, you won't be able to reach this goal.
- 株式相場がこれからどう動くか予測し得ない。It is unpredictable how the stock market will change in the future.
えない is a written language, but ありえない is often used in spoken language.