If you have a cat, you can train it to be a hunter, and then after so long of gaining a higher hunting skill, the cat can capture ghosts. Or you can buy the invention table in Build & Buy mode and invent things until your invention skill is very high, making you able to invent the ghost zapper.
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You can't
No, it is like any normal sims game and you cant have a pet, but you can have babies.
as far as i know you cant
There Shouldn't Be Any Trait Restriction For Any Profession Or Job In The Sims 3. Traits Just Change The Way Sims Act.
Your pet will need to have 10,000 Lifetime Points saved up before you can purchase the Pet Hygienator from the Lifetime Rewards tab.