Let's say you have a bag with infinite apples inside. You begin taking out the apples randomly. There is one (maybe more) particular apple that will never be taken out.
>>16
They are the system's definition. You can derive theorems true within that system and prove falsity of statements.
If you think Peano's axioms are wrong, you're free to show where the contradiction lies. If some axiomatic system leads to contradiction, it's trivial and can be discarded if you so wish.