In Peano's axioms, the successor to any integer n is n+1.
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It was the successor to the successor of the number 0.
According to Peano's axioms, in either system, it is the successor to the successor to the successor to the successor to the successor to 0.
No, there is no such number.
It is the successor to 22. Its successor is 24.
You can get its successor, which is 583.