Can every bijection of natural numbers be defined with a closed form formula

Question

I am interested in the existance of closed form formulas for bijections on natural numbers.

With the term closed form is lose. Any information on formulas that represent permutations on N are welcomed. The following are some additional questions:

1. Can any permutation be represented with a formula for elementary function on real numbers? (or finitly many different such formulas for finitly many different cases)
2. Can any permutation be expressed just using finitly many cases and formulas from peano arithmethics?
3. What if we include all injective functions?

EDIT: As noted by @arturo-magidin, the set of formulas over a finite sign is countable, therefore 2. is false.

EDIT Formulas could be derived from an uncountable alphabet. (For example real numbers with $+$ and $\cdot$ signs. ).