Make all positive integers, starting from $1$, using:
- Exactly four $4$'s (and no other digit), concatenation being allowed,
- Any standard mathematical symbol(s) and operation(s) not listed below.
But not using:
- Addition, subtraction, multiplication, division, modulo reduction,
- Logarithms, exponentiation, root extraction,
- Floor, ceiling, fractional part,
- Integral, differential, binomial coefficients,
- English, Greek, or Hebrew or any other language letters (Note that trig ratios require the use of English letters, and hence are not allowed),
- Any function or operation generated by you.
(And yes, it's possible.)
Enjoy! I'll be awaiting creative responses.