Can we use finite combination of factorials, surds and floor function on a combination of single number, say 4 to form every number?

Question

So the title is much pretty much the question, one thing to note that is you strictly have to use factorials, surds and floor function.

Edit:

The problem was a little bit ambiguous earlier. So, here are more details:

1. You can use only one number $4$.
2. We can apply any number of factorials and nth roots, and floor functions. Also, if it's not possible with factorials, is it with gamma functions?