Can every function which can be described by words, be formulated as well?

Question

Almost one year ago i was amused when i saw this page. It was the generation of the prime numbers using the floor function, mostly. I became more interested about the things we can do with the floor function. For instance to calculate IsPrime$(n)$, we can use Wilson's Theorem like this (Wilson's theorem states that n is prime if and only if it divides $(n - 1)! + 1$):
IsPrime$(n)$=IsInteger$(\frac{(n-1)!+1}{n})$=$[\frac{(n-1)!+1}{n}]+[\frac{(n-1)!+1}{-n}]$+1 which i know calculating it will take more time than to see if it is prime or not with the prime numbers definition, but that's not my point now.
To find another example, I came up with a idea to formulate $d(x)$ which is the number of divisors of $x$. I will give the necessary information of how i found it:
By a definition, $d(x)=(a_1+1)(a_2+1)...(a_k+1)$ where $a_1,a_2,\cdots,a_k$ are all of the powers of the distinct prime factors in the prime factorization of $n$. What i should do now is to find a formula for the highest power of each prime factor of $n$. I first calculate which powers of every prime number are in the factorization of $n$, so that counting them up will give us the highest power of $p$ in $n$. This is a formulated version of what i just said: PowerofP$(p,n)=\sum_{i=1}^{[\log_px]}{\left([\frac{x}{a^i}]+[\frac{-x}{a^i}]+1)\right)}$
And now using the definition of $d(x)$ we can formulate the overall function like this: $\prod_{\substack{ p=1 \\ p \in \mathbb P}}^\infty{(\text{PowerofP}(p,n)+1)}$
I even managed to formulate functions like: "Number of ways to write $2n$ as sum of $2$ primes" or "IsSquareFree$(n)$". Now my question is: Can every function which can be described by words, be formulated as well?
Restrictions:1) In our formulas, everything i used in my examples are allowed. Limits are not allowed. 2)Mapping and plotting are not counted as the functions i stated in my question, as one can find easy counterexamples for them.

Answer 1

I don't think the question is well-formed, really, but since you mentioned Wilson's theorem and are asking about notions vs. notation, perhaps you would enjoy this quotation:

In his one-page proof of the long-unproven Wilson’s prime number theorem, first published by Edward Waring, Gauss noted that “neither of them was able to prove the theorem, and Waring confessed that the demonstration seemed more difficult because no notation can be devised to express a prime number. But in our opinion truths of this kind should be drawn from notions rather than from notations.”

Answer 2

If you allow limits (hence infinite sums and products), nearly anything is possible (though typically not as easy as the examples you have provided). If you only allow closed form expressions, which necessarily have finitely many operations, the answer is clearly no. For instance, the function mapping a polynomial with complex coefficients to its roots clearly has no closed form.

Another idea you might read up on is generating functions.