In QFT (quantum field theory), a wave-function $\Psi$ takes in a scalar field ($\phi: \mathbb{R}^3 \to \mathbb{R}$) as input, and returns a complex number ($\mathbb{C}$) as output. Thus:
$$\Psi: (\mathbb{R}^3 \to \mathbb{R}) \to \mathbb{C}$$
According to the Beth number article on Wikipedia, the set all functions $f$ from $\mathbb{R}^m$ to $\mathbb{R}^n$ has a cardinality of $2^{\mathfrak{c}} = 2^{2^{\aleph_0}}$:
$$f: \mathbb{R}^m \to \mathbb{R}^n$$
I want to know the number of wave-functions $\Psi$ possible; that is, the cardinality of the set of all wave functions $\Psi$.