The “number” of complete vector fields in a manifold

I was wondering about the cardinality of complete vector fields in relation to the cardinality of general vector fields.

Let $$\mathcal{A}=\{X:M\rightarrow TM| X \text{ is a vector field}\}$$ and $$\mathcal{B}=\{Y\in\mathcal{A}| Y \text{ is complete}\}$$

Can we say anything about $$|\mathcal{A}|$$ and $$|\mathcal{B}|$$? Obviously $$|\mathcal{A}|\geq|\mathcal{B}|$$, since $$\mathcal{B}\subset\mathcal{A}$$.

I'm trying to think as an analogy of the cardinality of continuous vs. all real functions of a real variable (in which case it differed as the former is just $$|\mathbb{R}|$$ whereas the latter, being the power set of $$\mathbb{R}$$ necessarily has a larger cardinality).

I've thought about it but the furthest I've gotten is realising that $$|M|=|\mathbb{R}|$$, since the charts are bijections between $$M$$ and $$\mathbb{R}^n$$ and we also have $$|\mathbb{R}^n|=|\mathbb{R}|^n=|\mathbb{R}|$$.