Let $\exists N\in\mathbb{R}:\forall x>N:A(x)$ for a statement $A(x)$. Do I need to define $x$ to be in $\mathbb{R}$ or is this already given by $x>N$?

Question

Let $\exists N \in\mathbb{R}:\forall x>N:A(x)$ for some statement $A(x)$. Do I need to define $x$ to be in $\mathbb{R}$ or is this already given by the fact that $x>N$?

Answer 1

It depends on how your formal language defines bounded quantifiers. If we are careful, we only use unbounded quantifiers $\forall x$ and $\exists x$ and spell out restrictions further down. And all bounded quantifiers can (and should) be defined in terms of these workarounds (and always with the intention of more legible and less convoluted sentences).

If we work in set theory, bounded quantifiers a la $\forall x\in y\colon\phi\equiv\forall x\colon x\in y\to \phi$ and $\exists x\in y\colon \phi\equiv \exists x\colon (x\in y\land \phi)$ suggest themselves.

If your domain of discourse allows so, other similar constructs may come in handy - and can be used once they are defined accordingly. Given suitable definitions, we For example, it may be useful to have bounded quantifiers involving the order relation and then you may want to define these accordingly; it seems most useful to define $\forall x>y$ to mean "for all real numbers $x$ that are greater than the (real) number $y$".

In other (number thoretical) context, we may want introduce bounded quantifiers in terms of divisibility and for example write $$ \forall n>1\colon (n\,\text{prime}\lor \exists d\mid n\colon (1<d\land d^2<n))$$ with the interpretation "every natural number greater than $1$ is prime or has a proper divisor $\le \sqrt n$".

Answer 2

The condition $ x>N $ does not define precisely the variabe or the parameter $ x$.

It doesn't say if $ x $ is in $\Bbb N $ or $ \Bbb Q $ or elsewhere.

If $ x $ is a real, Your proposition should be written as

$$(\exists N\in \Bbb R)\;:\;(\forall x\in \Bbb R)$$ $$\Bigl(x>N \implies P(x)\Bigr)$$

Or in an equivalent way $$(\exists N\in \Bbb R)\;: $$ $$\Bigl((\forall x\in \Bbb R \;:\; x>N)\; \; P(x)\Bigr)$$