Q:

predicate logic syntax when universe of discourse is natural numbers/everything

From my course text:

If the universe of discourse is composed of the natural numbers then
we can write:
∀X (odd(X) ⇒ (∃Y (X = 2 x Y)))
If the universe of discourse is everything then we can write:
∀X ∃Y(odd(X) ⇒ (X = 2 x Y))

I know that the difference is the position of the "∃Y". But I do not understand why this occurs. Can someone explain to me why "∃Y" needs to be shifted to the front?
Thanks!

A:

It doesn't need to be shifted as far as I can see. The formulas you gave are logically equivalent, by which I mean, the first is satisfied by a model M, if and only if the second is also satisfied by the model M.

