Translate sentence to symbolic (Every real number is greater than some integer)

Question

I'm not sure how to translate the following into a symbolic form.

"Every real number is greater than some integer".

I see two possible answers;

1) There exists an integer x, such that for any real number y, y is greater than x

$∃x∊\mathbb{Z}$ such that $∀y∊\mathbb{R},y>x$

2) For any real number y, there exists an integer x such that y is greater than x

$∀y ∊\mathbb{R}, ∃x ∊\mathbb{Z}$ such that $y>x$

I'm aware that the first statement is false, while the second is true, however the question I have been posed is to translate the sentence, then explain whether it is valid or invalid (thus the answer could be an invalid statement.)

I'm leaning towards the first but I'm not certain. I'd greatly appreciate any help on this. Thanks.

Answer 1

Your two sentences are quite different. $1$ is false while $2$ is true. The difference is the order of the quantifiers, which corresponds to the order of selecting values for the variables. In $1$ you have to choose an $x$ which will be less than any number I name, which I get to do after you choose $x$. You can't do that. In $2$ I have to choose an $x$, then you get to choose a $y$ that is less. That you can do.

English is not as precise as math. I think the intended sense of the sentence is $2$, but there are other nouns where I would think the same structure points to $1$.

Answer 2

The second interpretation is the correct one.

The first suggests that there exists at least one integer that is smaller than every real number, which is clearly untrue, since both $\mathbb{R}$ and $\mathbb{Z}$ are unbounded. Note that your statement is trying to attach a sort of boundedness to $\mathbb{Z}$, which leads to it being false.

The second is perfectly sound (note that it suggests an unboundedness of $\mathbb{R}$ and $\mathbb{Z}$).