Quantifier statement explanation

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I'm having trouble understanding what the above statement means in plain English.

Does it mean: "For any Natural number there exist a number greater than it". I don't get what the right part of the implication means.

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You are correct. Literally translated, the statement means "for all $x$ such that $x$ is a natural number, there exists a number $y$ such that $y$ is a natural number and $x$ is less than $y$".