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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

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

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

$∀y \in \mathbb{R}, ∃x \in \mathbb{Z}$ 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.

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  • $\begingroup$ "For any real number y, there exists an integer x such that x is greater than y" I think you've made a small typo in swapping $x$ and $y$ $\endgroup$
    – Hadi
    Commented Mar 16, 2020 at 23:51
  • $\begingroup$ Thanks, fixed!! $\endgroup$
    – Mike
    Commented Mar 17, 2020 at 0:00
  • $\begingroup$ Whoever gave you this question is either trying to get you to appreciate the difference between precise mathematics (such as the symbolic logical forms) and imprecise statements in natural language or does not even realize that the question is imprecise. $\endgroup$
    – user21820
    Commented Mar 17, 2020 at 4:23

2 Answers 2

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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 $1$.

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  • $\begingroup$ I agree, the sentences are significantly different (1 being false while 2 is true), however the problem I have is which one is the correct translation? To exaggerate the sentence, "Every single real number that exists is greater than some specific integers". Do you see how that might imply that the statement is wrong? (I haven't been told if the statement is true or false) $\endgroup$
    – Mike
    Commented Mar 16, 2020 at 23:38
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    $\begingroup$ I would say the way to word it to clearly make $2$ the correct translation is "Each real number is greater than some integer." A change in the nouns like I suggested at the end is "Every positive number is greater than some negative number". This would make the first interpretation true and I would feel it is the correct translation. I think the context suggests $2$ for your sentence, but it is not clear. Unless the point of the problem is to show the ambiguity of English, I would downgrade the problem setter for an unclear question. $\endgroup$ Commented Mar 17, 2020 at 0:13
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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 $\mathbb{Z}$).

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    $\begingroup$ I completely agree the first statement is untrue, however the question I've been asked is to translate the sentence then explain whether it is valid or invalid (thus the statement could be invalid). To exaggerate the sentence, "Every single real number that exists is greater than some specific integers". Do you see how that could imply the false statement? Correct me if I'm being nonsensical. $\endgroup$
    – Mike
    Commented Mar 16, 2020 at 23:56
  • $\begingroup$ If I'm understanding you correctly, the original sentence would be valid. A sentence is valid as long as all possible interpretations of it are valid (just like using truth tables in propositional logic). The false statement is indeed false, but it's also not a correct translation of the original sentence, so it doesn't impact the validity of said original sentence. Since you found a correct translation (the second one) and you know it's valid, then the original sentence would be valid too. $\endgroup$
    – Hadi
    Commented Mar 17, 2020 at 0:05

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