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I have here an explanation for the difference between the two statements but I don't understand something in it. The first statement says that for each positive integer $x$, there is a larger positive integer $y$. The second statement says there is a single positive integer $y$ that is greater than every positive integer $x$. What I don't understand is that why we can't say that in the first statement for each positive integer $x$ there is a single positive integer $y$. Why he said there is a single $y$ in the second and didn't say the same thing in the first ?

Out Of Bounds
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  • It is in the second one that we should not say "there is a single." The "single" has been added there in an attempt to emphasize, but it could cause confusion, and did. – André Nicolas Sep 18 '14 at 03:25
  • But $\exists y$ means there exists y. So indeed it's single. Why on the first one it says for each x there is y. By saying that, y is no more single. We gonna have multiple x's with multiple y's – Out Of Bounds Sep 18 '14 at 03:28
  • $\exists y$ means there exists a $y$. If I say there exists a $y$ such that $y$ is a dog, I do not intend to claim that there is in the universe a single dog. – André Nicolas Sep 18 '14 at 03:30
  • It is only a problem of translation into natural language. The correct reading of the first one is (as you said) "for each positive integer $x$, there is at least a positive integer $y$ such that : $y$ is larger than $x$". In the same way, for the second one : "there is at least a positive integer $y$ such that, for every positive integer $x$ : $y$ is larger than $x$". The "single" creeps in because in this specific example (i.e. with integer numbers) we implicitly assume that if such an $y$ exists, it must be "the greatest" integer number. – Mauro ALLEGRANZA Sep 18 '14 at 07:19

2 Answers2

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It is not correct to interpret $\exists y\forall x (y\gt x )$ as meaning that there is a "single" (unique) positive integer $y$ which is greater than every positive integer $y$. It is regrettable that the "single" wording was used to interpret the second sentence, since it led you to try to use the same interpretation in the first sentence. In neither case should "single" be used.

Let us use a different example, where the variables range over the integers, positive, negative, and $0$. Consider the sentence $\exists y\forall x(y\mid x)$. Here $u\mid v$ means that $u$ divides $v$.

The sentence is true, for $1$ divides $x$ for all $x$. But $-1$ also divides $x$ for all $x$.

André Nicolas
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The first says that there's always a bigger fish. (for any given $x$, we can find a $y$ that's bigger than it)

The second says that there's a biggest fish. (there's a particular $y$ such that, no matter what $x$ you pick, $y$ is bigger than $x$.) note that we don't have to know what $y$ is -- or even for there to be just one -- to be able to say this.

The second is, without qualification on the domains of $x$ and $y$, false based on the standard definition of $>$: $y$ would have to take on a value that $x$ simply cannot be, or $y>x$ would be invalidated for $x=y$.

Dan Uznanski
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