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There is no greater real number.

There is no positive integer that is greater than any other positive integer.

I was wondering if the negative sign is necessary here or not. Are there many different ways of writing the two statements?

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You can write these by using that $$\neg \exists x (P(x)) = \forall x (\neg P(x)).$$ –  JavaMan Oct 30 '12 at 20:34
    
The literal translation of the mathematical English into symbols would use negation. For the first problem, we can say it using the equivalent "for any real number $x$, there is a real number $y$ such that $x \lt y$." However, negation cannot always be dispensed with. –  André Nicolas Oct 30 '12 at 21:00
    
@AndréNicolas ... especially not in the statement "negation cannot always be dispensed with" :) –  Hagen von Eitzen Oct 30 '12 at 21:02
    
I don't understand why you can write y > x, because it could also be interpreted as there is a greater real number y. –  hjggh Oct 30 '12 at 21:43
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1 Answer

There are many ways of writing these statements using first order logic. As said by @JavaMan and @AndréNicolas, you can use the logical equivalence between the quantifiers.

There is no greater real number.

I assume that, as you're using "greater", there must be some number, lets say $n$, whom no real number can be greater than.

$G(x,y)$ means $x$ is greater than $y$. And assume $x,y,n \in \mathbb{R}$.

$\forall x ~\neg G(x,n)$ or $\neg \exists x ~G(x,n)$

There is no positive integer that is greater than any other positive integer.

$P(x)$ means $x$ is positive. In this case $x,y \in \mathbb{Z}$.

$\neg \exists x ~\forall y~P(x) \wedge G(x,y)$

I'll let you do the other ones.

Hope it was helpfull.

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