Formalization of a simple statement regarding intervals I have the following statement:
"The open interval (0,1) contains no 'largest' number"
Which I am asked to "formalize" (I hope I am using the correct mathematical jargon here).
I think the most correct way to formalize it is:
$$\forall a (((a>0) \wedge (a<1)) \rightarrow \exists b ((a<b) \wedge (b<1)))$$
I would like to hear your opinion about it.
But is the following also correct? If not, why?
$$\forall a \exists b((a>0) \wedge (a<1) \wedge (a<b) \wedge (b<1))$$
 A: If you translate this to human language the first expression says:

*

*"for all $a$ in $(0,1)$ there exists a $b$ between $a$ and one" (perfect).

The second expression sounds odd:

*

*"for all $a$ [which ones?] there exists a $b$ such that $a$ is in $(0,1)$ and $b$ is between $a$ and one."

It is certainly not true that any $a$ is in $(0,1)$.
A: 
$$\forall a\;(((a>0) \wedge (a<1)) \rightarrow \exists b\;((a<b) \wedge (b<1)))$$ $$\forall a \exists b\:\:((a>0) \wedge (a<1) \wedge (a<b) \wedge (b<1))$$

Let's call the two statements First and Second. Their domain of discourse is $\mathbb R.$

*

*First accurately formalises the given verbal statement, and is a true statement.

*Second claims that whatever $a$ is, we can find some $b$ such that $a∈(0,1)$ and $a<b<1;$ the counterexample $a=100$ clearly shows
that Second is a false statement.

*First is logically equivalent to $\forall x \exists y\;(((x>0) \wedge (x<1)) \to ((x<y) \wedge (y<1))).$
(Why?)

*Second logically implies First (why?), but not vice versa (why?); that is, Second is a stronger statement than First.

