Between two different numbers exists another different number

Question

Formalize the following statement "between two different numbers exists another different number". I know that the answer is $$∀x∀y((x<y)→∃z((x<z)∧(z<y)))$$

What I can't understand is why the right answer can't be: $$∀x∀y((x<y)∧∃z((x<z)∧(z<y)))$$

Why there is a need to use the "→" symbol instead of "∧"?

Think about that second formula carefully - it is a false statement because not every pair (x,y) of real numbers satisfies x<y. You can express this without implies, but not like that — Matthew Towers, Aug 15 '20 at 11:52
@MatthewTowers that should probably be put into an answer. I would but you deserve the reputation. — user400188, Aug 15 '20 at 23:56

score 1 · Accepted Answer · answered Aug 16 '20 at 14:15

The second formula is false because not every pair $(x, y)$ of real numbers satisfies $x<y$. It can't therefore be a formalisation of a true statement.

It's not necessary to use $\rightarrow$ if you don't want to: you can use the logical equivalence $p \rightarrow q \equiv (\neg p) \vee q$ to eliminate it.

Between two different numbers exists another different number

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