I was trying to convert a simple proposition concerning the real numbers to Prenex Normal Form but arrived at a logical statement that didn't appear equivalent to what I started with.
The proposition I began with:
If two real numbers $a,b$ can get arbitrarily close to one another, then they are equal.
As a logical formula:
if $|a-b|<ε$ for all $ε>0$, then $a=b$.
Entirely with logical symbols:
$ (∀ε>0, (|a-b|<ε)) ⇒ (a=b) $
This logical statement appears identical in meaning to the original proposition above.
Now I perform the following steps to convert this to Prenex Normal Form (PNF) by applying the instructions in https://www.csd.uwo.ca/~lkari/prenex.pdf.
Eliminate occurrences of conditionals and biconditionals from the formula. This means removing the $⇒$ symbol in this situation.
Applying the fact $(p⇒q) ⇔ (¬p∨q)$ gives:
$ ¬(∀ε>0, (|a-b|<ε)) ∨ (a=b) $
Observe that the conditional is removed.
Move all negations inwards so that negations only appear as part of literals.
There's only one negation to move and moving it inwards gives:
$ (∃ε>0, ¬(|a-b|<ε)) ∨ (a=b) $
The PNF can now be obtained by moving all quantifiers to the front.
So we just have to shift the existential quantifier outside and we're done:
$ ∃ε>0, ( ¬(|a-b|<ε) ∨ (a=b) ) $
Observe that the formula that comes after the existential quantifier can be rewritten as a conditional since it's in the "$(¬p∨q)$" form. Rewriting the formula as a conditional gives:
$ ∃ε>0, ( (|a-b|<ε) ⇒ (a=b) ) $
This result does not appear equivalent to what I began with:
If two real numbers 𝑎,𝑏 can get arbitrarily close to one another, then they are equal.
Related phrases:drinker's paradox, drinker paradox, drinker's theorem, Smullyan's Drinker's principle, Raymond Smullyan's "What Is the Name of this Book?"