I mean, the other way around is also true, but the other way around is obvious. Obviously going from a weaker logic to a stronger logic can't eliminate contradictions.

What's surprising is that in this particular case (the Peano axioms), the reverse is also true; you won't be able to eliminate contradictions by passing to this weaker logic.

Note, all I said is, if you can prove a contradiction classically, then you can prove a contradiction constructively! I didn't say, if you have a classical proof of a contradiction then it's a valid constructive proof of a contradiction. Obviously not! But it will still be true that you'll be able to prove a contradiction constructively; it just won't necessarily be the same proof.

If you want to know more about the particular transformation, well, here's a relevant Wikipedia article: https://en.wikipedia.org/wiki/Double-negation_translation

Thank you for your explanation and pointing me to the double negation translation.

I know this is off-topic but if you have a book recommendation which teaches about intuitionistic logic/constructive math and also mentions the double negation translation, I'm interested. Most books I've read seem to either concentrate on only constructive or classical math.

I don't, sorry. I'm not really a logician, just a mathematician who thinks the foundations are worth understanding. :) I don't know what proper references on this material would be.

Classical logic can be embedded into intuitionistic logic. The double-negation translation N [0] turns any tautology of classical logic into a tautology of intuitionistic logic, so if θ ∧ ¬θ is provable with classical logic, N(θ) ∧ ¬N(θ) is provable with intuitionistic logic.

[0]: https://en.wikipedia.org/wiki/Double-negation_translation