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Since a while I've immersed myself in studying some math, and I must say that I started to enjoy pure mathematics. However, sometimes I don't really feel comfortable with some of the reasoning; for instance, when using proof by contradiction.

As you may have guessed, my question is the following: For you guys who have studied, or are studying mathematical logic, do these types of questions get answered in there?

Also, does mathematical logic show that the very foundations of mathematics are sound, and that all of the typical mathematical reasoning is air-tight?