Deductive systems are either incomplete or inconsistent. Meaning

• Inconsistent: they contain contradictions. Statements can be true and false in the same deductive system.
• Incomplete: Statements can be found that can not be proven to be true or false.

Gödel proved this for us.

Wittgenstein formulated something similar:

The truth is built of true facts and untrue facts: facts that are not based on a system of observation yet are true anyway. Nevertheless, Wittgenstein seems to disagree with Gödel’s incompleteness theorem. Food for a lasting scientific debate. Anyway, Wittgenstein was looking at language and philosophy, not at mathematics.

Final remarkable mathematical truth for now from Cantor.

Cantor proved that one infinity is not the same as the other infinity. He developed a way to compare infinite sets and describe how infinite sets with different characteristics exist.

As an example, Cantor proved that real numbers are more numerous than the set of natural numbers. While both are infinite. He also invented a way to operate on infinite sets.

Cantor ended up in a mental hospital, which seems to be viewed as as heroic achievement among mathematicians—an opinion I do not share.

I recall reading The Mystery of the Aleph by Amir D. Aczel about Cantor. Unfortunately, I have lost my notes and the book. This book was very accessible, I do recall that.