This note is a part of my Zettelkasten. What is below might not be complete or accurate. It is also likely to change often.
Strange loops are self-referencing ideas which are often paradoxes. Strange loops can be seen in the music of Bach (Cannons, and their casual cousin the Fugue, are self-referencing pieces of music). These loops can also be seen in the designs of Escher (Endless staircase and the infinite waterfall). These strange loops have been observed in maths/logic too.
Consider the Epimenides paradox, Epimenides was a Cretan who made this statement: "All cretans are liars". Its a paradox, if you assume true or false, the statement self-contradictory.
In set theory, we have Russel's paradox, which defines two kinds of sets - run-of-the-mill sets, like the set of walruses, and self-swallowing sets - set of all things except walruses, which contains the set itself too. Which category would a set of all run-of-the-mill sets fall into? Its not self swallowing - since its a set of all run-of-the-mill and if it were self-swallowing, it would have to contain itself too. Its not run-of-the-mill, since then it would have to contain itself (and therefore become self-swallowing). Either choice leads to a paradox.
Godel's incompleteness theorem states that any formal system will contain paradoxes which cannot be proved or disproved using that formal system. Also, that any formal system cannot prove itself to be consistent.
Finally, there is Zeno's Paradox, which tries to disprove that motion (and time) itself doesn't actually exist - its created by the senses. It examines a scenario where Achilles is racing a tortoise. Since Achilles is the greatest and fastest of the greek warriors, the tortoise gets a head-start of ten metres (the unit doesnt matter). By the time Achilles has reached the ten metre mark, the tortoise has moved forward a little bit, say one metre. By the time Achilles reaches this point, the tortoise has moved forward a little bit more. And this can continue an infinite number of times. So Achilles can never catch up to the tortoise.