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Entry about Model Theory in the Stanford Encyclopedia of Philosophy by Wilfrid Hodges. 

Marcus du Sautoy discusses Gödel's Incompleteness Theorem on Numberphile. 

Magic: The Gathering (popular trading card combat game) is Turing complete. Based on this paper by Churchill et al. 

Gödel, Escher, Bach: an Eternal Golden Braid, by Douglas Hofstadter. Pulitzer Prize-winning book on a range of topics in mathematics and philosophy. 

TED talk on Hilbert’s Paradox of the Grand Hotel; a thought experiment on infinite sets. 

Wikipedia page on the Topologist’s Sine Curve, an example of “pathological behaviour” that arises in analysis but not in o-minimality.