If you are not interested in the foundations of mathematics (and of computer science, I think), then you may disregard this post. For those of you that are interested but are unsure where to start: I hope this will help you get started with the foundations movement at the turn of the twentieth-century!
- From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (Jean van Heijenoort, editor)
This source book is definitely worth purchasing for your own library and use. It contains highly influential primary texts in logic by, among others, Cantor, Peano, Frege, Russell, Zermelo (yes, the one of ZF fame), von Neumann, Hilbert, Brouwer, and, of course, Gödel. It is sorely missing Tarski and Church and Marcus, but you can make up for that elsewhere. Frege’s Begriffsschrift alone justify purchasing this, as do Gödel’s papers.
- The Search for Mathematical Roots, 1870-1940 (Ivor Grattan-Guinness, author)
Grattan-Guinness, sadly now deceased, was a highly regarded historian of mathematics (he also wrote a book on the history of calculus). This book offers a good sketch of the history of the foundations of mathematics movement. It is not exhaustive, but it covers a good enough selection to contextualize whatever else you go on to read.
A combination of those two should put you in a position to pursue whatever else you find interesting about the foundations of mathematics movement at the turn of the 20th century. I am, of course, happy to recommend further items. Just get in touch!