It is easy to see that the rational numbers are countable, and a lovely, short proof of Cantor shows that the real numbers are not countable. This led Cantor to ask if the "infinite size" of the real line is "one bigger than that of the rational numbers". The first part of the project is to understand what this means, and to express it clearly in mathematical terms.
The "answer" is one of the most famous results in mathematics (and maybe philosophy) of the 20th century: it can be proved that this question cannot be answered. The second part of the project is to understand what this means, and to explain it well.
The project needs some knowledge of the real numbers, as covered in 2011 = Real analysis 2, for example. It also needs a little background in logic, as covered in 2040, for example; the module 3123 on Set theory would be very helpful.
There are many popular accounts of this topic.
For details on the borders of analysis and logic, see J.K. Truss, Foundations of Mathematical Analysis, OUP, 1997.
The proof of the "independence of the continuum hypothesis" is given in the book "An introduction to independence for analysists", H.G. Dales and W.H. Woodin, CUP. The project does not require a full account of this proof (or a treatment of "forcing"), but should conclude with some account of how such a proof could work.