Things can make sense

Mathematical logic is an essential part (if not the essential part) of the foundations of mathematics

See Discrete mathematics, Theoretical computer science, Logic..

http://us.metamath.org/index.html

Video lectures:

Mathematics - Mathematical Logic

NPTEL Computer Sc - Discrete Mathematical Structures

Metamathematics

Boolean logic

Boolean algebra

Set theory

Theory of types. An axiomatization of set theory by Russell and Whitehead, that creates a hierarchy of sets to avoid Strange loops and Paradoxes

Proof theory

Mathematical structure

Formal systems

http://us.metamath.org/mpegif/mmset.html#overview

Metamath Proof Explorer: A Modern Principia Mathematica

http://ghilbert-app.appspot.com/ – http://math.stackexchange.com/questions/1111758/is-there-a-canonical-database-of-theorems