Modal logic extends classical logic by adding operators for necessity (□) and possibility (◇). This powerful framework allows us to reason about not just what is true, but what must be true and what could be true.

The basic axioms of modal logic (in the system K) are:

1. □(p → q) → (□p → □q) [Distribution Axiom]
2. If p is a theorem, then □p is a theorem [Necessitation Rule]

From these simple beginnings, we can develop rich systems for reasoning about:

The Axiom of Choice (AC) is perhaps the most controversial axiom in mathematics. It states that given any collection of non-empty sets, we can select exactly one element from each set to form a new set.

While this seems intuitive for finite collections, its application to infinite collections leads to some surprising and counterintuitive results, such as the Banach-Tarski paradox, which proves that it’s possible to take a solid ball, cut it into a finite number of pieces, and reassemble those pieces to create two identical copies of the original ball!

The Peano axioms, formulated by Giuseppe Peano in 1889, provide a rigorous foundation for the natural numbers. These elegant axioms define the natural numbers using just a few simple principles:

1. 0 is a natural number
2. Every natural number has a successor
3. 0 is not the successor of any natural number
4. Different natural numbers have different successors
5. If a property holds for 0 and holds for the successor of every number that has it, then it holds for all natural numbers

From these simple axioms, we can build all of arithmetic! Addition, multiplication, and even more complex operations can be defined using just these fundamental principles.

Boolean algebra, developed by George Boole in the 19th century, provides the mathematical foundation for digital computing and logical reasoning.

The fundamental operations in Boolean algebra are:

• AND (conjunction)
• OR (disjunction)
• NOT (negation)

These simple operations, combined with the values True and False, form a complete system for logical reasoning. This system is not just theoretically interesting - it’s the basis for all digital computer operations, from simple calculations to complex decision-making algorithms.

Kurt Gödel shook the mathematical world in 1931 with his incompleteness theorems. These revolutionary results demonstrated fundamental limitations of formal mathematical systems.

The First Incompleteness Theorem states that for any consistent formal system F within which basic arithmetic can be carried out, there are statements that can be formulated in F that can neither be proved nor disproved within F.

This means that mathematics contains true statements that cannot be proved within the system itself - a profound and somewhat unsettling result that changed our understanding of mathematical truth and formal systems forever.