Axiomatic Systems

Mathematics is not built on thin air; it is built on Axioms. An axiom is a statement that is taken to be true, to serve as a premise or starting point for further reasoning.

1. ZFC: The Foundation of Set Theory

Almost all modern mathematics is built on Zermeio-Fraenkel Set Theory (ZFC). Think of these as the “low-level assembly instructions” of math.

The Axiom of Extensionality

Two sets are equal if they have the same elements.

• Example: $\{1, 2\}$ is the same as $\{2, 1\}$. The order and repetition don’t matter.
• Code Analogy: In Python, a set behaves exactly like this.

# Demonstrating Extensionality
set_a = {1, 2, 3}
set_b = {3, 2, 1, 1, 2}

print(f"Set A: {set_a}")
print(f"Set B: {set_b}")
print(f"Are they equal? {set_a == set_b}")

The Axiom of Infinity

There exists an infinite set. Without this, we couldn’t formally prove that the set of Natural Numbers ($\mathbb{N}$) exists in its entirety. It allows us to build the infinite sequence $\{0, 1, 2, \dots\}$.

The Axiom of Choice (the C in ZFC)

If you have a collection of bins, you can choose one item from each bin. This seems obvious, but for infinitely many bins, it leads to strange results like the Banach-Tarski Paradox, where you can disassemble a sphere and reassemble it into two identical spheres.

2. Russell’s Paradox

Before ZFC, mathematicians used “Naive Set Theory.” Bertrand Russell showed this was broken with a simple logical trap:

Consider the set $S$ of all sets that do not contain themselves. Does $S$ contain itself?

# The Logic of Russell's Paradox
def check_membership(contains_self):
  # Definition of S: x is in S if x does NOT contain itself.
  # If we check if S is in S:
  return not contains_self

# There is no stable answer!
print(f"If we assume True -> {check_membership(True)}")
print(f"If we assume False -> {check_membership(False)}")

ZFC fixes this by making it impossible to define “the set of all sets.” You can only create sub-sets of existing sets (the Axiom of Specification).

3. Why Formalize?

We formalize mathematics to ensure that our proofs are not just “convincing” but mechanically certain. If our foundational axioms have a contradiction, everything we build on top of them (calculus, physics, computer science) would collapse.

Exercises