Axiomatic Set Theory and the ZFC Framework

Set theory is often called the “language of mathematics.” Virtually every mathematical object—numbers, functions, manifolds, operators—can be formally defined as a set.

Russell’s Paradox and the Necessity of Axioms

In Naive Set Theory, one could define a set through any property $P$: $S = \{x \mid P(x)\}$. Bertrand Russell asked: if we define $R = \{x \mid x \notin x\}$ (the set of all sets that do not contain themselves), does $R$ contain itself?

class RussellSet:
  def __contains__(self, item):
      # A set that contains itself only if it DOES NOT contain itself
      return not (item in item if hasattr(item, "__contains__") else False)

r = RussellSet()
try:
  print(f"Does r contain r? {r in r}")
except RecursionError:
  print("Infinite recursion: Russell's Paradox detected!")

The Axioms of ZFC

To resolve such paradoxes, the Zermelo-Fraenkel axioms with the Axiom of Choice (ZFC) were developed.

1. Extensionality: Sets are equal if they have the same elements.
2. Empty Set: $\emptyset$ exists.
3. Power Set: For any set $x$, there exists a set $\mathcal{P}(x)$ containing all subsets of $x$.

from itertools import chain, combinations

def power_set(iterable):
  s = list(iterable)
  return chain.from_iterable(combinations(s, r) for r in range(len(s)+1))

s = {1, 2, 3}
ps = list(power_set(s))
print(f"Set: {s}")
print(f"Power Set size: {len(ps)}")
print(f"Subsets: {ps}")

Constructing the Universe: The Von Neumann Hierarchy

Using these axioms, we can build the natural numbers $\mathbb{N}$:

• $0 = \emptyset$
• $1 = \{0\} = \{\emptyset\}$
• $2 = \{0, 1\} = \{\emptyset, \{\emptyset\}\}$

def von_neumann_ordinal(n):
  if n == 0:
      return set()
  prev = von_neumann_ordinal(n - 1)
  # n = prev U {prev}
  return prev | {frozenset(prev)}

for i in range(4):
  print(f"{i}: {von_neumann_ordinal(i)}")

Ordinals and Cardinals

• Ordinals: Describe the order type (position).
• Cardinals: Describe the size (quantity). Two sets have the same cardinality if there exists a bijection between them.

The Axiom of Choice (AC)

AC states that given a collection of non-empty sets, we can select one element from each. While intuitive, it leads to non-constructive results like the Banach-Tarski Paradox (decomposing a ball into two identical balls).