Real Numbers and Continuity

The real numbers $\mathbb{R}$ extend the rationals $\mathbb{Q}$ by filling in the “gaps,” allowing for a continuous number line.

Dedekind Cuts

A Dedekind cut $(A, B)$ is a partition of $\mathbb{Q}$ such that $A$ has no largest element and everything in $A$ is less than everything in $B$. This construction formally defines irrational numbers like $\sqrt{2}$.

def is_in_sqrt2_cut(q):
  # A = {q in Q | q < 0 or q^2 < 2}
  if q < 0: return True
  return q**2 < 2

test_rationals = [1, 1.4, 1.41, 1.414, 1.42, 1.5, 2]
for q in test_rationals:
  print(f"{q} in A: {is_in_sqrt2_cut(q)}")

Completeness Axiom

The Completeness Axiom states that every non-empty set of real numbers that is bounded above has a Least Upper Bound (Supremum) in $\mathbb{R}$. This is what distinguishes $\mathbb{R}$ from $\mathbb{Q}$.

import numpy as np

def find_supremum_approximation(func, bounds, steps=1000):
  x = np.linspace(bounds[0], bounds[1], steps)
  y = func(x)
  return np.max(y)

# Find supremum of -x^2 + 4 on [0, 5]
# (Should be 4 at x=0)
f = lambda x: -x**2 + 4
print(f"Approximated Supremum: {find_supremum_approximation(f, [0, 5])}")

Archimedean Property

The Archimedean Property states that for any $x \in \mathbb{R}$, there exists an $n \in \mathbb{N}$ such that $n > x$. This implies that there are no “infinitely large” real numbers.

The Cantor Set

The Cantor Set is constructed by repeatedly removing the middle third of line segments. It is a classic example of a set that is uncountable but has “measure zero.”