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Continuum — Definition, Formula & Examples

The continuum refers to the set of real numbers R\mathbb{R}, viewed as a complete, unbroken line with no gaps. Its cardinality, denoted c\mathfrak{c} or 2ℵ02^{\aleph_0}, is strictly greater than the cardinality of the natural numbers.

In set theory, the continuum is the set R\mathbb{R} with cardinality ∣R∣=2ℵ0|\mathbb{R}| = 2^{\aleph_0}, where ℵ0\aleph_0 is the cardinality of N\mathbb{N}. The Continuum Hypothesis (CH) asserts that there is no set whose cardinality is strictly between ℵ0\aleph_0 and 2ℵ02^{\aleph_0}, i.e., 2ℵ0=ℵ12^{\aleph_0} = \aleph_1. Gödel (1940) and Cohen (1963) proved that CH is independent of the ZFC axioms.

Key Formula

∣R∣=2ℵ0=c|\mathbb{R}| = 2^{\aleph_0} = \mathfrak{c}
Where:
  • R\mathbb{R} = The set of all real numbers
  • ℵ0\aleph_0 = The cardinality of the natural numbers (the smallest infinite cardinal)
  • 2ℵ02^{\aleph_0} = The cardinality of the power set of the natural numbers
  • c\mathfrak{c} = Standard notation for the cardinality of the continuum

How It Works

The key idea is that while both N\mathbb{N} and R\mathbb{R} are infinite, R\mathbb{R} is a "larger" infinity. Cantor's diagonal argument proves that no bijection exists between N\mathbb{N} and R\mathbb{R}, so R\mathbb{R} is uncountable. The power set P(N)\mathcal{P}(\mathbb{N}) has the same cardinality as R\mathbb{R}, which is why we write ∣R∣=2ℵ0|\mathbb{R}| = 2^{\aleph_0}. In real analysis, the continuum property—that R\mathbb{R} has no gaps—is formalized by the least upper bound property: every nonempty subset of R\mathbb{R} bounded above has a supremum in R\mathbb{R}.

Worked Example

Problem: Show that the open interval (0,1)(0,1) has the same cardinality as R\mathbb{R}, confirming both represent the continuum.
Step 1: Define a function from (0,1)(0,1) to R\mathbb{R}. A standard choice is:
f(x)=tan⁡ ⁣(πx−π2)f(x) = \tan\!\left(\pi x - \frac{\pi}{2}\right)
Step 2: Verify this is a bijection. As x→0+x \to 0^+, f(x)→−∞f(x) \to -\infty; as x→1−x \to 1^-, f(x)→+∞f(x) \to +\infty. The tangent function is strictly increasing and continuous on (−π/2,π/2)(-\pi/2, \pi/2), so ff is a bijection from (0,1)(0,1) onto R\mathbb{R}.
Step 3: Since a bijection exists, the two sets have the same cardinality:
∣(0,1)∣=∣R∣=c|(0,1)| = |\mathbb{R}| = \mathfrak{c}
Answer: The interval (0,1)(0,1) and R\mathbb{R} both have cardinality c=2ℵ0\mathfrak{c} = 2^{\aleph_0}, so both represent the continuum.

Why It Matters

The continuum is foundational in real analysis, topology, and measure theory. Understanding its cardinality is essential when studying Lebesgue measure, probability spaces on R\mathbb{R}, and the structure of function spaces. The independence of the Continuum Hypothesis from ZFC is also a landmark result in mathematical logic.

Common Mistakes

Mistake: Assuming all infinite sets have the same size, so ∣N∣=∣R∣|\mathbb{N}| = |\mathbb{R}|.
Correction: Cantor's diagonal argument proves ∣R∣>∣N∣|\mathbb{R}| > |\mathbb{N}|. The continuum is a strictly larger infinity than countable infinity.