1. The Simple Question with a Complex Answer
The question “What comes after one?” seems straightforward, but in mathematics—especially when dealing with infinity—it unravels profound truths about the nature of numbers, order, and logic.
2. In the Real Numbers: No “Next” Number
- Standard Ordering: In the usual real number line (e.g., 1, 1.1, 1.01, …), there is no immediate successor to 1. Between any two real numbers, you can always find another (e.g., 1.001, 1.0001, etc.). This property is called the density of the real numbers.
- Why? Real numbers form a continuum—an uncountably infinite set where numbers flow seamlessly without gaps. Unlike integers, which have discrete successors (2 follows 1), real numbers defy such simplicity.
3. Cantor’s Revolution: Different Sizes of Infinity
- Countable vs. Uncountable: Georg Cantor showed that not all infinities are equal.
- Countable Infinity: Sets like natural numbers (ℕ), integers (ℤ), and rationals (ℚ) can be “listed” (paired with ℕ).
- Uncountable Infinity: The real numbers (ℝ) are too vast to list. Cantor’s diagonal argument proved ℝ cannot be paired with ℕ—there’s no way to write all real numbers in a sequence.
- Implication: ℝ is a “bigger” infinity than ℕ. This shattered the ancient view that infinity is a single, monolithic concept.
4. The Axiom of Choice and Well-Ordering
- Cantor’s Unfinished Dream: Cantor believed even uncountable sets like ℝ could be well-ordered—arranged so every subset has a “first” element. But he couldn’t prove it.
- Zermelo’s Breakthrough: Using the Axiom of Choice (AC), Ernst Zermelo showed ℝ can be well-ordered.
- AC Allows “Choosing”: From infinitely many sets, AC lets you pick one element from each simultaneously—even with no rule.
- Non-Constructive Order: A well-ordering of ℝ exists abstractly, but we can’t write it down or visualize it. In this ordering, every number (including 1) has a “next” number, but it bears no relation to the usual order of ℝ.
5. Paradoxes and the Price of Choice
The Axiom of Choice is powerful but controversial. It enables “proofs” that defy intuition:
- Vitali Sets: Non-measurable collections of numbers with no coherent length—they break basic geometry.
- Banach-Tarski Paradox: A sphere can be split into 5 pieces, rotated, and reassembled into two identical spheres. These pieces are non-measurable “ghosts” of mathematics.
- Why Accept AC? Despite paradoxes, AC is essential for modern math. It simplifies proofs in analysis, topology, and algebra. As mathematician Jerry Bona joked:“The Axiom of Choice is obviously true, the Well-Ordering Theorem is obviously false, and who can tell about Zorn’s Lemma?”
6. Gödel and Cohen: You Choose Your Math
- Independence Results: Kurt Gödel (1938) and Paul Cohen (1963) proved AC is independent of standard set theory (ZF). You can:
- Accept AC: Embrace well-ordering, streamlined proofs, and paradoxes.
- Reject AC: Work in a stricter universe where everything is “constructible,” but lose key theorems (e.g., every vector space has a basis).
- It’s a Choice: Like choosing Euclidean vs. non-Euclidean geometry, mathematicians pick axioms based on the math they want to do.
Final Answer
- In the Standard Real Numbers: Nothing comes immediately after 1—there’s always another number in between.
- With the Axiom of Choice: A “next” number exists in a non-constructive, well-ordered sequence, but it’s a purely abstract artifact of set theory.
The axiom of choice bridges intuition and paradox, revealing mathematics as a tapestry of logic woven from axioms we choose to accept. As for what comes after one? It depends on the rules you play by.
