Numerical Sets: ℕ, ℤ, ℚ, ℝ
Axiomatics, cardinality, completeness — a first-year perspective
Why do we need different number sets?
In school, number sets are studied as natural extensions: to solve $x+5=3$, natural numbers are not enough; to solve $2x=5$ — integers are insufficient; to solve $x^2=2$ — rationals fail. In the first year of university, the perspective changes: we no longer merely describe numbers, but axiomatically define them and prove rigorous theorems — about density, supremum, and countability. This material is a bridge between the school-level and university-level understanding of number systems.
What is new compared to school: field and order axioms, the completeness axiom, concepts of supremum and infimum, cardinality of a set, countability of ℚ, uncountability of ℝ, algebraic and transcendental numbers, the Archimedean property, density of ℚ in ℝ.
ℕ Natural Numbers
Natural numbers are the most intuitive set — the numbers used for counting.
Definition:
$$\mathbb{N} = \{1, 2, 3, 4, 5, 6, \ldots\}$$
Peano Axioms (how ℕ is constructed)
In rigorous mathematics, natural numbers are defined via five Peano axioms (1889):
- $1 \in \mathbb{N}$ (one is a natural number).
- Every $n \in \mathbb{N}$ has a unique successor $n' \in \mathbb{N}$.
- $1 \neq n'$ for any $n$ (one is not the successor of any natural number).
- If $n' = m'$, then $n = m$ (different numbers have different successors).
- Axiom of induction: if $1 \in M \subset \mathbb{N}$ and $n \in M$ implies $n' \in M$, then $M = \mathbb{N}$.
From the fifth axiom, the method of mathematical induction follows directly — the principal tool for proving statements about natural numbers.
Closure of operations in ℕ
| Operation | Closed? | Example |
|---|---|---|
| Addition ($+$) | ✓ Yes | $3 + 5 = 8 \in \mathbb{N}$ |
| Multiplication ($\times$) | ✓ Yes | $4 \times 7 = 28 \in \mathbb{N}$ |
| Subtraction ($-$) | ✗ No | $3 - 5 = -2 \notin \mathbb{N}$ |
| Division ($\div$) | ✗ No | $5 \div 2 = 2.5 \notin \mathbb{N}$ |
The impossibility of subtraction motivates the transition to ℤ.
Natural numbers on the number line
Principle of induction — example
Prove: $1 + 2 + \ldots + n = \dfrac{n(n+1)}{2}$.
Base case: $n=1$: $1 = \tfrac{1 \cdot 2}{2}$. ✓
Inductive step: assume the formula holds for $n$. Then
$1 + \ldots + n + (n+1) = \tfrac{n(n+1)}{2} + (n+1)$
$= \tfrac{(n+1)(n+2)}{2}$. ✓
By the axiom of induction, the formula holds for all $n$. ∎