Number Systems¶
Recall (doc/postdoc). Not a first course.
Job: nested algebraic structures; each extension solves a universal/completion problem.
Core¶
\[
\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\subset\mathbb{C}
\]
(\(0\in\mathbb{N}\): convention — state it.)
| Set | Structure | Construction sketch | Still missing |
|---|---|---|---|
| \(\mathbb{N}\) | monoid \((+,0)\) | Peano / set-theoretic | additive inverses |
| \(\mathbb{Z}\) | ordered ring | differences / Grothendieck | multiplicative inverses |
| \(\mathbb{Q}\) | ordered field | \(\operatorname{Frac}(\mathbb{Z})\) | completeness; \(\sqrt2\notin\mathbb{Q}\) |
| \(\mathbb{R}\) | complete ordered field | Cauchy / Dedekind | \(x^2+1=0\) |
| \(\mathbb{C}\) | algebraically closed field | \(\mathbb{R}[i]/(i^2+1)\) | order compatible with mult. |
Rational criterion: \(x\in\mathbb{Q}\) iff decimal expansion terminates or eventually periodic.
Cardinality: \(|\mathbb{N}|=|\mathbb{Z}|=|\mathbb{Q}|=\aleph_0\); \(|\mathbb{R}|=2^{\aleph_0}=|\mathbb{C}|\).
Used later for¶
- Real analysis: completeness axiom, Cauchy sequences
- Complex analysis: \(\mathbb{C}\) as complete field, FTA
- Set theory: cardinality, continuum
- Galois theory: algebraic closures, extensions
Check¶
Is \(\mathbb{R}\) uniquely characterized up to unique ordered-field isomorphism?
Yes — any complete ordered field is uniquely isomorphic to \(\mathbb{R}\) (as ordered field).