Practice question · Put in order
Order the layers of mathematical architecture, from the assumed foundation upward.
- Theorems: proven from axioms and definitions
- Definitions: name a new idea in terms of what is already available
- Axioms: assumed without proof
- Corollaries: follow quickly from a theorem just proven
Hints
- Only one layer is assumed rather than derived.
- A corollary depends on a theorem, so it cannot come first.
Show the answer
- Axioms: assumed without proof
- Definitions: name a new idea in terms of what is already available
- Theorems: proven from axioms and definitions
- Corollaries: follow quickly from a theorem just proven
Why
Axioms are the assumed base; definitions fix vocabulary; theorems are earned by proof; corollaries are cheap consequences of theorems. Each layer may use only what sits below it, which is what keeps the structure free of circularity.
Practise The Structure of Mathematical Reasoning
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