Practice Completeness (Intuition) in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
The property of a mathematical system where every true statement that can be expressed in the system can also be proved within it.
A complete system has no hidden truths that are provably beyond reach — there are no true statements you cannot prove from the axioms.
Example 1
easyThe real numbers \mathbb{R} are 'complete' while the rationals \mathbb{Q} are not. Illustrate this by finding a sequence of rationals that converges to an irrational number.
Example 2
mediumCheck that a proof by induction for P(n) is complete: what cases must be covered? Use P(n): 'n^2 \ge n for all n \ge 1' as an example.
Example 3
easyA student proves a statement for all even integers but forgets odd integers. Is the proof complete? What is missing?
Example 4
mediumState whether \mathbb{Q} or \mathbb{R} is a better domain for solving x^2 = 2, and explain in terms of completeness.