Practice Completeness (Intuition) in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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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.

Showing a random 20 of 50 problems.

Example 1

medium
An incomplete system has a true statement GG it cannot prove. Can ¬G\neg G be proved in a consistent such system?

Example 2

easy
A logical system is complete if every true statement in it can be _____.

Example 3

easy
Is the set of integers closed under multiplication?

Example 4

medium
Are the rationals complete under the operation 'take a square root of a positive element'? Decide with 9\sqrt9 vs 2\sqrt2.

Example 5

easy
By Godel, can a sufficiently powerful, consistent system be complete?

Example 6

hard
Find sup⁡{n/(n+1):n∈N≥1}\sup \{ n / (n+1) : n \in \mathbb{N}_{\ge 1}\} and decide whether it is attained.

Example 7

hard
Does the set S={x∈Q:x2<2}S = \{x \in \mathbb{Q} : x^2 < 2\} have a supremum in Q\mathbb{Q}? In R\mathbb{R}?

Example 8

medium
Does adding i=−1i=\sqrt{-1} make the complex numbers complete for ALL polynomial roots (not just x2+1x^2+1)?

Example 9

easy
Can a system be consistent but incomplete?

Example 10

easy
If a system is incomplete, can there be a true statement it cannot prove?

Example 11

medium
Is the set of 2×22 \times 2 invertible matrices closed under matrix multiplication?

Example 12

hard
Determine whether ∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2} converges in R\mathbb{R}.

Example 13

medium
Does every bounded increasing sequence of reals converge? Name the property that guarantees it.

Example 14

medium
Closure of {1}\{1\} under addition is the set of all _____.

Example 15

medium
Does the polynomial x5−3x+1x^5 - 3x + 1 have at least one real root? (Use a completeness/continuity argument.)

Example 16

challenge
Explain why a consistent, sufficiently strong formal system cannot prove its own consistency (state which theorem).

Example 17

medium
Show that Q\mathbb{Q} is dense in R\mathbb{R} but R\mathbb{R} is 'larger' (uncountable). Sketch the argument that completeness fills in 'gaps'.

Example 18

medium
Why is Q\mathbb{Q} not 'complete' as an ordered field? Give a Cauchy sequence in Q\mathbb{Q} that does not converge in Q\mathbb{Q}.

Example 19

medium
Are the natural numbers complete under subtraction (is 3−53-5 a natural number)?

Example 20

challenge
The Bolzano-Weierstrass theorem says every bounded sequence in R\mathbb{R} has a convergent subsequence. Sketch why this fails in Q\mathbb{Q}.
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Related Concepts

Consistency (Meta)