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

G

it cannot prove. Can

\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

\sqrt9

\sqrt2

Example 5

easy

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

Example 6

hard

Find

\sup \{ n / (n+1) : n \in \mathbb{N}_{\ge 1}\}

and decide whether it is attained.

Example 7

hard

Does the set

S = \{x \in \mathbb{Q} : x^2 < 2\}

have a supremum in

\mathbb{Q}

? In

\mathbb{R}

Example 8

medium

Does adding

i=\sqrt{-1}

make the complex numbers complete for ALL polynomial roots (not just

x^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 \times 2

invertible matrices closed under matrix multiplication?

Example 12

hard

Determine whether

\sum_{n=1}^{\infty} \frac{1}{n^2}

converges in

\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\}

under addition is the set of all _____.

Example 15

medium

Does the polynomial

x^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

\mathbb{Q}

is dense in

\mathbb{R}

but

\mathbb{R}

is 'larger' (uncountable). Sketch the argument that completeness fills in 'gaps'.

Example 18

medium

Why is

\mathbb{Q}

not 'complete' as an ordered field? Give a Cauchy sequence in

\mathbb{Q}

that does not converge in

\mathbb{Q}

Example 19

medium

Are the natural numbers complete under subtraction (is

3-5

a natural number)?

Example 20

challenge

The Bolzano-Weierstrass theorem says every bounded sequence in

\mathbb{R}

has a convergent subsequence. Sketch why this fails in

\mathbb{Q}

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Related Concepts

Consistency (Meta)