Completeness (Intuition) Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Completeness (Intuition).

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A system is complete if every statement it can express is settled — provably true or provably false — with no true-but-unprovable gaps.

Common stuck point: The procedure for completeness (intuition) is the easy part; the trap is confusing completeness with consistency. Asking "Does this system prove every true statement it can express, leaving no true-but-unprovable gaps?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does this system prove every true statement it can express, leaving no true-but-unprovable gaps?

Worked Examples

Example 1

easy

The 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.

Answer

1, 1.4, 1.41, 1.414, \ldots \to \sqrt{2} \in \mathbb{R} \setminus \mathbb{Q}

First step

Consider the decimal approximations of

\sqrt{2}

1, 1.4, 1.41, 1.414, 1.4142, \ldots

Full solution

2
Each term is rational (a terminating decimal). The sequence converges — each term is closer to $\sqrt{2}$ than the last.
3
But $\sqrt{2} \notin \mathbb{Q}$ . In $\mathbb{Q}$ , this sequence has no limit — the limit 'falls through a gap.'
4
In $\mathbb{R}$ , $\sqrt{2}$ exists, so the limit exists. $\mathbb{R}$ is complete; $\mathbb{Q}$ is not.

Completeness of

\mathbb{R}

means every Cauchy sequence of reals converges to a real number. The rationals have gaps at irrational numbers, which is why

\mathbb{Q}

is not complete.

Example 2

medium

Check 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

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 4

medium

By Godel's first incompleteness theorem, can a sufficiently strong, consistent, recursively axiomatized theory (like Peano Arithmetic) decide every arithmetic statement?

Example 5

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 6

hard

Use the Monotone Convergence Theorem (a form of completeness) to show that the sequence

a_n = (1 + 1/n)^n

converges.

Example 7

hard

Explain why a proof by induction that omits the base case is incomplete, using

P(n)

: '

n = n + 1

' to illustrate.

Example 8

hard

Why does the Extreme Value Theorem (continuous functions on

[a,b]

attain max/min) fail on the open interval

(a,b)

? Give an example.

Example 9

challenge

State Godel's second incompleteness theorem in plain language, and explain why it implies no 'self-certifying' foundational system for mathematics.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A student proves a statement for all even integers but forgets odd integers. Is the proof complete? What is missing?

Example 2

medium

State whether

\mathbb{Q}

\mathbb{R}

is a better domain for solving

x^2 = 2

, and explain in terms of completeness.

Example 3

easy

Does every quadratic equation

ax^2+bx+c=0

(

a\ne0

) have a solution in the complex numbers?

Example 4

easy

Is the set of rationals 'complete' in the sense that

\sqrt2

is rational?

Example 5

easy

Does the real number line contain a number for the limit of

3,3.1,3.14,3.141,\ldots

(toward

\pi

Example 6

easy

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

Example 7

easy

Can a system be consistent but incomplete?

Example 8

easy

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

Example 9

easy

Are the integers 'complete' for division, i.e., is

7\div2

an integer?

Example 10

easy

Does the real number system have a least upper bound for the set

\{x : x^2<2\}

Example 11

medium

The rationals fail completeness because some bounded sets have no rational supremum. Give such a set.

Example 12

medium

Is the statement 'every even integer

>2

is a sum of two primes' (Goldbach) known to be provable in standard arithmetic?

Example 13

medium

Does adding

i=\sqrt{-1}

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

x^2+1

Example 14

medium

An incomplete system has a true statement

G

it cannot prove. Can

\neg G

be proved in a consistent such system?

Example 15

medium

Are the natural numbers complete under subtraction (is

3-5

a natural number)?

Example 16

medium

Is 'unprovable in system

S

' the same as 'false'? Decide using a true-but-unprovable Godel sentence.

Example 17

challenge

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

Example 18

challenge

The reals are complete (every Cauchy sequence converges). Use this to argue

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

converges to a real number, and find it.

Example 19

challenge

Is the theory of a 'dense linear order without endpoints' (like

\mathbb{Q}

) complete in the logical sense? State the known result.

Example 20

medium

Are the rationals complete under the operation 'take a square root of a positive element'? Decide with

\sqrt9

\sqrt2

Example 21

medium

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

Example 22

medium

Is the integer system complete for solving

2x=3

Example 23

easy

Are the natural numbers

\mathbb{N}

closed under subtraction (i.e., is

\mathbb{N}

'complete' for subtraction)?

Example 24

easy

Does

\mathbb{R}

have a least upper bound for the set

\{x \in \mathbb{R} : x^2 < 9\}

Example 25

easy

Is the set of even integers closed (complete) under addition?

Example 26

easy

Give one example of a real number that is in

\mathbb{R}

but not in

\mathbb{Q}

Example 27

easy

Is the set of integers closed under multiplication?

Example 28

medium

Does the polynomial

x^2 + 2x + 5 = 0

have a solution in

\mathbb{R}

? In

\mathbb{C}

Example 29

medium

Find

\sup\{1 - \tfrac{1}{n} : n \in \mathbb{N}_{> 0}\}

Example 30

medium

A proof of 'all primes

p > 2

are odd' must handle primes

\le 2

too. How does the statement avoid that gap?

Example 31

medium

Is the set

\{x \in \mathbb{R} : x < 5\}

bounded above? If so, what is its supremum?

Example 32

medium

Does the polynomial

x^5 - 3x + 1

have at least one real root? (Use a completeness/continuity argument.)

Example 33

medium

Is the set of

2 \times 2

invertible matrices closed under matrix multiplication?

Example 34

hard

Find

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

and decide whether it is attained.

Example 35

hard

Does the set

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

have a supremum in

\mathbb{Q}

? In

\mathbb{R}

Example 36

hard

Determine whether

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

converges in

\mathbb{R}

Example 37

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)

Background Knowledge

These ideas may be useful before you work through the harder examples.

consistency meta