Infinite
Infinite
Describes a set which is not finite. Formally, a set is infinite if it can be placed in one-to-one correspondence with a proper subset of itself.
See also
Infinity, countably infinite, uncountably infinite, aleph null, infinitesimal
Example
Problem: Show that the set of natural numbers {1, 2, 3, 4, ...} is infinite by demonstrating a one-to-one correspondence with a proper subset of itself.
Step 1: Choose a proper subset of the natural numbers. The set of even numbers {2, 4, 6, 8, ...} is a proper subset because every even number is a natural number, but not every natural number is even.
E={2,4,6,8,…}⊂N
Step 2: Define a function that pairs each natural number with exactly one even number. Multiply each natural number by 2.
f(n)=2n
Step 3: Verify the pairing is one-to-one: 1 maps to 2, 2 maps to 4, 3 maps to 6, 4 maps to 8, and so on. Every natural number gets a unique even number, and every even number is covered.
1↔2,2↔4,3↔6,4↔8,…
Step 4: Since we matched the full set of natural numbers one-to-one with a proper subset of itself (the even numbers), the formal definition of infinite is satisfied.
Answer: The natural numbers are infinite because they can be placed in one-to-one correspondence with the even numbers, which are only a proper subset of the naturals.
Another Example
Problem: Is the set {1, 2, 3} infinite?
Step 1: The set {1, 2, 3} has exactly 3 elements. Its proper subsets include {}, {1}, {2}, {3}, {1, 2}, {1, 3}, and {2, 3}.
Step 2: Every proper subset has fewer than 3 elements. A one-to-one correspondence requires both sets to have the same number of elements, so no proper subset can be matched one-to-one with {1, 2, 3}.
Step 3: Since no such correspondence exists, the set is finite, not infinite.
Answer: The set {1, 2, 3} is finite because it cannot be placed in one-to-one correspondence with any proper subset of itself.
Frequently Asked Questions
Is infinite a number?
No. Infinite is a property describing something without bound, not a specific number you can use in arithmetic. The related concept 'infinity' (often written ∞) is a symbol used to represent unboundedness, but it is not a real number. You cannot add, subtract, or divide with it using normal rules of arithmetic.
Are there different sizes of infinity?
Yes. Georg Cantor proved that not all infinite sets are the same size. The natural numbers {1, 2, 3, ...} are countably infinite, but the set of all real numbers is uncountably infinite — strictly larger. This means there are infinitely many 'levels' of infinity, each bigger than the last.
Infinite vs. Infinity
'Infinite' is an adjective describing a set or quantity that has no end — for example, 'the set of integers is infinite.' 'Infinity' (∞) is a noun or symbol representing the concept of unboundedness. You say a list is infinite; you write ∞ as a limit or endpoint in notation like limx→∞. One describes a property; the other names the abstract idea.
Why It Matters
The concept of the infinite is foundational to nearly all of higher mathematics. Calculus depends on infinite processes — limits, infinite series, and integration all rely on the idea of approaching or summing infinitely many terms. Understanding what it means for a set to be infinite also opens the door to set theory, which provides the logical framework for modern mathematics.
Common Mistakes
Mistake: Treating infinity as a real number and using it in arithmetic (e.g., writing ∞ − ∞ = 0 or ∞ / ∞ = 1).
Correction: Infinity is not a real number. Expressions like ∞ − ∞ and ∞ / ∞ are indeterminate forms, not defined values. You need limits or other tools to evaluate them.
Mistake: Assuming all infinite sets are the same size because they all 'go on forever.'
Correction: Infinite sets come in different sizes (cardinalities). The integers are countably infinite, but the real numbers are uncountably infinite — a strictly larger kind of infinity. The key test is whether a one-to-one correspondence can be established between two sets.
Related Terms
- Infinity — The symbol and concept of unboundedness
- Finite — The opposite property: having a definite size
- Set — A collection of objects that may be infinite
- Countably Infinite — An infinite set matchable with the natural numbers
- Uncountable — An infinite set larger than the natural numbers
- Aleph Null — The cardinality of the smallest infinite set
- One-to-One Function — Used to compare sizes of infinite sets
- Proper Subset — Central to the formal definition of infinite