The concept of infinity in mathematics can seem paradoxical. How can there be different sizes of infinity? Isn’t infinity just endlessly large? While the notion of infinity being quantified may seem counterintuitive, mathematicians have developed rigorous definitions and comparisons of different infinite sets.
What Does Infinity Mean in Mathematics?
In mathematics, infinity describes something that is boundless, endless, or immeasurable. The infinity symbol ∞ represents the idea of something continuing without limit. Infinity is not considered a real number, but it can be used in algebraic expressions and equations to represent a quantity or process that has no finite endpoint.
There are two main types of infinity in mathematics:
- Countable infinity – This refers to an infinite set that can be enumerated or counted. The set of natural numbers {1, 2, 3, …} is a simple example. Despite having no endpoint, the elements can be put in a one-to-one correspondence with the positive integers.
- Uncountable infinity – This describes an infinite set that cannot be enumerated or counted. The set of real numbers between 0 and 1 is an example. No matter how many decimal places are specified, there are still infinitely more real numbers in that interval.
Georg Cantor developed fundamental theories about different infinite sets during the late 19th century. He showed that although both countable and uncountable infinities are infinitely large, they are actually comprised of differing quantities of elements.
What is the Smallest Infinity?
When comparing sizes of infinite sets, the smallest possible infinity is a countable infinity. Sets that are countably infinite have a one-to-one correspondence with the positive integers. Uncountable infinite sets, on the other hand, have a larger cardinality and cannot be put in direct bijection with the natural numbers.
Some examples of countably infinite sets are:
- The set of natural numbers {1, 2, 3, …}
- The set of integers {…, -2, -1, 0, 1, 2, …}
- The set of rational numbers (all fractions)
- The set of algebraic numbers (solutions to polynomial equations)
These sets all share the quality that their elements can be sequentially counted and mapped to the positive integers, even though they stretch to infinity without end. Therefore, they exemplify the smallest possible cardinality of an infinite set.
Comparing Finite and Infinite Sets
To understand why countable sets represent the smallest infinity, it helps to compare finite and infinite sets:
| Finite Set | Infinite Set |
|---|---|
| A finite set has a definite number of elements. For example, the set {a, b, c} has 3 elements. | An infinite set has an endless number of elements. For example, the set of natural numbers {1, 2, 3, …} has no last element. |
| The number of elements in a finite set is called its cardinality. | The cardinality of an infinite set is called its cardinal number. |
| A finite set with n elements is said to have cardinality n. | Countably infinite sets have cardinality א0 (“aleph-null”). |
| There is no bijection between a finite set and an infinite set. | A countable infinite set has a bijection to the natural numbers. |
| Finite cardinalities get larger as you count higher. | Countable infinite sets all have the same smallest cardinality. |
This table summarizes how cardinality behaves differently with finite versus infinite sets. When mapping sets with a bijection, the cardinality gives a sense of the “size” of the set. Finite cardinalities get larger as you map to higher natural numbers. However, all countably infinite sets have the same smallest cardinality of א0.
Properties of Countable Infinity
The main properties that characterize countably infinite sets are:
- Enumeration – The elements can be numbered sequentially.
- One-to-one correspondence – Each element can be matched to a unique natural number.
- Countable union – The union of two countable sets is still countable.
- Same cardinality – All countably infinite sets have the cardinality א0.
These special properties distinguish the smallest “countable” infinity from higher uncountable infinities. Let’s look at each in more detail:
Enumeration
A countably infinite set can be sequentially enumerated by mapping its elements to the natural numbers {1, 2, 3, …}. For example, mapping the integers:
…, -3 ↦ 1, -2 ↦ 2, -1 ↦ 3, 0 ↦ 4, 1 ↦ 5, 2 ↦ 6, …
No elements are left out of the enumeration, even though it stretches to infinity in both directions.
One-to-One Correspondence
Every element in a countably infinite set can be paired with a unique natural number. This bijection demonstrates that the set has the same cardinality as the natural numbers, even though it is infinite. For example, the positive even integers map to the naturals:
2 ↦ 1, 4 ↦ 2, 6 ↦ 3, 8 ↦ 4, …
Despite being a proper subset of the natural numbers, the even numbers can be perfectly paired with the naturals in a one-to-one manner. This shows they have the same smallest cardinality.
Countable Union
Taking the union of two countable sets results in another countable set. For example:
Integers Z = Positive integers + Negative integers
Naturals + Integers = All integers
Since the naturals and integers are countable, their combined union set is also countably infinite. This is not true for uncountable sets.
Same Cardinality
All countably infinite sets share the same cardinal number א0. This can be shown by demonstrating a bijection between any two countable sets. For example, the integers have the same cardinality as the rational numbers, even though at first glance they seem like different magnitudes of infinity:
Integers: …, -2, -1, 0, 1, 2, … Rational numbers: 0/1, 1/1, -1/1, 2/1, 1/2, -1/2, …
Despite appearing very different, the elements in both sets can be enumerated and paired up. This means they actually have the same smallest possible cardinality.
Examples of Countably Infinite Sets
Some important examples of sets that are countably infinite include:
- Natural numbers {1, 2, 3, …} – This ordered, well-ordered set is the basis for countable cardinality.
- Integers {…, -2, -1, 0, 1, 2, …} – Includes both positive and negative numbers, with no gaps.
- Rational numbers – All numbers that can be written as fractions p/q for integers p and q.
- Algebraic numbers – Solutions to polynomial equations with integer coefficients.
- Computable numbers – Real numbers with algorithms to compute their digits.
Each of these important sets in mathematics can be mapped in a one-to-one manner to the counting numbers and thus have the cardinality of the smallest infinity א0. However, many other infinite sets are larger and uncountable…
Uncountable Infinities
While countably infinite sets represent the smallest possible cardinality, there are actually many sizes of infinity that are uncountably infinite. Some examples of uncountable infinite sets include:
- Real numbers
- Interval between 0 and 1
- Power set of the natural numbers
- Cantor set
- Functions from natural numbers to a finite set
These sets cannot be enumerated or placed in one-to-one correspondence with the natural numbers. They have a larger cardinality and represent higher orders of infinity.
Cardinal Hierarchy
Georg Cantor showed that there is actually an infinite hierarchy of infinite cardinal numbers. Each cardinal number represents the size and cardinality of an infinite set:
- א0 – Cardinality of the natural numbers. The smallest infinity.
- א1 – Cardinality of the reals. The next largest after א0.
- א2 – Cardinality of the power set of the natural numbers.
- א3, א4, … – Continues infinitely.
This cardinal hierarchy represents an infinite progression of higher and higher infinities. Although unintuitive, it shows that infinity comes in many gradations and the “smallest” infinity occurs with the countable sets.
The Continuum Hypothesis
A key unsolved question in set theory is the continuum hypothesis. Proposed by Cantor in 1878, it states:
There is no set whose cardinality is between א0 and א1.
In other words, there is no intermediate size of infinity between the countable numbers and the reals. This would mean the cardinal number א1 representing the reals is the second-smallest infinity.
The continuum hypothesis remains unproven despite intense effort by mathematicians over the decades. Its validity would have sweeping implications for mathematics. Most set theorists believe it is independent and impossible to prove from the standard ZFC axioms of set theory.
Conclusion
The notion of infinity comes in many sizes, and the smallest possible infinity occurs in countable sets. Sets like the natural numbers that can be mapped sequentially to the integers represent the lowest order of cardinality, denoted by א0. Despite being infinitely large, they have a smaller cardinal number than uncountable infinite sets like the reals. The concept of infinite hierarchies and cardinal numbers provides a rigorous way to compare notions of infinity that all seem paradoxically endless.