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Galileo's Paradox of Infinity Calculator

Created by Luciano Miño
Reviewed by Anna Szczepanek, PhD and Adena Benn
Last updated: Jan 18, 2024

This Galileo's paradox of infinity calculator will detail the ins and outs of a problem regarding the relationship between two infinite sets of numbers proposed by Galileo in his final work, "Two New Sciences".

The paradox refers to one's inability (according to Galileo) to compare the "sizes" of two infinite sets, in this case: the natural numbers and the perfect square numbers.

In this article, we will cover the following:

  • What Galileo's paradox of infinity is;
  • How to solve Galileo's paradox of infinity;
  • One-to-one correspondence between natural numbers and squares; and
  • The definition of cardinality.

What is Galileo's paradox of infinity?

Considering the set of natural numbers (including zero):

N={0,1,2,3,...}\N = \{0, 1, 2, 3,...\}

and the set of perfect squares (numbers which are the square of the naturals):

S={0,1,4,9,...}\text{S} = \{0, 1, 4, 9, ... \}

Galileo observed:

  • There are more naturals than squares. Some of the natural numbers are squares while others are not, so all of these numbers together must be more numerous than just the number of squares.
  • However, each natural has its corresponding square (11,24,391 \rightarrow 1, 2 \rightarrow 4, 3 \rightarrow 9, and so on), so there must be the same number of naturals as there are squares.

This led Galileo to state that the concepts of larger or smaller do not apply to infinite sets, and we simply cannot compare them.

We know that many things in math involving infinity can be solved, such as geometric series (covered in our sum of series calculator). So, how do we solve this problem? Using the concepts of cardinality and one-to-one correspondence.

🙋 Try our Galileo's Paradox of Infinity calculator to go through each of Galileo's reasonings and see if you can devise your own solution!

Cardinality definition

Cardinality, in simple terms, refers to a set's "size" or number of elements. A set B={0,1,2,4}B = \{0, 1, 2, 4\} has four elements and, therefore, a cardinality of four. We denote cardinality using B=4|B| = 4.

One-to-one correspondence between natural numbers and squares

But how do we measure an infinite set's size? Here's where the mind of Georg Cantor comes in. He proposed that infinite sets (such as naturals or squares) can be compared to each other based on the existence of a one-to-one correspondence between their elements.

🙋 A one-to-one correspondence or bijection means that each element from set AA is paired with exactly one element from set BB, and each element from BB is paired with exactly one element from AA.

More formally:

  • Two sets have the same size (cardinality) if and only if a bijective mapping (one-to-one correspondence) exists between them.

Therefore, if we can find a bijective mapping between any two sets (the set of squares and the set of natural numbers, for example), we can conclude that they are the same size!

💡 Luckily, we've already defined this bijective mapping since (as Galileo said): each natural has its corresponding square and each square has its corresponding root.

This bijective mapping can be written as f:NSf:\N \rightarrow S with:

f(n)=n2,  nNf(n) = n^2,\ \forall\ n \in\N

So, according to the previous proposition , the set of natural numbers, N\N, and the set of squares, SS, have the same size! There aren't more natural numbers than perfect squares.

Other infinite sets

There are many infinite sets ♾ with the same cardinality as the natural numbers. Even the rational numbers are the same size!

Below you'll find examples of some of these sets.

Integers and naturals (including zero)

Let's take the set of integers:

Z={0,1,1,2,2,3,3,...}\footnotesize{ \Z = \{0, 1, -1, 2, -2, 3, -3, ...\} }

and the set of natural numbers (including zero):

N={0,1,2,3,...}\footnotesize{ \N = \{0, 1, 2, 3, ...\} }

We may be inclined to think that the size of Z\Z is larger than N\N since the naturals include only non-negative numbers. But, based on our previous findings, if we find one bijective mapping (there could be more, we only need one) between these sets, they are the same size. Let's see that, in fact, that mapping exists.

Let's call g:NZg: \N \rightarrow \Z, as:

g(n)={n+12,if n is oddn2,if n is eveng(n) = \begin{cases} \frac{n+1}{2}, &\text{if $n$ is odd}\\ -\frac{n}{2}, &\text{if $n$ is even} \end{cases}

which is a bijection. Therefore, Z\Z and N\N are the same size! You can read more about integers in our integer calculator and this consecutive integers calculator.

Even numbers and naturals (including zero):

Now, let's consider the set of even numbers, 2N2\N, and again the set of naturals, including zero.

We can easily find the bijective mapping h:N2Nh: \N \rightarrow 2\N, with h(n)h(n) defined as:

h(n)=2n,  nNh(n) = 2n,\ \forall\ n \in \N

So, the set of even numbers also has the same size as the set of naturals.

Now try using our Galileo's paradox of infinity calculator. It will walk you through Galileo's steps when dealing with this paradox.

If you still have questions regarding infinite sets, perhaps our Hilbert's hotel calculator 🏨 will resolve these doubts with a similar paradox that is easier to visualize.


Are there more natural numbers than perfect squares?

No, the set of natural numbers and the set of perfect squares are equal in size. There is a one-to-one correspondence between natural numbers and perfect squares, so there is exactly one perfect square for each natural number, and vice-versa.

How many perfect square numbers are there between 0 and 100?

11 if you include 0 and 100, 9 if you do not. Perfect square numbers are those which are the square of some integer. Between 0 and 100 (including lower and upper bounds), there are 11 perfect squares: {0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100}.

How can I determine if two infinite sets are equal in size?

To determine if two infinite sets (A and B) are equal in size, you need to determine if a one-to-one correspondence exists between their elements. For that:

  1. Find an injective function f: A → B.
  2. Find an injective function g: B → A.
  3. By the Cantor–Bernstein theorem, there exists a bijective function h: A → B, and the sets are equal in size.

Can an infinite set be countable?

Yes. By definition, an infinite set is countably infinite if a one-to-one correspondence exists between their elements and the set of natural numbers N.

Luciano Miño
Galileo, logically, considered that the number of perfect squares must be less than the number of naturals (non-negative integers). Let's see why:
Input any two numbers and see how many natural numbers and perfect square numbers are between them.
Number 1
Number 2
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