The Paradox of the Infinite Series of Square Numbers by Galileo


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"Galileo Galilei" by cliff1066™

Galileo Galilei: Image by cliff1066™

Besides astronomy, Galileo provided a paradox about the infinite series of square numbers. His paradox gave great insight into a thorny problem in the mathematics of set theory.

Defining Galileo’s Infinite Series of Square Numbers Paradox

Galileo’s paradox is that there are “more” natural numbers than square numbers; but yet there must be the same number of each. How could Galileo do more with the same?

Prepare for Galileo with a Set of Coins

Let’s start with a set of the coins commonly circulated in Canada. The set C={1 cent, 5 cents, 10 cents, 25 cents, 100 cents (which is the $1 Loonie), 200 cents (or the $2 Toonie)}.

Canadian coins illustrate Galileo’s Paradox: Image by spcbrass

This set has 6 elements. Let’s make a set with only the values of these coins. The set V={1, 5, 10, 25, 100, 200}.

We can define two subsets, based on whether the number is even or odd. So VO={1, 5, 25} and VE={10, 100, 200}. Of course there are many other ways to define subsets of V.

Let’s define a much more interesting subset of V as the numbers which happen to be the square of a number found in V. So VS = {1, 25, 100} because 1^2=1, 5^2=25 and 10^2=100.

To no one’s surprise, VS is a proper subset of V. That simply means that there are fewer elements in VS than in V, but every member of VS is indeed a member of V.

Galileo’s Paradox of Counting Square Numbers

Let’s paraphrase what the famous astronomer Galileo Galilei stated in his “Dialogue Concerning Two New Sciences.” Consider the set of positive natural numbers, N={1, 2, 3, 4, 5, 6, 7, 8, 9…}.

Applying the “square” function to each number in N gives the set S={1, 4, 9, 16, 25, 36, 49, 64, 81…}, the set of all squares of positive integers. Clearly we could create the set of positive natural numbers which are not square numbers, so S={2, 3, 5, 6, 7, 8, 10, 11, 12…}.

Clearly, compared to the set N of all natural numbers, set S has “gaps”. This section listed the first nine elements of N, S and S. The set N only reached number 9, but the set S reached the number 81 as its ninth element. To quote the “Dialogue…”: “Therefore if I assert that all numbers, including both squares and non-squares, are more than the squares alone, I shall speak the truth, shall I not?”.

In other words, there are “more” integers than square numbers, since there are obvious gaps in S when we compare an initial sequence from each set.

Galileo Lowers the Boom

Galileo then lowered the boom on the paradox by noting that, despite these “gaps”, there are exactly as many square numbers as there are roots of square numbers. Every positive natural number in the set N can be squared, and each of these square numbers is a member of the set S.

With a one-to-one correspondence between the elements of sets N and S, Galileo concluded that the size of each of those sets is “infinite”, and that these two “infinities” could not be compared with “less than” or “equal to” operations.

Sparse or Dense but 1-to-1: Image by Mike DeHaan’

This spreadsheet shows how we can have a sparse 1-to-1 correspondence
from N to S, as well as a dense relationship between these same sets.”

How Galileo Applied His Paradox

In this point in his “Dialogue…”, Galileo was arguing that a short line segment could have “as many” points as a longer line segment. He wanted to divide the line segment an infinite number of times, so there would be an infinitesimal number of pieces.

Later mathematicians such as Cantor would demonstrate that the number of points, or “cardinality”, in a segment of the real number line is larger than the cardinality of the natural numbers.

In that sense, Galileo’s argument within his “Dialogue…” is flawed. However, he did indeed demonstrate the paradox of the infinite number of integers.

Who was Galileo Galilei?

"Galileo by Le Chevalier Falardoude" by Svadilfari

Galileo by Le Chevalier Falardoude: Image by Svadilfari

Born in 1564 in Pisa, Italy, Galileo Galilei began studying to become a priest, switched to pursue a medical degree, and left that incomplete to become immersed in mathematics.

He attained various university postings in mathematics. In Padua, he began pursuing astronomy. This led to studying the Copernican theory that the earth revolves around the sun; that assumption simplifies much of the mathematics and geometry required in astronomy.

In 1611, the Catholic Church condemned “On the Revolution of the Heavenly Orbs” by Copernicus. Galileo was warned to avoid defending the theories of Copernicus.

After Pope Urban VIII was elected, Galileo believed he could safely publish his “Dialogue…”. The next year, 1633, Galileo was condemned and placed under house arrest.

Galileo died in 1642.

Conclusion to Galileo’s Paradox of the Size of an Infinite Set

Galileo used square numbers to demonstrate the mathematical paradox in set theory that a “smaller” subset of an infinite set can, itself, be infinite. He did this by showing the one-to-one relationship between the initial infinite series of positive integers and an infinite subset, the series of square numbers.


Butler, T. Galileo’s paradox. Suitcase of Dreams. (2006). Accessed Sept. 3, 2011.
Crew, H. and de Salvio, A., translators of Galileo Galilei’s Dialogue Concerning Two New Sciences. MacMillan. (1914). Accessed Sept. 3, 2011.
Machamer, P. Galileo Galilei. Stanford Encyclopedia of Philosophy. (2005, 2009). Accessed Sept. 3, 2011.

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