Galileo's paradox
Encyclopedia
Galileo's paradox is a demonstration of one of the surprising properties of infinite sets.
In his final scientific work, the Two New Sciences
, Galileo Galilei
made two apparently contradictory statements about the positive whole numbers. First, some numbers are perfect square
s (i.e., the square of some integer, in the following just called a square), while others are not; therefore, all the numbers, including both squares and non-squares, must be more numerous than just the squares. And yet, for every square there is exactly one number that is its square root, and for every number there is exactly one square; hence, there cannot be more of one than of the other. This is an early use, though not the first, of a proof by one-to-one correspondence of infinite sets.
Galileo concluded that the ideas of less, equal, and greater apply to finite sets, but not to infinite sets. In the nineteenth century, using the same methods, Cantor
showed that while Galileo's result is correct as applied to the whole numbers and to the rational numbers, the general conclusion does not follow: some infinite sets are larger than others, in that they cannot be put into one-to-one correspondence.
While these two notions are equivalent in the finite case, they do not agree in the infinite case (as seen above). To wit, if A, B are finite sets with A ⊆ B then A ≠ B if and only if #A < #B, where # is the cardinality symbol. Or put differently (still in the case where A ⊆ B are finite) A = B ⇔ #A = #B.
In his final scientific work, the Two New Sciences
Two New Sciences
The Discourses and Mathematical Demonstrations Relating to Two New Sciences was Galileo's final book and a sort of scientific testament covering much of his work in physics over the preceding thirty years.After his Dialogue Concerning the Two Chief World Systems, the Roman Inquisition had banned...
, Galileo Galilei
Galileo Galilei
Galileo Galilei , was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism...
made two apparently contradictory statements about the positive whole numbers. First, some numbers are perfect square
Square number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
s (i.e., the square of some integer, in the following just called a square), while others are not; therefore, all the numbers, including both squares and non-squares, must be more numerous than just the squares. And yet, for every square there is exactly one number that is its square root, and for every number there is exactly one square; hence, there cannot be more of one than of the other. This is an early use, though not the first, of a proof by one-to-one correspondence of infinite sets.
Galileo concluded that the ideas of less, equal, and greater apply to finite sets, but not to infinite sets. In the nineteenth century, using the same methods, Cantor
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...
showed that while Galileo's result is correct as applied to the whole numbers and to the rational numbers, the general conclusion does not follow: some infinite sets are larger than others, in that they cannot be put into one-to-one correspondence.
Galileo on infinite sets
The relevant section of Two New Sciences is excerpted below:Resolution of the paradox
The paradox is quickly resolved by noticing that two different notions of being bigger are used in the paradox. For the sake of simplicity, let us write S for the set of square natural numbers. If one says that the natural numbers are more numerous than the squares, one means that the squares form a proper subset of the natural numbers (i.e. S ⊂ N) and thus N is bigger than S (this is the first notion of being bigger). On the other hand, one notices that the squaring function N → S, n → n2 is a bijection between N and S (i.e. these two sets have the same cardinality) and thus they are equally big (this is the second notion of being bigger).While these two notions are equivalent in the finite case, they do not agree in the infinite case (as seen above). To wit, if A, B are finite sets with A ⊆ B then A ≠ B if and only if #A < #B, where # is the cardinality symbol. Or put differently (still in the case where A ⊆ B are finite) A = B ⇔ #A = #B.
External links
- Philosophical Method and Galileo's Paradox of Infinity, Matthew W. Parker, in the PhilSci Archive