Galileo's paradox is a demonstration of one of the surprising properties of

infinite set In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Properties The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set ...

s. In his final scientific work, '' Two New Sciences'',

Galileo Galilei Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642), commonly referred to as Galileo Galilei ( , , ) or mononymously as Galileo, was an Italian astronomer, physicist and engineer, sometimes described as a poly ...

made apparently contradictory statements about the

positive integers In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...

. First, a square is an integer which is the square of an integer. Some numbers are

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

s, 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 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 the idea of

one-to-one correspondence In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivale ...

in the context of infinite sets. Galileo concluded that the ideas of ''less'', ''equal'', and ''greater'' apply to finite quantities but not to infinite quantities. During the nineteenth century

Cantor A cantor or chanter is a person who leads people in singing or sometimes in prayer. Cantor as a profession generally refers to those leading a Jewish congregation, although it also applies to the lead singer or choir director in Christian contexts. ...

found a framework in which this restriction is not necessary; it is possible to define comparisons amongst infinite sets in a meaningful way (by which definition the two sets, integers and squares, have "the same size"), and that by this definition some infinite sets are strictly larger than others. The ideas were not new with Galileo, but his name has come to be associated with them. In particular,

Duns Scotus John Duns Scotus ( ; , "Duns the Scot"; – 8 November 1308) was a Scottish Catholic priest and Franciscan friar, university professor, philosopher and theologian. He is considered one of the four most important Christian philosopher-t ...

, about 1302, compared even numbers to the whole of numbers.

Galileo on infinite sets

The relevant section of '' Two New Sciences'' is excerpted below:

References

External links

Philosophical Method and Galileo's Paradox of Infinity
by Matthew W. Parker – PhilSci-Archive {{Galileo Galilei Paradoxes of set theory Paradoxes of infinity

Paradox A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictor ...

Galileo on infinite sets

See also

References

External links