The Paradox of Galileo

The Galileo Paradox is an example illustrating the properties of infinite sets . In a nutshell: there are as many natural numbers as there are squares of natural numbers , that is, in the set 1, 2, 3, 4 ... there are as many elements as there are in the set 1, 4, 9, 16 ...

In his last work, “Two Sciences,” Galileo gave two conflicting judgments about natural numbers . First: some numbers are exact squares (that is, the squares of other integers); other numbers do not have this property. Thus, there should be more exact squares and ordinary numbers than just exact squares. The second proposition: for every natural number there is an exact square, and vice versa - for every exact square there is an integer square root , therefore there must be the same number of exact squares and natural numbers. This is one of the first, although not the earliest, example of using the notion of one-to-one mapping in the context of infinite sets.

Galileo concluded that the same number of elements can only be judged for finite sets . In the 19th century, George Cantor , using his theory of sets, showed that one can introduce the “number of elements” for infinite sets — the so-called cardinality of a set . Moreover, the cardinalities of the set of natural numbers and the set of exact squares coincided (Galileo's second argument turned out to be true). Galileo’s paradox came into conflict with the Euclidean axiom , which states that the whole is larger than any of its own parts (the own part means the part that does not coincide with the whole) ^[1] . It is remarkable to what extent Galileo anticipated subsequent work in the field of infinite numbers. He showed that the number of points on a short line segment is equal to the number of points on a larger segment, but, of course, did not know the Cantor proof that its power is greater than the power of the set of integers. Galileo had more urgent tasks. He dealt with contradictions in the Zeno paradox in order to clear the way for his mathematical theory of motion ^[2] .