# Galileo's Paradox and Purely Euclidean Numerosities

## Guillaume Massas

Galileo [2] famously asked whether there are more natural numbers than square numbers. On the one hand, every square number is a natural number, while the converse is false, suggesting that there are strictly more natural numbers than squares. On the other hand, there is an obvious way to define a one-to-one correspondence between the two collections, obtained by mapping any natural number to its square, suggesting that they are, in fact, equinumerous. Galileo’s paradox is a striking example of a clash between two intuitive principles about sizes of infinite collections [4]. According to the Part-Whole Principle **(PW)**, any proper subcollection of a collection \(A\) has size strictly less than the size of \(A\). According to the Bijection Principle **(BP)**, any two collections have the same size if and only if there is a one-to-one correspondence between them. While the modern notion of cardinality obeys both principles in the finite, Cantor famously adopted **(BP)** as the foundation of his transfinite arithmetic, thus rejecting **(PW)**. However, the recently developed theory of *numerosities* [1] has been presented as a viable alternative to the Cantorian picture that is based on **(PW)** rather than on **(BP)**.

In this talk, I will focus on numerosities for countable sets. I will argue that the current theory of numerosities faces some major issues because it mixes two distinct intuitions. The first one is the *Euclidean Intuition* that the whole is always greater than any of its proper parts. The second, which I call the *Density Intuition*, is the intuition that the size of a set of natural numbers is determined by the frequency with which its elements appear in the sequence of natural numbers. While the first intuition is compatible with a very natural invariance condition that one would want to impose on any adequate notion of size for sets, the second intuition (and therefore also the standard theory of numerosities) is not. I will propose an alternative theory that is based purely on the Euclidean Intuition and is, in some precise axiomatic sense, the best way to combine part-whole intuitions with invariance criteria. Time permitting, I will also give a semantic intuition for this theory of Purely Euclidean Numerosities that is based on possibility semantics for classical first-order logic [3].

## Bibliography

- Benci, Vieri and Di Nasso, Mauro,
*Numerosities of labelled sets: a new way of counting*,,vol. 173 (2003), pp. 50–67.*Advances in Mathematics* - Galilei, Galileo_Discorsi e dimostrazioni matematiche intorno a due nuove scienze_,Boringhieri, 1958.
- Holliday, Wesley H.,
*Possibility Semantics*,, vol. 2 (2021).*Selected Topics from Contemporary Logics* - Mancosu, Paolo,
*Measuring the size of infinite collections of natural numbers: Was Cantor’s theory of infinite number inevitable?*,,vol. 2 (2009), no. 4, pp. 612–646.*The Review of Symbolic Logic*