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In his Dialogue of the Two Sciences, Galileo famously considered the following puzzle. Is the set of all square natural numbers smaller in size than the set of all natural numbers, or do they have the same size? The issue comes down to the incompatibility of two intuitive principles. According to the Part-Whole Principle, if A is a proper subset of B, then the size of A is strictly less than the size of B. According to the Bijection Principle, two sets A and B have the same size if and only if there exists a one-to-one correspondence between them. Cantor famously based his definition of cardinality on the Bijection Principle, rejecting the validity of the Part-Whole Principle and thus laying the foundations of modern set theory. However, recent developments in mathematical logic, such as the theory of numerosities, have renewed the interest in the question whether Cantor’s choice was the only way to develop a coherent and viable theory of size for infinite sets. The question is relevant to both the epistemology of mathematics and the philosophy of set theory.

This talk will explore the non-Cantorian path through Galileo’s labyrinth. I will argue that we should distinguish between two intuitions that clash with the Bijection Principle: the Part-Whole Principle itself and a strengthening of it that I call the Density Intuition. Although modern non-Cantorian proposals such as the theory of numerosities are implicitly built around the latter, I will argue that the former is more likely to provide a well-motivated and coherent alternative to Cantorian sizes, and I will present such a theory for sets of natural numbers.