In this thesis, we study the concept of size when it comes to infinite sets. From the observation that the Part-Whole Principle (\"The whole is greater than the part\") and the commonly accepted measure of size in mathematics (Cardinality) disagree when it comes to infinite sets, we study a new way of defining the size of a set proposed by Benci and Di Nasso in \"Numerosities of labeled sets: a new way of counting\", and further explored in a series of papers1, and look at the motivations for what theory would statisfy the concept of size best.
The mathematical part of this thesis is motivated by the observation the observation that the account of size as cardinality yields some counter-intuitive results. With cardinality, we find that the set of (positive) even numbers and the set of natural numbers have the same \"size\" whereas the Part-Whole Principle yields that the natural numbers (the whole) are greater than the even numbers (the part). Moreover, it is intuitively plausible to believe that the natural numbers are greater since they occur twice as often, or are more dense than the even numbers. In that perspective, we focus on detailing Vieri Benci and Maurro Di Nasso's work on developing a new system to compare size of infinite sets of natural numbers. By considering suitable \"labelings\", they develop a theory for comparing infinite countable sets which is consistent with the usual properties of finite cardinalities. This theory, unlike Cantor's theory of cardinality, satisfies the Part-Whole principle – if X is a proper subset of Y, then the numerosity of X is strictly smaller than the numerosity of Y.
In the philosophical part of this thesis, we analyze Galileo's paradox and related puzzles in order 1) to gain a better sense of the options available to us in comparing the sizes of infinite collections and 2) to reflect more explicitly on what considerations are relevant in choosing one option over the others. From the need to choose one of these options as the tool we will use to measure size, we list them and try to judge of their credibility. In doing so, we come across Emmanuel Maignan who argues, as opposed to Galileo, that infinite collections can be compared, than we can say of an infinite collection that it is greater than another. We analyze his argument and discover some more elements underlying the concept of size, notably an idea of density and a more generalist Euclidean approach, therefore widening our range of options. We then use Matthew Parker's Method of Conceptual Articulation to gain a better perspective on how to decide how we should define our theory of size. We then explain that the full extent of the powers of a theory and the interests that it serves are exhibited only once the theory has been made into one. We cannot judge of a potential theory while it is still only at the stage of principle like the Part-Whole Principle is when by itself or like one-to-one correspondence was before being developed in Cantor's theory of cardinality.
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