Equinumerosity and the definition of finiteness
What does it mean for two sets to have the same size? What does it mean, exactly, to say that a set is finite or that it is infinite? Can we provide a precise mathematical definition of finiteness?
Before the dinner party, young Gottlob happens to pass by the dining room prepared for the guests, and he observes at a glance, without counting, that the number of plates on the dining table is the same as the number of knives.
How does he do this without counting? He simply notices that every place setting has one plate and one knife. By thus associating every plate with its counterpart knife at that setting, he has formed a one-to-one correspondence between the plates and knives, which implies that their numbers will be the same without any need to count them. It is not necessary to know how many plates or knives there are in order to know the numbers are the same.
In the lecture hall I might similarly tell at a glance that the number of people at the lecture is the same as the number of noses, while counting neither people nor noses.
I can do this simply by placing the people into a one-to-one correspondence with the noses, associating every person with their own nose. If indeed everyone has a nose and there are no dreadful extra noses in someone's pocket, then indeed this is a one-to-one correspondence, which shows that the two numbers are the same, whatever they are, with no need to count them.
In general, we say that sets A and B are equinumerous if there is a one-to-one correspondence between their elements.
Every individual a of the first set A is matched with a corresponding individual b in the second set B in such a way that all individuals have exactly one partner in the other set. The shepherd counts off his sheep on his fingers.
The main idea here is that equinumerosity seems to provide us with a criterion for saying that two sets have the same number of elements, in a manner that is conceptually prior to our having a suitable concept of number or a method of counting. For example, we can speak coherently of equinumerosity with infinite sets, even if we may not yet have figured out an appropriate notion of counting or numbering them. This idea has come to be known as the Cantor-Hume principle, expressing our expectations for how numbers should behave, whether in finite or infinitary contexts—the principle expresses a necessary requirement on whatever number concept we may eventually settle upon.
Cantor-Hume Principle. Two sets have the same number of elements if and only if those sets can be placed in a one-to-one correspondence.
The Cantor-Hume principle is also widely known simply as Hume's principle, in light of David Hume's brief statement of it in his Treatise of Human Nature (1739):
When two numbers are so combin'd, as that the one has always an unite answering to every unite of the other, we pronounce them equal; (I.III.I)
The idea has come up many times in the philosophy of numbers and infinity. Long before Hume, Galileo had in effect considered equinumerosity in his treatment of several confounding matters of infinity in his final scientific work, Dialogues Concerning Two New Sciences (1638). After Hume, at the end of the 19th century, Gottlob Frege founded his theory of arithmetic directly upon the Cantor-Hume principle. Around the same time or slightly before Cantor had placed equinumerosity at the very center of his spectacular treatment of infinite and especially uncountable cardinalities. It is evidently a very natural idea.
Before discussing Galileo, let us roll Aristotle's wheel across the page. Consider the large blue circle as shown here centered at A with radius AC, containing the smaller red circle with radius AB within it.
A swift kick will set the wheel rolling from left to right, bringing its center from A to D, smoothly completing exactly one revolution. During the journey, every point on the circumference of the blue circle is therefore matched with its point of contact on the ground, which changes continuously as the wheel rolls. The blue circumference is thus laid out onto the segment CF. In particular, the circumference of the blue circle is exactly the same as the length of segment CF.
A more troublesome observation, however, concerns the smaller red circle.
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