On Multiverses and Infi nite Numbers

Abstract

A multiverse is comprised of many universes, which quickly leads to the question: How many universes? There are either finitely many or infinitely many universes. The purpose of this paper is to discuss two conceptions of infinite number and their relationship to multiverses. The first conception is the standard Cantorian view. But recent work has suggested a second conception of infinite number, on which infinite numbers behave very much like finite numbers. I will argue that that this second conception of infinite number is the correct one, and analyze what this means for multiverses. When it comes to infinite number, the overarching question that I will address is: Which objects are the infinite numbers? How do I mean this question? In the finite case, you have the finite whole numbers, also called the natural numbers, numbers like 7, 13, and 106. I am interested in the question: How should these numbers, the finite natural numbers, be extended into the infinite? I will address this question, and then talk about how that conception of infinite number bears on multiverses. As a preview, if you ask the question -How many universes are there? -then I think that any sensible finite answer is constrained to a finite natural number. You have to say that there are 7 universes, or that there are 106 universes. It doesn’t make any sense to say that there are 2.5 universes, or that there are -2 universes, or that there are 3+2i universes. Now similarly I am going to argue that if you give an infinite answer to the question -How many universes are there? -then any sensible infinite answer is constrained to an infinite natural number. So that is why I think it is important to answer the question: Which objects are the infinite natural numbers? That is a bit of an overview. Let us turn to the concept of infinite number. I am going to tell a bit of a story. This is a story in which the whole numbers were not moved into the infinite in the correct way. The standard view of infinite number is a Cantorian view and I am going to tell a story, involving concrete objects, that tries to highlight the sort of mistake that I think was made. Consider the following items (and concepts): male birds, female birds, and black cars in the United States. And somebody comes along, and he (his concepts) carves up the world as follows. When he sees a male bird, he says “There is a male bird.” But he thinks that the black cars in the U.S. are the female birds. So when he sees a black car in the U.S. he says, “There is a female bird.” Here is a table summarizing this: Object Concept/Calls Them Male Birds “Male Birds” Female Birds Black Cars in the U.S. “Female Birds” [wrong conception] Now, I think that this person would run into all sorts of questions and problems. For example: Why can’t male birds and female birds produce viable offspring? Why are female birds so large? Why, when female birds cross the border into Canada, do they disappear? There would be a whole host of * This paper is based on a talk, available at: https://ryecast.ryerson.ca/67/watch/3043.aspx The paper will appear in an edited collection: God and the Multiverse, edited by Klaas Kraay.