Approaching Infinity

The title is the title of a book by Michael Huemer, which I read most of, but not all. It’s good in parts. It addresses infinity in several ways. He presents many paradoxes involving infinity and gives his solutions for some of them. He discusses Cantor’s diagonal arguments (see my previous post for one). He doesn’t seem to wholly agree with them, but he doesn’t argue against them either.

He does challenge Cantor concerning Galileo’s paradox. “The puzzle is that there is a compelling argument both that the two sets are equally numerous (the one-to-one correspondence argument), and that one set is larger than the other (for the squares are a proper subset of the natural numbers). Cantor embraces the first of these arguments and rejects the second. His only justification for this is fiat: he proposes to simply define the relations ‘equal to’, ‘greater than’, and ‘less than’ using the one-to-one correspondence criterion.

     Of course, one could not consistently brace both of the arguments Galileo mentions, since they entail contradictory results. But Cantor’s decision to embrace only the one-to-one correspondence argument is not the only alternative. One could embrace the proper-subset argument while rejecting the one-to-one correspondence argument. Or one could, like Galileo, reject both arguments and hold that ‘greater than’, ‘less than’, and ‘equal to’ relations do not apply to pairs of infinite sets. Cantor does not argue for his alternative over the others” (6.9.4).

Huemer did not say one could embrace the proper subset argument for comparing the integers and the rationals.

The weakest parts of the book in my opinion were about the empty set and geometric points.

Huemer misses the strongest reason for having the concept of an empty set. The two major operations on sets are union and intersection. The first forms a new set by combining the members of two or more sets. The second forms a new set by identifying the common members of two or more sets. Say we have set A = {1, 2, 3} and set B = {3, 4, 5}. The intersect of A and B is {3}. But what if B = {4, 5, 6}? Then A and B have no common members. In other words, the intersect of A and B is nothing, i.e. the empty set { }, often symbolized as {∅}. In other words, the empty set is needed to make defined operations complete. It is similar to a need for the number zero, for example 5 – 5 = 0.

Huemer is frustrated with geometric points. He says they are unimaginable, because they have size zero, yet they are supposed to be the building blocks of other geometrical objects (11.2).

I regard the building block perspective as from the wrong direction. Start with 3-dim space, e.g. a cube. Then disregard one dimension to get 2-dim space (a square plane), disregard another dimension to get 1-dim space (a line), and finally to 0-dim space (a point). Alternatively, each intersection of two lines of the square is a point.

Later in the book there is a section (14.6.3) about points as locations. He does not have any arguments against this view, but he doesn’t endorse it either and says he doesn’t understand it. 

In my opinion it is the best view. We often hear people say things like how to get from point A to point B. More concretely it might mean driving a route from one location, such as a town, to another location, another town. In the context of geometry, a point is a location in Cartesian or polar coordinates. For example, the point (2,3) in 2-dim Cartesian coordinates means 2 units to the right of the origin (0, 0) or y-axis and 3 units above the x-axis. Of course, Euclid’s Elements did not have this perspective, since Cartesian coordinates were invented much later. 

Infinity contra-Cantor #2

Georg Cantor also famously claimed that the cardinality of the set of rational numbers equals the cardinality of the set of integers because he could devise a 1-1 correspondence or mapping between them as shown here and here.

Relatedly, the continuum hypothesis is: There is no set whose cardinality is strictly between that of the integers and the real numbers (rationals and irrationals).

Accepting certain assumptions Paul Cohen showed that the continuum hypothesis is neither true nor false. I think it’s false, the quantity of rational numbers being greater than the integers, but less than the reals (rationals and irrationals combined). I base this on two  perspectives different from that of Cantor and Cohen. The first is the real number line. There is an unlimited count of rational numbers between 0 and 1, between 1 and 2, and so forth.

The second is part-whole logic. As shown in the above links, a 1-1 mapping can be made. However, note the first column in the array in either link, which is the integers. A different 1-1 mapping or function f(x) = x can also be made. Accordingly, the integers comprise a proper subset of the set of all rational numbers, implying the set of rational numbers is larger than the set of integers.

“Cantor was concerned to combat the Aristotelian view that there cannot be an actual infinity, mainly because Cantor believed that God was infinite.” – Michael Huemer, Approaching Infinity, p. 71.

Infinity contra-Cantor #1

In the late 19th century Georg Cantor chose the method of 1-1 correspondence to decide the different quantities of infinite sets. He called such quantities their “cardinality.” If the members of one set can be paired 1-1 with the members of another set, then they have the same “cardinality” (link). 

As the Wikipedia page shows, some famous mathematicians such as Kronecker and Poincaré objected to some of Cantor’s ideas. But most people who have an opinion about it accept Cantor’s arguments and consider that the cardinality of the positive integers equals that of the even positive integers, as explained on the linked page.

I was astounded when I became aware of that many years ago. Years later I realized Cantor's error. The assumed method of 1-1 correspondence implicitly rules out 2-1 correspondence, N-1 correspondence, and part-whole logic. 

A 2-1 correspondence or mapping can be formed from the natural numbers (integers) to the even numbers as follows: 1, 2 → 2;   3, 4 → 4;   5, 6 → 6;   7, 8 → 8;   9, 10 → 10; and so on. If this mapping did not include “and so on”, then the finite quantity of natural numbers would surely be regarded as being twice that of the even numbers. (See above link, Exercise 3). However, many abandon that idea when “and so on” is included. I do not. From a part-whole logic perspective, imagine having a container with all the integers in it, then removing the odd integers from the container. Wouldn’t the container then have half the quantity of numbers as before? Or partition the container in half vertically, putting the odd integers on one side of the partition and the even integers on the other side. Wouldn’t there be twice as many numbers in the whole container as there are on one side of the partition?