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.
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