Notes on "Pocket Set Theory"
In the course of teaching a math for liberal arts course, I noticed
that there are really only two infinite cardinals that one can
encounter in real life, aleph-null and c. The alternative set theory
of Vopenka also has only these two cardinals.
Rudy Rucker observes that the Continuum Hypothesis has a certain
analogy to an axiom of von Neumann's set theory with proper classes:
von Neumann asserted that all proper classes are the same size, which
has the nice effect of establishing replacement and choice "by magic".
If one restricts oneself to the hereditarily countable sets, this
axiom would assert that the collection of all HC ordinals (size
aleph-one) is the same size as the universe of hereditarily countable
sets (size c). The alternative set theory also exploits this.
Thus my modest proposal of "pocket set theory" which follows:
It has primitive predicates of membership and classhood and the
following axioms:
I. Classes with the same members are the same; non-classes (if there
are any) have no members.
We define "set" as "class which is a member".
II. Any condition whatever that we can express on elements (including
non-class elements) defines a class.
This is, formally speaking, an impredicative class comprehension
scheme; it allows definitions of classes of sets (or sets and
non-class elements) which may involve quantification over all classes.
Notice that we get a unique empty class in addition to any
non-classes.
III. There is a set I and relation P such that "for each x in I,
there is exactly one y in I such that Pxy" and "for each y in I, there
is at most one x such that Pxy" and "for some x, for all y, not Pyx".
I is a Dedekind infinite set. It is straightforward to prove that I
has any concrete finite number of distinct elements.
We define "proper class" as "class which is not a set".
IV. All proper classes are the same size.
Formally, we introduce a predicate ClassSize with axiom "for all A, B,
if A and B are proper classes then for each x in A there is exactly one y in B
such that ClassSize(A,B,x,y)".
Definition: the class omega of all finite ordinals is defined as the
intersection of all classes which contain the empty class (if it is a
set) and contain x U {x} if it is a set and x is in the class.
Theorem: The empty class is a set.
Proof: Suppose not. Then every proper class would be empty by axiom
II. But the Russell class is nonempty by Axiom III; it must contain
{I}, a nonempty class which is not an element of itself (because I has
more than one element).
Theorem: There is a set with one element.
Proof: Suppose otherwise. Then every proper class would have one
element. But the Russell class has at least two elements, {} and
{I,{}} (I has at least three elements). Note that we have
actually proved that every singleton is a set.
Theorem: For any a,b, {a,b} is a set.
Proof: If {a,b} were not a set, it could not be placed in one-to-one
correspondence with the Russell class, since we can produce elements
{}, {{}}, {{{}}}, {{{{}}}}, of the Russell class _ad libitum_ using
the previous theorem.
As soon as 2 is shown to be a set, we have pairing, and the ability to
represent relations as sets of ordered pairs and quantify over
relations.
Theorem: Each finite ordinal is a set.
Proof: If n were not a set, we could produce at least n+1 elements of
the Russell class: the singletons of each element of n plus {{{}}}.
It should be straightforward to prove by induction that any finite
ordinal (including a proper class finite ordinal) cannot be placed in
one-to-one correspondence with a class with one more element.
Theorem: The universe can be well-ordered.
Proof: It is the same size as the proper class of all von Neumann
ordinals.
Definition: A class or set is said to be infinite if it is not the
same size as any finite ordinal.
Our last axiom
V. The infinite sets are exactly the classes such that there are
class bijections witnessing the fact that they are the same size as I.
Theorem: The class of finite ordinals is a set.
Proof: It is clearly an infinite class. Suppose it were not a set; it
would then be the same size as the Russell class by axiom II. The
universe can be mapped one-to-one into the Russell class by sending
each x with one element to {x,0} and each other x to {x}; by
composition we could map the universe one to one into the finite
ordinals, from which it is clear how to map the universe one to one
onto the finite ordinals. Any infinite set would similarly map into
the finite ordinals one-to-one, so onto the finite ordinals, and it
would follow that the infinite set was the same size as an infinite
proper class, which contradicts our axioms.
We now have the set of natural numbers, and can construct the sets of
integers and rationals in a well-understood way. The class of reals
can then be defined as cuts in the rationals (individual reals are
sets).
Theorem (CH): All infinite classes of real numbers are either
countable or the same size as the set of reals.
The reals are uncountable and so are a proper class by V. Each
infinite subclass of the reals is either a countable set or a proper
class with the size of the reals, since these are the only
alternatives allowed by axiom V.
It is clear that the reals can be realized as sets using cuts in the
rationals. Continuous functions from R->R can be realized by
restricting them to the rationals. Intervals in the reals can be
coded, and so can Borel sets at each stage in their hierarchy. The
countable ordinals are all sets; the proper class ordinal is
aleph_one.
This theory is acceptable from a moderate Platonist standpoint; it is
the most Platonist theory one can get with any reasonable case that
the mathematical universe is no larger than the physical universe. It
is also strong enough to consider the existence of models of any
first-order theory we can express from a Platonist standpoint; if one
is committed to pocket set theory, one believes that there is an
answer to the question as to whether it is consistent that there is a
measurable cardinal.