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NOTE: "?" denotes greek or hebrew mathematical
symbol that is unable to be displayed.
How can we (or can we at all)
justify claims about the concept of infinity, so that we may
incorporate it in mathematical theory? Firstly, the concept
of infinity poses several paradoxes which need to be resolved
in order to form a concrete understanding of what it is (or
what is it not) to be infinite, if we wish to claim any particulars
at all. The continuum hypothesis, symbolically represented
as 2?0 = ?1 roughly translates to mean that the infinite realm
of rational numbers is "one level of infinity" smaller
than the infinite realm of real numbers which includes both
rational and irrational numbers. A further claim of the continuum
hypothesis is that the infinite realm of real numbers can
be paralleled to the infinity of the infinitely small, because
both are infinites by division. Likewise it correlates the
infinite realm of rational numbers with the infinity of the
infinitely big, since both are infinities by addition. My
paper will propose a comprehensive understanding of the motivation
for, and context of the continuum hypothesis which will be
necessary to address the creditability of the claims the continuum
hypothesis makes concerning comparable infinities.
The motivation for the continuum
hypothesis began an attempt to resolve the paradoxes posed
by the infinitely big. Consider the statement "there
are as many even natural numbers as there are natural numbers."
Intuitively one would disagree with this statement based on
the knowledge that an even natural number occurs after every
odd natural number, making the set of all natural numbers
double the size of the set of even natural numbers. But we
also know that the set of natural numbers extends on infinitely.
So if we were to draw two sets of numbers, no matter how long
we went on listing numbers there would always be a one to
one correspondence between the sets (making them equal to
each other). This deduction indicates the paradox posed by
the infinitely large, since it is contradictory to say that
something that is boundless is larger than something else
that is boundless.
Avoiding this paradox the infinitely
large poses was the motive Cantor used to develop his criteria
for sets. He postulated that sets could be compared in a correlative
sense, or in a subset sense. In the correlative sense, one
asks if the members of a set can be paired to members of another
set, while the subset sense asks if the members of a set belong
to another set. With this criterion we can now state that
the set of even numbers is a subset of all numbers, thus smaller
than the set of natural numbers in the subset sense. Cantor
had developed a methodology of referring to a set with an
infinite number of members as a whole, and promoted the idea
that the concept of infinity was set-theoretic.
Cantor's set theory did not come
without problems. One of the problems that Cantor encountered
can be illustrated with Russell's paradox which states "Let
R be the set of all sets which do not belong to themselves"
returning the contradiction that "R belongs to itself
if and only if R does not belong to itself." This seems
to show that addressing the paradoxes of the infinitely big
with set theory, causes the paradoxes of the one and the many
to arise.
To address the paradoxes of the
one and the many, Cantor further developed diagonal arguments.
These diagonal arguments expand on Cantor's correlation criterion,
adding that any set A is bigger than set B when all members
of B can be paired with some members of A, but not with all
of them. A result (of Cantor's diagonal arguments) showed
that there are more real numbers between 0 and 1 than there
are natural numbers, which became the fundamental motivation
to develop the continuum hypothesis. The proof for this claim
is illustrated by creating two sets of numbers where you pair
natural numbers with decimal expansions of real numbers between
0 and 1. You can then always construct a real between 0 and
1 that is not listed in the table by applying the following
conditions:
"Create a new decimal by following the table diagonally
and for each decimal place where X occurs write Y, and write
X where every number that is not X occurs. "
Doing so will always construct a real that is not listed in
the table. But the initial table constructed contains all natural
numbers, so there are members of the set of real numbers that
cannot be paired within the set of natural numbers. This makes
the set of real numbers larger in proportion to the set of natural
numbers, even though both sets are infinite. Cantor further
expanded his diagonal arguments to show that there are more
sets of natural numbers than there are natural numbers. To compensate
he applies the concept of a powerset, in which the powerset
of A is the set of all subsets of A. The concequence to the
statement that the powerset of A is larger than the initial
set A is that now there is no limit to how large an infinite
set can be. This leads to the non-trivial question Cantor posed
that the continuum hypothesis addresses that says "The
set of natural numbers is infinite, but its powerset (the set
of real numbers) is larger. So how much larger is the set of
real numbers than the set of natural numbers?"
There are two possibilities to answer
"Cantor's unanswered question." One is that the set
of real numbers is the "next infinite size up" from
the set of natural numbers, the other is that there are intermediate
sizes of infinity between the set of real numbers and the set
of natural numbers. The continuum hypothesis is the claim that
the answer to Cantor's question is that the set of real numbers
is the next size of infinity up from the set of natural numbers.
Motivated to talk about levels of
infinity, Cantor developed a hierarchy of sets. Levels in the
hierarchy are measured in the Ordinal numbers, which specifically
measure the "length" of a well ordered set in correlation
sense. The ordinals are also in itself, well ordered. The first
ordinals on the hierarchy are the sets of natural numbers. The
ordinal that succeeds the set of natural numbers is Omega (?)
and so on up to Kappa (? = ??) which is the first cardinal that
can only be named by the ordinal it is (due to it's massive
"size"). To avoid the paradox posed by the fact that
there is no limit to how big an infinite set can be, we can
assume that there is a limit to how small a set can be. So by
definition, cardinal numbers allows us to measure the "size"
of sets, and are the smallest ordinals of a given set size.
The finite cardinals include the natural numbers, and measure
the size of finite sets. The first infinite cardinal number
is the set of all natural numbers (?0), making the second cardinal
number, the set of all real numbers (?1).
Now that there is a system for referring
to infinite sets (the ordinals and cardinals) we can now look
at several transfinite arithmetic equations, specifically ones
that are involved in the continuum hypothesis. Suppose that
Kappa is a cardinal number, logically it would be less than
its powerset (2?) which follows Cantor's claim that the powerset
of any given set is larger than the initial set. Therefore the
set of natural numbers is less than the powerset of natural
numbers (?0 < 2?0). Further, we can now claim that the set
of real numbers is larger than the set of natural numbers by
one cardinal number since cardinals measure the "size"
of sets and the set of natural numbers is less than its powerset,
and its powerset is equal to the set of real numbers (2?0 =
?1).
A "continuum" has two
basic properties: that between any two points there is another
point (denseness), and that there are no gaps between points.
The set of rational numbers is dense, being that between any
two rationals there is another rational number, but there are
gaps where irrational numbers occur. The set of real numbers
includes both irrational and rational numbers, which is represented
as ?1 and measures the "size" of a continuum since
it measures how close together points on a line are. This illustrates
that the set of the infinitely divisible (small) has members
that cannot be paired with members of the infinitely addable
(big) since there are gaps between natural numbers where irrational
numbers (accountable by division) occur. Although technically
we can neither prove or disprove the continuum hypothesis within
any formal set theory (ZF).
In conclusion I don't believe that
the argument in favor of the continuum hypothesis is a strong
one. Firstly CH cannot justifiably identify the set of real
numbers with the infinitely divisible. The infinite by division
only states that there exists another point between any two
points on a line. This only identifies with the property of
denseness for which we only need the rational numbers, and not
the real numbers. And the size of the set of rational numbers
is ?0 and not 2?0. Secondly, the continuum hypothesis cannot
justifiably identify the set of rational numbers with the infinitely
addable, since we can extend addition to ordinals beyond omega
?. Although we now have a formal theory of sets, which enable
us to talk and refer to objects and instances in the infinite
realm, it is unconvincing to claim that the continuum hypothesis
is true. |
