This is the first part of the forthcoming trilogy in four parts. The trilogy will be devoted to the understanding of the Continuum Hypothesis by looking at some statements that are equivalent to it.
In this post I just expose some philosophical background concerning the foundations of mathematics, the Continuum Hypothesis and its independence. So if you are a set theorist, then you should probably do something else.
Since there are plenty of expositions of this subject, say here, here and here, I will focus merely on my personal experience and point of view.
Truth?
When I became interested in mathematics, I believed (or wanted to believe) that mathematics can be regarded as an Absolute Truth. This provoked some questions, such as “what does it mean to be true?” and also philosophically easier questions such as
“What implications do mathematical truths have?”
Let me give an example: It is true that
in the field of real numbers there is a solution to the equation
[tex]x^{51}-30=2.[/tex] (#)
Now, what implications does this have to my understanding of the real world? I can never even know the precise value of this number, since it is irrational. To appreciate the difficulty note that there is not obvious which one is bigger: the solution to (#) or to [tex]x^{52}-33=2[/tex]. Moreover real numbers is an uncountable (very infinite) set, so what implications may it possibly have to my finite world?
If I look carefully at the used notions, I will notice that (#) is just an assumption, or follows from the assumptions. To be more precise, what do I mean by the real numbers? By them I mean a collection of axioms, the axioms of real numbers. (#) follows from these axioms. The completeness axiom is of special importance here. Namely the completeness axiom implies that a solution to the equation in (#) is exists, because it can be approximated by rational numbers. Thus a meaningful interpretation of (#) is that the solution can be approximated.
Of course my understanding of the real numbers is not limited to the axioms; I probably visually imagine some line segments, points and motion when I think about real numbers. If I imagine them in a way which is consistent with the axioms of real numbers, I will possibly be able to understand some consequences of these axioms without deducing the consequences formally. Also I might make mistakes, since consistent does not imply follows. On the other hand my understanding of the axioms of real numbers is deeply biased by my everyday experience, and so I use more assumptions than just the axioms of real numbers. What are those assumptions?
ZFC
In developing foundations of mathematics, people have tried to justify, what are the basic principles which are present in our rational thought and what are the basic principles necessary to develop the existing mathematics.
An incredible success has been gained in this field. One of the standard approaches nowadays is ZFC.
What is ZFC? It is a list of axioms about sets, or collections. The axioms of ZFC are chosen in a fortunate way and correspond very tightly to the way people tend to think. The latter fact makes some people feel that the axioms of ZFC are true. Indeed, isn’t it true that if I have a collection (a set) of objects A and another one, say B, then I can think of a single collection C which consists of all the objects from A and B. I am talking about union here. Thinking of trees and bushes makes it possible to think of trees-and-bushes.
Take some amount of such “obvious” axioms and you get ZFC. And suddenly you can interpret almost all known mathematics just as a consequence of these axioms. This is pretty striking. Or is it?
One is able to construct an infinite group in ZFC, but it does not make group theory any easier. On the other hand you cannot construct an infinite group using only axioms of group theory (of infinite groups). Already something. Moreover, in ZFC one can construct a field of real numbers and show that this is actually the field of real numbers (i.e. ZFC implies that it is unique).
Many properties of mathematics are decided by ZFC. But not all. Some properties still need new axioms in order to be decided. As an example featuring this and the following few posts is the Continuum Hypothesis.
Continuum Hypothesis
I am not going to present here any introductory course on the subject. Those who do not know what does it mean for two sets to have the same size or the same cardinality, can check it out. The Continuum Hypothesis is the statement
(CH) There is no set whose cardinality is strictly greater than that
of the integers and strictly less than that of the real numbers.
The first responce of a child to this is “rational numbers”, but the rationals turn out to have the same size as the integers. :-(
CH is provably independent from ZFC, provided that ZFC is consistent. This may require some explanation. Independence of a statement from ZFC means that this statement cannot be deduced from ZFC but also the negation of that statement cannot be deduced. As a simple example of this situation:
From the assumptions “A is delicious” and “A is cold” it cannot be deduced that “A is ice cream”, because there are also other cold and delicious things.
Consistency of ZFC means that there is no sentence [tex]\varphi[/tex] such that
both [tex]\varphi[/tex] and its negation [tex]\lnot\varphi[/tex] can be deduced from ZFC. It is not known whether ZFC is consistent and, unfortunately, if it is then we will never know that. But most people act as it were consistent. Similarly as most people act as if they knew that the Sun will rise next morning, though there is no proof that the Sun won’t transformate into a pink elephant.
At first, it is unclear what implications might CH have and why is it of any importance. And it has to be admitted that a large part of mathematics is not concerned with this problem. The best way to understand the connections of CH to other mathematics is probably to find mathematical statements which are equivalent to it.
Next three posts will be devoted to that. Wednesday Problems do not count of course.
Consistency and Independence
In what follows I reveal in very rough terms the idea of how independence of CH is proved. So if you are a set theorist but still reading this, it is now good time to do something else..
Formally, ZFC is an infinite (countable) collection of axioms, since it contains a Comprehension Axiom for each formula [tex]\varphi[/tex]:
for each set A, the subset B of elements satisfying [tex]\varphi(x)[/tex],
[tex]B=\{x\in A\mid \varphi(x)\}[/tex]
exists
and a similar Relpacement Axiom scheme.
The rough idea of the CH’s independence proof is the following. Suppose that ZFC is consistent. Let [tex]T\subset\textrm{ZFC}[/tex] be any finite list of axioms of ZFC. Now it is possible to prove in ZFC that there exists a model [tex]M[/tex] for these axioms. (In the same but more general manner as one can prove that there is a group (a model of group theory).) Using Gödel’s ideas, one can produce also a model [tex]M_1[/tex] for [tex]T\cup\{\textrm{CH}\}[/tex] and using Cohen’s ideas in addition to Gödel’s, a model [tex]M_2[/tex] for [tex]T\cup\{ \lnot \textrm{CH}\}[/tex]. Now let us make a counter assumption: either [tex]\textrm{CH}[/tex] or [tex]\lnot\textrm{CH}[/tex] can be deduced from ZFC. Both cannot be deduced by our consistency assumption. Suppose that CH is the one which can be deduced. Because the deduction is finite, there should be a finite list T of axioms of ZFC from which CH already follows. Let [tex]\varphi[/tex] be the formula which says
there is a model M for [tex]T\cup\{ \lnot \textrm{CH}\}[/tex].
By the above (“Cohen’s ideas”), in ZFC one can prove [tex]\varphi[/tex]. But on the other hand, since T proves CH, it is easy to show in ZFC that such M satisfies CH. Thus M is a model which satisfies both: CH and [tex]\lnot \textrm{CH}[/tex] which is a contradiction and is easily transformed into a contradiction from ZFC (e.g. now one can prove that ZFC implies [tex]\varphi[/tex] and [tex]\lnot\varphi[/tex]), but we assumed that there is no such contradiction which is a contradiction. Thus CH cannot be deducible from ZFC. Similarly one argues (this time with Gödels arguments but not necessarily Cohen’s) that [tex]\lnot \textrm{CH}[/tex] cannot be deduced from ZFC.
Note that I skipped the most important parts, namely Gödel’s and Cohen’s ideas. It takes years for some people to understand them.
