Continuum Hypothesis

Solved or unsolved? Can you prove it true? No. Can you prove it false? No. But why??

Doesn’t this excite you to know more? LET’S TALK ABOUT IT!

INTRODUCTION

Before jumping into the details of the hypothesis, let’s give you some idea of the basic terms to understand this. As the continuum hypothesis and set theory are somehow related, we say there are a few kinds of sets, such as finite sets, infinite sets, countable sets, and uncountable sets. Related to these types of sets, the term ‘cardinality’ basically gives the idea of the number of elements in a set, or let’s say the size of a set. For example, the set

$$ \{-1,8,6,0\} $$ has cardinality equals 4, as its number of elements is 4. So, the two sets which are in one to one correspondence with each other are said to have the same cardinality. We represent the cardinality of countably infinite set with $\mathbf{\aleph_0}$ (aleph naught) and We denote the cardinality of set of integers as $|\mathbb{Z}|$. Also, $|\mathbb{Z}| = |\mathbb{N}| = |\mathbb{Q}| = \aleph_0$. Similarly, for the uncountable infinite sets, $\mathbf{\aleph_i}$ , $i=1,2,3\dots $ denotes the cardinality. For example, Set of real numbers is not countable i.e. $|\mathbb{R}| = \aleph_1 $ .The Continuum Hypothesis is from set theory. It says that there is no set whose cardinality is strictly between that of the integers and the real numbers. Mathematically, the Continuum Hypothesis can be stated in the form of the following equation:

$$ 2^{\aleph_0} = \aleph_1 $$

where $\aleph_0$ is the cardinality of an infinite countable set (Ex, set of natural numbers), and the cardinal numbers of larger “well-orderable sets” are $\aleph_1, \aleph_2, \dots, \aleph_{\alpha}, \dots $ are the ordinal numbers. The cardinality of the continuum can be shown to equal $2^{\aleph_0}$; thus, according to the continuum hypothesis, there can’t exist a set of size intermediate between the natural numbers and the continuum. That was a basic introduction to the continuum hypothesis. Now, let’s travel through time researching the continuum hypothesis and let’s see more about it in the motivation ‘section’ below.

MOTIVATION

The Continuum Hypothesis(CH) has been known to us since the birth of set theory. In 1873, a famous mathematician named Cantor was unable to resolve the problem of CH. Let me mention again the statement of continuum hypothesis so that you don’t miss the context. The continuum hypothesis is reduced to the question, how big is infinity? After Cantor’s failed attempt on CH, Hilbert and Cohen took this problem into their hands and they reached to a conclusion that it can’t be resolved through the existing set of axioms of mathematics. They were only able to show that these axioms are not enough to settle CH. In-fact, the CH came out to be independent of these axioms of choices from Zermelo-Frankel set theory (ZFC).

It was this hypothesis which led Cantor to develop the set theory during the attempt to solve the continuum hypothesis. But why do they say “You can neither prove it wrong nor true”? Does this mean it doesn’t have a solution? There can be a few ideas to understand the scenario. Either the mathematicians don’t have the required principles yet or they are missing the key principles or are there principles to find that will enable them to find whether the continuum hypothesis is true or false? Or maybe, there is this much stronger version of Cohen’s theorem, that shows that there really is no answer. In mathematical terms, the two concepts are there in mind of a mathematician:

The mathematical infinity
The universe of sets

The fact that it’s all a human creation points the finger of doubt towards this concept of infinity being a fiction. If the Continuum Hypothesis is all fiction, then so is the set theory and the concept of infinity and the other concepts that are derived from it. Then the question pops up into our mind that it could all just be an extrapolation of something which is not true, not being able to answer the continuum hypothesis is a direct challenge to this. That’s not all.

Past the time of Hilbert and Cohen, the next 30 years were of Gödel’s. Following one line of inquiry or an attempt to find new axioms, he showed there was a kind of ultimate solution along this line and speculated then in a series of articles that the continuum hypothesis was false. In Fact there was exactly one infinity in between the infinity of a counting numbers and infinity of real numbers and in any case he showed that there was an optim seal solution, but then that solution ultimately doesn’t fit into understanding the entire universe of sets, that is the continuum hypothesis is just once instance of an infinite sequence of questions and it makes no sense to answer the problem with continuum hypothesis without having some global solution to the universe of sets. So that became a problem, what could that global solution be? Now as the set theorists have proved that the continuum hypothesis can’t be disproved from the above set of axioms. So, according to them the only option left is to either consider the continuum hypothesis as the new axiom or the concept of infinity will be thought as a fiction.

Well, the key question is whether these conjectures can be proved or not? If they can be proved, we’ll have one new principle which when added to the traditional axioms of the set theory will settle the continuum hypothesis true. It will also resolve all instances of the questions which the Cohen’s method today has been used to show are unsolvable. One possibility is to refute these conjectures, then it is utter chaos. Many people have said that the fact that the continuum hypothesis could not be solved is an indication of ‘this is all just human imagination gone wild’. On the other hand if the other picture unfolds and we prove these conjectures, maybe we can do it in the next few years, maybe never, who knows. If that happens one day, that will be a remarkable event because set theory doesn’t see it, it’s not like we look for a particle and we find it physically in some region. There is no technology which could determine whether or not this continuum hypothesis is true or false. You don’t find the answers to the continuum hypothesis by some clever proof, somehow there has to be a solution that forms our intuition, there have to be clues, but there is no reason to expect that will happen. It really could just all be fiction. So there’s this thing, and we see beyond it, does it exist? The one key question: if it exists, the continuum hypothesis has an answer and we need to find it or we could go the cynic route and show once and for all that the conception of mathematical infinity is just that a fiction and there is no answer, so those are the two futures.

There is still no evidence on what basis the hypothesis should be kept and on what basis the hypothesis should be rejected. Let’s wait to see what happens next…

A hypothesis about the possible sizes of infinite sets

INTRODUCTION

MOTIVATION

REFERENCES

CONTENTS

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