Continuum Hypothesis Kyle Rodriguez

How did the Continuum Hypothesis Start?

It all started when Georg Cantor said that the numbers 1,2,3 to infinity is less than the numbers in between 0 and 1 as the numbers in between these two have a infinite amount as well. We look at all natural numbers which goes to infinite cardinals values on the list, but in the real number set, which is the continuum, there is a uncountable infinite number of values when it comes to its cardinality. This would mean that the uncountably infinite set has a larger cardinality as the countably infinite set. Georg Cantor’s argument on the continuum hypothesis is that, there is no infinite set with a cardinality lesser than natural numbers existing in between these two sets.

With Georg Cantor creating set theory, he also created the problem which we now call the continuum hy- pothesis. When Georg Cantor showed that there was a one-to-one relationship between algebraic and natural numbers with the product of his work showing that there is a higher level of infinity when it comes down to real numbers. With most objects involved in mathematics are defined as infinite, Georg Cantor’s set theory is just an assumption of infinity.

How does this relate to the continuum hypothesis? As stated in the previous paragraph, the continuum hypothesis can simply be stated as ”how many points are on a line.” Essentially, there are infinite points on a line as you can continue making new spots on a line and this goes with Cantor’s set theory involving infinity. Which says that there are more points on a line to count than all whole numbers we know today. Who is Georg Cantor and What are his Accomplishments

A famous and well know mathematician by the name of Georg Cantor was a great contributor to the world of mathematics that we know and love today. Cantor made and proved his own theorems on infinite set, but never gained the respect of other workers in the field until he passed away. Cantor was a necessity to allow us, future generations, to continue our research in mathematics which would not have been possible if Cantor chose not to contribute to this field.

On March 3, 1845, Cantor was born in St. Petersburg, Russia. His pursuit in mathematics stemmed from his father guiding him onto the path of engineering. Cantor always had a passion for mathematics and eventually got his father’s approval to allow him to pursue the mathematics instead engineering. Which lead to the invention of a the simplest, yet unsolvable continuum hypothesis.

His theories were used by many, but his idea on transfinite numbers was also very impactful on society. Though again, when it came to other workers in his field, there was always someone there to try and put Cantor to shame. With someone always there to judge his work, Cantor struggled with nervous breakdowns as a result. Due to his work, he was later accredited by the London Mathematical Society, but by that time he had already passed away.

Cantor first started dealing with the theory of numbers. This was his first choice as this area of mathematics was a fascination of his. He then proceeded to work in the theory of trigonometric series in 1870, showing how a function can only be represented one way through a trigonometric series. Two years later, he went on to ”define irrational numbers in terms of convergent sequences of rational numbers (quotients of integers)” and then proceeded to begin his most well-known field, the concept of transfinite numbers and the theory of sets. What Does the Continuum Hypothesis Mean

We already know that the Continuum Hypothesis revolves a lot around the term infinity. We also find out that there are different sizes to infinity as well! With this being shown when it comes to counting the infinite amount of even numbers versus counting the infinite amount of all real numbers. They are both clearly infinite, but most would say that the infinite amount of all real numbers is bigger than the infinite amount of just even numbers as it involves all numbers. This is the Continuum Hypothesis. Which infinite set has the greater cardinality when compared to another infinite set.

This means that there are different levels of infinity. With the lowest level being called ”countable infinity” and the higher levels being called ”uncountable infinities.” Countable infinity represent’s all natural numbers and all real numbers represent’s uncountable infinity. Given the definitions, behind all sets of real numbers either, end, go on for as long as the list of natural number goes, or it has the same amount of numbers as the set of real numbers. Then, any infinite set of real numbers are either the same length as the amount of natural numbers, or has the same amount of numbers as the set of real numbers. Lastly, when it comes down to the uncountable set of real numbers has to have the same amount of numbers as the entire set of real numbers. Hilbert’s Hotel

Now we can dive into how the Continuum Hypothesis works by understanding the story of the ”Hilbert Hotel.” The story behind Hilbert’s hotel is a story that involves a hotel with an infinite amount of rooms for an infinite amount of people. Now, it is not possible to have an infinite amount of anything in the real world as infinity is just a term used to describe a limitless amount of something and everything we use in our daily lives eventually runs out.

Though with Hilbert’s hotel, it begins with a driver that is extremely tired and sees this hotel but it has a sign saying there are no available spots. With the man being extremely tired, he still tests his luck by walking in and asking if there are any available spots. To his surprise, the clerk comes up with this idea that involves moving every single person to the room right next to them so the first room becomes vacant. Now, the man has a place to sleep but soon realizes he will not be able to sleep as someone else continues to ask for a vacant spot and he will continue to move rooms. By the time he is ready to leave, he is in room 18 and it is time to pay, but when he goes to pay, the man in room 19 tells him that the bills is on him as his bill is being covered by the person in room 20.

This story shows a hypothetical situation with countable infinite amount of rooms and can show a one-to-one relationship between natural numbers and rational numbers as every room is occupied by a single person. Though the sign said there were no more available spaces as all rooms are occupied, it does not say that there is no more space for new guest. That alone is paradoxical as all rooms are occupied, yet it is possible for new guest to continue coming in for a place to stay. This hypothetical situation was constructed by the famous twentieth-century mathematician, David Hilbert. Other Mathematicians That Have Attempted to Prove This Hypothesis

At the beginning of the twentieth century, mathematicians tried to come up with a system of set truths that would serve as the foundation of all mathematics. The purpose of trying to devise such a system was so every mathematician’s work could be proven correct from just these simple rules. The reason behind coming up with a rulebook in the first place was because of an assumption that said ”it should be possible to derive every mathematical truth from these axioms, using only the rules of logic.”

Two mathematicians named Ernst Zermelo and Abraham Faenkel, developed a basis for the creation of the system mentioned in the paragraph above. The system the created is now known as ZF, named after both the mathematicians. The only problem being that there is no solid way yet to be able to individually grab a certain element in a set. The system of ZF essentially says if you are able to create a new set by ”picking the left shoe from each pair.” With another version being called ZFC with the introduction of the axiom of choice.

2 Going from shoes to socks now, if you have an infinite amount of socks that are completely alike. The axiom of choice is the principle of being able to choose a single element from sets that are infinite and identical. Without the principle there would be no exact way of being able to take an element from a set if the set is infinite and indistinguishable. Which puts the principle into the equation of being able to now say ”choose one sock from each pair.”

Another well know mathematician by the name of Kurt Gdel, also tried to prove this hypothesis and was the first to show progress was being made on the topic in 1940. He simply showed that the continuum hypothesis worked with ZF and could not be disproven with the standard ZF theory even if the axiom of choice was implemented. Though this is still a very far way from proving the hypothesis correct, it still gave confidence on being able to progress on the topic.

Paul Cohen was another mathematician who invented the technique of forcing new sets that obeyed ZF and could also be manipulated to work with other variables as well, but it did not satisfy the axiom of choice. Gdel showed that the axiom of choice did not contradict ZF and actually worked with the system, what Cohen did was the complete opposite and showed that the continuum hypothesis and axiom of choice are both independent of ZF. Thus showing that there could be two different situations with the hypothesis is true and one where it is not. Which now creates two world of mathematics as proving it right and wrong are both possible. Therefore showing that the continuum hypothesis is currently impossible to prove with the rules of mathematics we have today. He was then rewarded with the Fields Medal in 1966 and the National Medal of Science in 1967. Continuum Hypothesis Work

In 1877 Cantor proposed the Continuum hypothesis which was then published a year later. Fast forward thirty years, a German mathematician proposed GCH which stands for ”Generalized Continuum Hypothesis.” Stating that if a set that is infinite has a cardinality that is in between the infinite set S and the power set of S, then it must have the same cardinality as the set of S or the power set of S.

Then in 1934, Paul Cohen was born and twenty-nine years later shared his notes about the independence of the continuum hypothesis. Many people were involved in the process of finding a definite answer to the continuum hypothesis, but Paul Cohen began working on the continuum hypothesis with the help of Gdel’s work and use of the ZF system. Cohen began his task on showing that the continuum hypothesis and axiom of choice were independent from ZF, which he eventually did in the 1960’s. Remarks

Researching the continuum hypothesis was a fun journey as I learned more about axioms and how they are directly related to Zermelo and Fraenkel’s system. With the most interesting part to me being that Cohen essentially found out the answer at the young age of twenty-nine. With the process starting from Gdel showing the connection between the ZF system and the continuum hypothesis, and started the race between mathematicians who could make a model where the continuum hypothesis fails.

Which Cohen rose to the challenge in the early 60’s and within three years produced a model where the continuum hypothesis is independent from ZF by using a technique he created called ”Forcing”. Forcing is a way to add new reals to a model of the mathematical universe. This new technique lead to many people not believing Cohen’s work at first, until Gdel went over the proof and gave it his seal of approval. References http://plato.stanford.edu/entries/continuum-hypothesis/

3 https://www.ias.edu/ideas/2011/kennedy-continuum-hypothesis https://math.berkeley.edu/ kpmann/CH.pdf https://arxiv.org/pdf/1403.0059.pdf https://www.math.rutgers.edu/ cherlin/History/Papers2000/brazza.html https://www.britannica.com/biography/Georg-Ferdinand-Ludwig-Philipp-Cantor http://www.ii.com/math/ch/ https://plus.maths.org/content/os/issue47/features/elwes1/index https://plus.maths.org/content/cantor-and-cohen-infinite-investigators-part-ii

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