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\title{Cantor's Paradise}
\author{Mahmoud A Elshinawy \\ ID: 900183926 \\ The American University in Cairo}
%\email{@aucegypt.edu}
%\address{The American University in Cairo}


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\textcolor{red}{}

\textbf{Abstract.}

\textcolor{blue}{Georg Cantor was one of the most significant mathematicians throughout history. This paper will introduce summary of his life and his works in mathematics and theorems, introduces some definitions about set theory and in the end, it proves one of Cantor's theoroms.}

\section{Cantor's Life}

\textcolor{blue}{Grorg Cantor was a great violinist, and his surname “Cantor” in Latin means the “singer” or “musician”. He named his doctoral thesis “In mathematics, the art of asking questions is more valuable than solving problems”. He also introduced some new ideas to answer the question about how big infinity is, which formed a new branch of mathematics, namely set theory.\\\\
Early Life (1845-1860)\\
    Cantor was lucky to be born in the period in which he was born. He was born in Saint Petersburg on the 3rd of March 1845 to Danish parents. Her mom was called Marie from a family whose origin was Russian musicians. Her father, Georg Woldemar, was a successful business man. Although Cantor loved music, he was influenced by his father more than his mother as his father had some philosophical interests and always advised him all his life whether in his childhood or even in college. As a child, there were signs of a great mathematician to come. However, Cantor was forced to study engineering for two years because at that time, it had more profession prestige.\\\\
    Education (1860-1869)\\
    He joined a private school in Frankfort, then went to the Wiesbaden Gymnasium in 1860. In 1862, he joined Höheren Gewerbschule to study engineering, then he shifted to the Swiss Federal Politechnic to study mathematics. After a year, his father died and he owned much money. He used this money to shift his studies of mathematics to the University of Berlin. In Berlin, he attended lectures by some lecturers who had a great interest in arithmetics. In 1866, he took a summer semester at the University of Göttingen. In 1867, he did his dissertation and after two years, he did his habilitation regarding number theory. He introduced a solution to a problem related to the Legendre’s equation, $ax²+by²+cz² = 0$. After he received his PH.D., he worked as a lecturer at Halle University while his friend worked under Eduard Heine, the professor of mathematics there.\\\\
    Early Career (1870-1873)\\
    His first publication after graduation was called “Cantor’s Uniqueness Theorem” which states that every function from the set of real numbers to the set of real numbers has exactly one representation by a trigonometric series.  This theorem was built on functions of complex variables done by Reimann before. In 1872, he introduces an improved version of the previous one talking about the boundary points and their derivatives. After that, he established a theory stating that any real number is an infinite series of rational numbers. In the paper, Cantor also compared this definition to what is mentioned in the previous papers. Later in the same year, Cantor met a mathematics professor who published a paper explaining the structure of the set of real numbers. Cantor and the professor exchanged letters for many years. On November 29th, 1873, Cantor sent a letter to the professor to ask whether or not the collection of natural numbers and the collection of positive real numbers “can be corresponded so that each individual of one collection corresponds to one and only one individual of the other?”. He replied to Cantor’s letter that he did not know.\\\\\\
    Set Theory\\
    Couple of years later, Cantor published a paper about the collection of real numbers. This paper mainly described the uncountability of the collection. The results of the paper were as following: the first was that the set of real numbers is uncountable. The second is that in every interval, there are infinitely many numbers not included in any sequence and thus the set of real numbers are uncountably infinite. Cantor defined a set as a collection of things that can be thought of as one.\\\\
    Countability\\
    What is meant by a countable set is a set with the same cardinality as some subsets of the set of natural numbers. Countability means listability, that is, we can list the elements inside the set. Another definition of the countability is to be able to put in one to one correspondence with the elements in the set of the natural numbers.  This property enables us to compare the cardinality of the sets whether they are finite or infinite without counting any elements in sets.\\\\
    The Countability of Rational Numbers (1873)\\
    In 1873, Cantor proved that the set of rational numbers is countable using the snake argument or by putting them into a specific order in which it can be listed.\\\\
    The Countability of Real Algebraic Numbers (1874)\\
    In 1874, Cantor proved that the real algebraic numbers are countable. Real algebraic numbers satisfy the equation of the form $aₒ ωᵘ + a¹ωᵘ⁻¹ + … + aᵤ= 0$. He stated that the collection of all algebraic reals can be written as an infinite sequence.\\\\
    The Uncountability of Real Numbers (1874)\\
    In 1874, Cantor also proved that the set of the real numbers is uncountable, which makes the set of real numbers is the first set shown as uncountable. He stated that any real number can be written as infinite decimal representation.\\\\
    Cantor’s Diagonal argument (1891)\\
    Cantor published a paper in 1891 describing what he called the diagonal argument which is applied to prove the uncountability of the set of real numbers. That reveals the fact that there are not enough elements in the set of natural numbers to make the one-one correspondence. That means the set is uncountable.\\\\
    Infinite sets\\
    For any set to be infinite, there must be an infinite subset of the set can be put into one-to-one correspondence. For two infinite sets contain the same number of elements when there is one-to-one correspondence between them and the size of any whole must be greater than that of any of its parts.\\\\
    Cardinal numbers\\
    In 1878, Cantor turned to what he called “powers” or “Cardinal numbers”. Using this property, Cantor was able to answer whether he can map a square set to a one-to-one correspondence of the points on each.\\\\
    Infinite cardinal numbers\\
    Using the existence of powers, point-sets and the continuum, he introduced the absolute infinite that is bigger than any quantity either finite or transfinite, but the absolute is unincreasable.\\\\
    The Continuum Hypothesis (1878)\\
    This was infamous hypothesis, but so linked to his life work. He stated that no set has a cardinality more than that of the natural numbers and less than the cardinality of the real numbers.\\\\
    Attempted proofs\\
    Cantor spent his life struggling to prove that the continuum hypothesis is true. Cantor continued the process of taking derivatives into the transfinite as it does not always finish after a countably infinite number of times.\\\\
    Mental health\\
    In 1884, Cantor begin to suffer his first serious mental breakdown. Many historians said that this happened due to an ongoing dispute at the University of Berlin. Cantor’s breakdown occurred just as he had returned from a trip where he met other mathematicians. Cantor’s work on set theory was rejected by Kronecker as it lacked examples of the properties introduced in the paper.\\\\
    Final years\\
    When his son died in 1899, Cantor lost his passion for mathematics, and when someone after that tried to disprove the transfinite of the set theory, Cantor considered it as a grave public humiliation. However, the paper disproving the theory was invalid a day later. He spent the last 20 years of his life in a state of chronic depression and suffered from malnourishment during World War I. In 1918, he died of heart attack.\\\\
    Paradise lost?\\
    In 1900, the continuum hypothesis was considered as one of the greatest problems in the 20th century. In 1940, the Austro-Hungarian logician Kurt Gödel confirmed the consistency of the continuum hypothesis by showing that it could not be disproved from the other axioms of the set theory. Twenty-three years later, American mathematician Paul Cohen established its independence by showing that the continuum hypothesis could not be proved from the other axioms of set theory. The realization of the existence of this and other unprovable statements changed the nature of mathematics as a rigorous, logical discipline, prompting Hilbert in 1926 to proclaim in defence of Cantorian set theory.} \cite{Veisdal}.

\textcolor{red}{}

\section{Cantor's Theorem}
We start by defining a set.

\begin{definition}\rm
A \textit{set} is a collection of objects.
\end{definition}

\begin{definition}\rm
Let $A$ and $B$ be sets. We say that $A$ is a \textit{subset} of $B$, and write $A
\subseteq B$ if \textcolor{blue}{$\forall x (x\in A \implies x\in B)$ }
\end{definition}

\begin{example}\rm The following are examples of subsets.
\textcolor{red}{}
\begin{enumerate}
    \item The set of natural numbers is a subset of the set of integers.
    \item The set of integers is a subset of the set rational numbers
    \item The set of rational numbers is a subset of the real numbers
\end{enumerate}
\end{example}

The \textit{power set} $\mathcal{P}(S)$ of a set $S
We are now ready to introduce the notion of a power set.
\begin{definition}\rm$ is \textcolor{blue}{the set that contains the all the subsets of the subset $S$\\ $\mathcal{P}(S)=\{x\mid x\subseteq S \}$}.
\end{definition}

\begin{example}
\textcolor{blue}{let $A=\{1,2\}$ then $\mathcal{P}(A)=\{\{1\},\{2\},\phi, \{1,2\}\}$ \\
                      $B=\{1\}$ then $\mathcal{P}(B)=\{1,\phi \}$  }
\end{example}

\begin{definition}\rm Let $A$ and $B$ be sets.
\begin{itemize}
    \item We say that the cardinality of $A$ is equal to the cardinality of $B$, and write $|A|=|B|$, if there is a \textcolor{blue}{ $bijection$ function f \colon A \to B}

    \item We say that the cardinality of $A$ is less than or equal to the cardinality of $B$, and write $|A|\leq |B|$, if there is an \textcolor{blue}{ $injection$ function f \colon A \to B }

    \item We say that the cardinality of $A$ is strictly less than the cardinality of $B$, and write $|A|<|B|$, if \textcolor{blue}{$|A|\leq|B| \hspace {0.1cm} and \hspace {0.1cm} |A|\neq|B|$}.

    \item We say a set is \textit{countably infinite} if \textcolor{blue}{$|\mathbb{N}|=|S|$ (there is a bijection function from the set of natural numbers $\mathbb {N}$ to the set)}
\end{itemize}
\end{definition}

We will now show that the set of integers is a countably infinite set.

\begin{theorem}
The set of integers $\mathbb{Z}$ is countably infinite.
\end{theorem}

\begin{proof}
\textcolor{blue}{Let Assume the domain of a function is $\mathbb {N}$ and its codomain is $\mathbb {Z}$, we can obtain a bijection function where $f \colon \mathbb {N} \to \mathbb {Z} $ as following: \\
\(f(n) =\[ \begin{cases}
      \frac{-n}{2} & $n is even$\\
      \frac{n+1}{2} & $n is odd$\\
   \end{cases}
\]\\
It is clear that the two branches of the function are both bijective keeping in mind the domain and the codomain, so there is a bijection from $\mathbb {N}$ to $\mathbb {Z}$, which means that the set of integers is countably infinite.
\end{proof}
}
We will now present the main theorem of the article, Cantor's Theorem, which states that it is impossible to have a bijection between any set and its powerset.

\begin{theorem}[Cantor's Theorem]
Let $S$ be any (finite or infinite) set. Then $|S|<|\mathcal{P}(S)|$.
\end{theorem}
\begin{proof}
Let $S$ be any set.

First, we will show that $S\leq \mathcal{P}(S)$ by constructing an injective function $g:S\to \mathcal{P}(S)$ where f is the set of identity function:  $\(f(x)=\{x\}$ and as no element is repeated in the domain and the codomain (power set) contains all the elements of the set, every element in the domain has exactly one image.

\textcolor{red}{}

Second, we will prove that $|S|\neq |\mathcal{P}(S)|$. For the sake of contradiction, assume that there is a bijection $h:S\to \mathcal{P}(S)$.

\textcolor{blue}{Suppose that h is bijective for the sake of contradiction, so it is surjective. Let assume a set containing all the elements in the set itself, but it is not in an image of the function where:\\
$B=\{s\in S \land s\notin h(x)\}$, so $B\in\mathcal{P}(S) $ and as h is surjective, $B=h(x)$ for some elements in the domain, and this contradicts our assumption that $B$ contains only the elements which is in $S$ and not in the range of the function, so $h$ is not surjective}

Therefore, we have shown that $|S| < |\mathcal{P}(S)|$. \cite {website}
\end{proof}



From Cantor's theorem we can deduce the following consequences.



\begin{corollary}
The power set of the natural numbers is uncountable.
\end{corollary}

\begin{proof}
\textcolor{blue}{To prove that a set is uncountable we first prove that there is no bijection from the set of the natural numbers to the set. Using Cantor's theorom, $|S| < |\mathcal{P}(S)|$. Then $|\mathbb{N}|<|\mathcal{P}(\mathbb{N})|$, which means there is injection and there is no bijection $f\colon \mathbb{N} \to \mathcal{P}(\mathbb{N})$, therefore the power set of the natural numbers is uncountable.}
\end{proof}


\begin{corollary}
There is an infinite sequence $A_0, A_1, A_2, \ldots$ of infinite sets such that $|A_i|<|A_{i+1}|$ for all $i\in \mathbb{N}$. In other words, there is an infinity hierarchy of infinities.
\end{corollary}
\begin{proof}
\textcolor{blue}{Using Cantor's theorom, $|S| < |\mathcal{P}(S)|$. Therefore,  the power set of the power set of the set has cardinality that is strictly bigger than the power set of the set, and we keep doing this to have infinte secuence where every element is the power set of the previous element.\\
$|S| < |\mathcal{P}(S)|<|\mathcal{P}(\mathcal{P}(S))|<|\mathcal{P}(\mathcal{P}(\mathcal{P}(S)))|<........$\\
Therefore, we proved that there is an infinite sequence $A_0, A_1, A_2, \ldots$ of infinite sets such that $|A_i|<|A_{i+1}|$ for all $i\in \mathbb{N}$
}
\end{proof}


\begin{thebibliography}{9}

\bibitem{Veisdal}
Jorgen Veisdal. \textit{The Nature of Infinity - and Beyond}. Medium, 2018.
\bibitem{website}
Retrieved from:\\
https://www.whitman.edu/mathematics/higher_math_online/section04.10.html

\end{thebibliography}

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