The Life and Works of Georg Cantor

Georg Ferdinand Ludwig Philip Cantor, shortly Georg Cantor, was born on March 3, 1845 in Saint Petersburg, Russian Empire. Cantor was the eldest son of Maria Anna Bohm and Georg Waldemar Cantor among their six children. Cantor performed well in his studies both at boarding and trade school. Back then, he already excelled in mathematics and was good at playing violin. But as time went by, Cantor realized that he wanted to commit his life to mathematics. He thought that it was in the said field that he considered as his calling and would be the reason for his own success. He began his secondary studies at Rehlschule in Darmstadt. His exceptional skills in mathematics were noted as he graduated with distinction. He continued his studies at Technische Universität Darmstadt and Swiss Federal Polytechnic. Cantor transferred to study at the University of Berlin after receiving inheritance upon the death of his father. He was taught under the knowledge of the great German mathematicians namely Ernst Kummer, Leopold Kronecker, and Karl Weierstrass. He then focused on studying number theory. After receiving his doctorate degree at University of Göttingen, he continued his career as a teacher and a researcher mastering mathematics. For a short time, he taught in a girls' school located in Berlin. It was later that he took a position as a lecturer at the University of Halle, where most of his entire career was spent. During his time, Cantor made notable accomplishments in the field of mathematics which are widely used and studied today. First was the set theory. In the late 19th century he founded and started to develop the mathematical theory of sets, which aims to investigate well-defined collections of objects. He called these collections sets and the objects included in it are called its elements. Second was the transfinite numbers. In 1915, Cantor demonstrated that these numbers are infinite which are larger than all finite numbers, yet they are not necessarily absolutely infinite and are called transfinite numbers. These also include the transfinite ordinals, numbers used to provide an ordering of infinite sets and the transfinite cardinals, numbers used to quantify the size of infinite sets. Cantor introduces the symbol aleph naught or aleph null to represent the first transfinite cardinal number and omega to designate the least transfinite ordinal number. Third was the continuum hypothesis. In 1878, the continuum hypothesis was proposed and originally formulated by Cantor, which states that an infinite set with a cardinal number between a small infinite set of integers and an infinitely large set of real numbers doesn't exist. Meaning, an infinite set cannot contain numbers that are bigger than natural numbers and at the same time, smaller than the real numbers. It is the renowned problem of set theory. Many mathematicians attempted to prove this hypothesis but no one succeeded as it shows that it cannot be proved to be neither true nor false from the usual axioms of set theory after many various discoveries. Up to the present day, continuum hypothesis remains open. Lastly, the It is a mathematical argument to show that if a given set is a set of integers, real or any natural numbers, then it cannot be used in a one-to-one correspondence. Meaning, it's an argument to prove that a set of real numbers is uncountable. In his 1891 article entitled Contributions to the Founding of the Theory of Transfinite Numbers, he discussed it further and provided proof to support this argument. It is undeniably true that Georg Ferdinand Ludwig Philipp Cantor was a great mathematician. With perseverance, passion in mathematics and faith in God, he was able to discover and formulate many theorems that from his time and until this modern era are using and applying it. Despite the objections and rejections he received, he chose to pursue committing his life in mathematics and bring new perspective and knowledge generation by generation. With his knowledge, we are able to understand set theory, which is considered as a foundation of mathematics as we know it today. May we all find inspiration from the father of set theory, Georg Cantor. Let us widen our knowledge like how he himself described a set. Many that allows itself to be thought of as a one.