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Politics and Culture During Cantor's Time

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Since Cantor mostly published papers, made large mathematical insights and continued to work on mathematics for the majority of his life starting around 1869, a cultural and historical look at Germany from 1871-1918 will suffice, since Cantor died in 1918. On January 18,1871, the German Empire was founded. This was due to three successful wars that were won by the North German state of Prussia. Within seven years, Denmark and France were destroyed in very brief conflicts. The Germanic empire was not created through nationalism of the people, but through traditional diplomacy and agreement by the leaders of the states of the North German Confederation which was headed by Prussia. Prussia was so prevalent and the most obvious choice of the leader because it had more than three-fifths of the area of Germany as well as about three-fifths of the population, so it was the dominant force. It remained so until the empire’s demise at the end of yet another war in 1918. In the aspect of...
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Father of Set Theory (and Infinity Infinities) Part 3

Cantor’s work from 1874 to 1884 begins the real origin of set theory. Since set theory has become a fundamental part of modern mathematics and its basic concepts are utilized throughout multiple branches of mathematics, it is definitely worth noting this work. Sets had be used implicitly since the beginnings of mathematics (even back to Aristotle) however those ideas of sets had only covered finite sets. For distinction, the “infinite” was kept separate and was often a topic of philosophical and not mathematical discussion. Cantor showed that there could be infinite sets of different sizes some of which are countable and some of which are uncountable. Throughout the 1880s and 1890s his set theory was refined where he defined well-ordered sets and power sets. During this time he also introduced the concepts of ordinality and cardinality. Cantor was also attributed to the arithmetic of infinite sets. What is today known as Cantor’s theorem, states generally that, for any set A , the pow...
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Father of Set Theory (and Infinity Infinities) Part 2

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Last time we talked about how Cantor began exploring infinities by saying they could behave like integers in the fact that they could be added together and that one infinity could be larger than another infinity. He applied rules of a one-to-one basis on numbers to show that the sets of infinity were the same sizes of the infinite sets in the fact they had the same number of elements. Cantor realized that this could not be applied to decimal numbers which included irrational numbers. He had to think of several clever arguments in order to prove this, one of them being the “diagonal argument.” This argument proved that it was always possible to construct a new decimal number that was missing from the original list and thus makes decimal numbers infinite in nature, even more infinite than natural numbers. The diagonal argument is as follows and corresponds to the picture above: Imagine an infinite set of numbers made up of an infinite pattern of just two digits (the example ...
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Father of Set Theory (and Infinity Infinities) Part 1

Although Cantor was known to dabble into various subjects of mathematics and advance certain fields of mathematics such as trigonometry and transfinite numbers, his accomplishment which makes in a great mathematician is the founding of set theory. Specifically, Cantor established the importance of one-to-one correspondence in set theory, defined infinite and well ordered sets, and proved that real numbers are far more numerous that natural numbers. What most people note about Cantor’s work is that he found the existence of infinity infinities, which was of philosophical debate in his time. Through the founding of set theory, Cantor really found out a notion in mathematics that truly makes him remarkable, the idea of infinity. For a little bit of background on the idea of infinity, it is important to know who else has studied infinity before Cantor. Galileo (around the 17th century) had to confront the idea of infinity and he had numerous infinities presented to him. Galileo was suc...
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Georg Cantor: Personal Life/History

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Georg Ferdinand Ludwig Philipp Cantor Born March 3, 1845 in St. Petersburg, Russian Died: January 6, 1918 in Halle, Germany Georg Cantor was born in 1845, his father was Georg Waldemar Cantor and mother was Maria Anna Böhm. Georg Waldemar was born in Denmark and was described to have a passion for culture and arts. Georg Waldemar was a successful merchant in St. Petersburg and then became a broker in the St. Petersburg Stock Exchange. Maria Anna was Russian with strong taste for music and the arts as well. Therefore, Georg became talented at a young age by becoming a superb violinist. Georg’s father was Protestant and his mother was Roman Catholic; Georg himself was brought up as Protestant. Georg was educated in his early days by a private tutor and attended a primary school in St. Petersburg. When Georg was eleven years old, his family moved to Germany. It was said that Georg held Russia with nostalgia in his heart and never really felt at ease in Germany, even th...
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