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 successful in presenting there is one to one correspondence to all the natural numbers and all of the squares of natural numbers to infinity which in turn suggests that are as many square numbers as integers even though every integer is not a square number. This concept was known to be Galileo’s paradox. Sadly, Galileo shied away from the issue since most of his work could only be solved using a finite amount of numbers. To Cantor, this was a great challenge that he would overcome due to his dissatisfaction of the infinity presented by Galileo.
Cantor started off by saying that we can add two same integers together, such as 1 and 1 or 25 and 25. So, he thought it should be possible to add (as well as subtract) infinities. By this assumption, then there are more than one infinity, and one infinity can be larger than the other infinity and thus more infinities exist beyond that. Cantor showed that they may exists infinitely many sets of infinite numbers where the size of those sets are larger than others. These were his overall findings, his process was a bit more complicated. What Cantor did specifically to come to this conclusion is he took two sets of numbers: the natural numbers and an infinite series of the multiples of 10. Even though the multiples of 10 was a subset of natural numbers, a one-to-one correspondence can be made within the two sets to prove that they were the same “sizes” of infinite sets such that they had the same number of elements.
This occurred through other subsets of natural numbers too, as Cantor found out and was even able to find pairings with fractions with all of the whole numbers this way thus showing that rational numbers were also the same sort of infinity as the natural numbers despite having the gut feeling that there are way more fractions than whole numbers. When this method was applied to decimal numbers (which includes numbers such as π and square root of 2) his method then began to break down. The next blog post will cover his method to approach this breakdown and how he was still able to make findings on the subject of infinity, which was a very controversial philosophical subject for his time.
**Edit: Transitive was changed to transfinite since that is what the author of the post meant to say. Transfinite will be defined in the next post since it applies to the story of Cantor and his idea of infinity.
Mastin, L. (2010) 19th Century Mathematics-Cantor. Retrieved from: https://www.storyofmathematics.com/19th_cantor.html
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 successful in presenting there is one to one correspondence to all the natural numbers and all of the squares of natural numbers to infinity which in turn suggests that are as many square numbers as integers even though every integer is not a square number. This concept was known to be Galileo’s paradox. Sadly, Galileo shied away from the issue since most of his work could only be solved using a finite amount of numbers. To Cantor, this was a great challenge that he would overcome due to his dissatisfaction of the infinity presented by Galileo.
Cantor started off by saying that we can add two same integers together, such as 1 and 1 or 25 and 25. So, he thought it should be possible to add (as well as subtract) infinities. By this assumption, then there are more than one infinity, and one infinity can be larger than the other infinity and thus more infinities exist beyond that. Cantor showed that they may exists infinitely many sets of infinite numbers where the size of those sets are larger than others. These were his overall findings, his process was a bit more complicated. What Cantor did specifically to come to this conclusion is he took two sets of numbers: the natural numbers and an infinite series of the multiples of 10. Even though the multiples of 10 was a subset of natural numbers, a one-to-one correspondence can be made within the two sets to prove that they were the same “sizes” of infinite sets such that they had the same number of elements.
This occurred through other subsets of natural numbers too, as Cantor found out and was even able to find pairings with fractions with all of the whole numbers this way thus showing that rational numbers were also the same sort of infinity as the natural numbers despite having the gut feeling that there are way more fractions than whole numbers. When this method was applied to decimal numbers (which includes numbers such as π and square root of 2) his method then began to break down. The next blog post will cover his method to approach this breakdown and how he was still able to make findings on the subject of infinity, which was a very controversial philosophical subject for his time.
**Edit: Transitive was changed to transfinite since that is what the author of the post meant to say. Transfinite will be defined in the next post since it applies to the story of Cantor and his idea of infinity.
Sources:
Editors (17 October 2017). Georg Cantor Biography. Retrieved from: https://www.thefamouspeople.com/profiles/georg-cantor-519.php
Mastin, L. (2010) 19th Century Mathematics-Cantor. Retrieved from: https://www.storyofmathematics.com/19th_cantor.html
What are transitive numbers? Do you mean transfinite?
ReplyDeleteIndeed, what I meant to convey was transfinite! This will be explored within the next blog post. Stay tuned!
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