by Sarnaz Hossain
The idea of infinity has been around for decades yet most of the time it has
been disregarded as a topic which is reliant on philosophical views and more of
a topic for religion because of the implications of god than a mathematical
concept worthy of research and discussion. This may be one of the reasons
infinity is not really a topic of discussion even nowadays as it could bring up
different view points and beliefs. This implication for infinity is one that
has been around for hundreds of years. However there have been a few which have
delved deeper into this topic which brings me into my main focus for this
article.
Greorg Cantor is a German mathematician whose work in mathematics has been incredible from his work in set theory to his work with infinity, both if which will be explored here. Firstly Cantor was born in 1845 in Germany. This era in mathematics was the time where people elsewhere were very critical in the fundamental aspects of mathematics and were always questioning the validity of mathematics including core mathematical theories such as calculus. People wanted rigorous proofs and as Cantor was fascinated with concepts such as infinity which were quite difficult to explain and prove, he came across a lot of criticism from other influential mathematicians of the time. Sadly, later in his life he passed away due to mental health issues and was rumoured that this may have been the effect of him being ridiculed by other people at the time even though we now realise that his ideas were ingenious and extremely influential in the future that we live in todays.
So what were his ideas and theories? Well to being to discuss these theories we need to understand simple set theory. This is the theory that numbers can be expressed in a group a.k.a a set. For example the set of 3 4 and 5 can be shown as {3,4,5}. And the numbers from 1 to 100 can be expressed as {1,2,...,100} and so on. These set that I have shown clearly are finite set with clear end points. However things get more interesting when we think about infinite sets which can also be show like the set of natural numbers {1,2,3,4,...} or the set of even numbers {2,4,6,8,...}. Cantors main point was that if you can put each number of an infinite set to correspond with each number in the set of natural numbers the size of the sets are the same. This idea is hard to grapple with as even some of the great mathematicians such as Galileo couldn’t get past how let’s a the set of square numbers has the same size as the set of natural numbers even though there are many natural numbers that aren’t in the set of square numbers. This brang a whole new aspect to mathematics as people thought about trying to put all different kinds of numbers into 1-1 correspondents such as fractions which was done successfully and decimal number too.
However when trying to find the proof for these decimal numbers Cantor came across a huge problem. He couldn’t put the set of real numbers into a one to one correspondence. This is because if you think of a set of all the real numbers you can always create a number which isn’t in this set. You can do this by creating a number where the first decimal place is different to the first decimal place in the first number in your set. The second decimal place will be different to the second decimal in the set and so on until there is a number which isn’t in the set of real numbers and so cannot be put into 1-1 correspondence and therefore the size of the set is actually bigger than the infinity of set of natural numbers. The proof about the real numbers is shown also in the diagram below.
This was a revolutionary discovery that there were different sizes of infinity which Cantor coined the theory of transfinite sets. Even to this day cantors core ideas are helping in many different fields of mathematics and modern day technology ranging from things such as the research into black holes to things like the difference between artificial intelligence and the human brain. These implications of Cantors key theories show exactly how powerful his mathematics was and how even through ridicule and hardship trusting yourself and having belief in your work can result in great things. The “philosophical” ideas that were discouraged are now widely respected.
Greorg Cantor is a German mathematician whose work in mathematics has been incredible from his work in set theory to his work with infinity, both if which will be explored here. Firstly Cantor was born in 1845 in Germany. This era in mathematics was the time where people elsewhere were very critical in the fundamental aspects of mathematics and were always questioning the validity of mathematics including core mathematical theories such as calculus. People wanted rigorous proofs and as Cantor was fascinated with concepts such as infinity which were quite difficult to explain and prove, he came across a lot of criticism from other influential mathematicians of the time. Sadly, later in his life he passed away due to mental health issues and was rumoured that this may have been the effect of him being ridiculed by other people at the time even though we now realise that his ideas were ingenious and extremely influential in the future that we live in todays.
So what were his ideas and theories? Well to being to discuss these theories we need to understand simple set theory. This is the theory that numbers can be expressed in a group a.k.a a set. For example the set of 3 4 and 5 can be shown as {3,4,5}. And the numbers from 1 to 100 can be expressed as {1,2,...,100} and so on. These set that I have shown clearly are finite set with clear end points. However things get more interesting when we think about infinite sets which can also be show like the set of natural numbers {1,2,3,4,...} or the set of even numbers {2,4,6,8,...}. Cantors main point was that if you can put each number of an infinite set to correspond with each number in the set of natural numbers the size of the sets are the same. This idea is hard to grapple with as even some of the great mathematicians such as Galileo couldn’t get past how let’s a the set of square numbers has the same size as the set of natural numbers even though there are many natural numbers that aren’t in the set of square numbers. This brang a whole new aspect to mathematics as people thought about trying to put all different kinds of numbers into 1-1 correspondents such as fractions which was done successfully and decimal number too.
However when trying to find the proof for these decimal numbers Cantor came across a huge problem. He couldn’t put the set of real numbers into a one to one correspondence. This is because if you think of a set of all the real numbers you can always create a number which isn’t in this set. You can do this by creating a number where the first decimal place is different to the first decimal place in the first number in your set. The second decimal place will be different to the second decimal in the set and so on until there is a number which isn’t in the set of real numbers and so cannot be put into 1-1 correspondence and therefore the size of the set is actually bigger than the infinity of set of natural numbers. The proof about the real numbers is shown also in the diagram below.
This was a revolutionary discovery that there were different sizes of infinity which Cantor coined the theory of transfinite sets. Even to this day cantors core ideas are helping in many different fields of mathematics and modern day technology ranging from things such as the research into black holes to things like the difference between artificial intelligence and the human brain. These implications of Cantors key theories show exactly how powerful his mathematics was and how even through ridicule and hardship trusting yourself and having belief in your work can result in great things. The “philosophical” ideas that were discouraged are now widely respected.
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