Infinity and Beyond?

There once lived a man named George Cantor (1845-1918), and if you do not recognize the name and if you are not familiar with his work this may be irrefutable proof that you are a normal person.  Nevertheless, in the world of mathematics Cantor is extremely important, specifically for his work involving infinity. He invented set theory, which I won’t get into except to say that a revelation he had about something called one-to-one correspondence (between numbers of a set) lead him to very profound conclusions. I am not a mathematician so I am not going to embarrass myself trying to get into the details. The main conclusion Cantor drew was that some infinities can be larger than others. When I first heard (or read) this I almost dropped the book (ok, not really). It sounds ridiculous, does it not? Cantor’s reasoning went something like this…

Take your good old infinity, the integers {1,2,3,4,5…}. This series of numbers can of course go on forever, hence infinity. Although, this series of endless integers can in fact be counted (a job for a motivated undergrad!). This type of infinite set is known as either a countable set or denumerable sets.

In accordance with 1-to-1 correspondence, Cantor envisioned a set of numbers that included decimals. For example, (if you are not familiar with decimals) {1.0, 1.1, 1.2, 1.3, 1.4……}. However, as you probably know, you can add decimals to make your number more accurate, indeed you can add infinite decimals if you so choose. So, imagine a set of numbers like this: {1.1, 1.12, 1.123, 1.1234, 1.12345, 1.123456…}. This type of set is said to be uncountable or nondenumerable. Because you can really never make progress with this second set of numbers, it is said to be a larger infinity than the first. Cantor apparently proved this mathematically although I am not even going to look at his proof.

One thing (many things really) does not make sense to me. I have heard that there are as many odd integers as theres are integers. If you keep track of all the odd numbers and normal numbers, there will always be another odd number to correspond with the normal number (normal numbers just means odd and even numbers ex: 1,2,3,4,5,6…).  Of course the odd numbers will grow larger (it will approach 2x as large I think) but who cares, it’s infinite. But this fact seems to contradict Cantors notion of infinities of different sizes.

 

Why is it not the case that the countable infinity can just add another integer in its list to correspond to the uncountable series? The countable series will also grow bigger (infinitely bigger??) than the uncountable series. The answer to this most likely lies in the proof Cantor wrote which is why I am having difficulty grasping the concept. Another titan of this realm of mathematics named David Hilbert came up with something called Hilbert’s Hotel which aims at explaining this concept of infinities in a more concrete way.

I will consult with the infinitely intelligence over at Reddit and report what I find.You can read more about Cantor’s life and work at his wikipedia page here.

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About ben james

"The stars keep me up at night" -some song I heard one time Human. Studying the intersection of Neuroscience and Computation. Wanna be (astro) physicist.
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