Introducing infinity can be done many ways, in this essay it will go over the definitions, history, past discussions, and if it is useful. There are several definitions and infinity has been a very controversial topic over the years. This also explains why there are past discussions of it and the main discussion that will be used is Hibert’s discussion of the “Grand Hotel”. Including all of this is the idea of if it even is useful. This is how this essay will be introduced.
To describe infinity, I would put it as a very big amount, but it has many definitions such as a far distance, endless value, or a number that is greater than any quantity. Infinity is an idea that has been defined as many things, the main definition for mathematics though is a number that is greater than any quantity. Although this contradicts just being a large amount or even an endless value so figuring out the true definition will end up being part of the discussion. If you were to say it was an endless value tho, it could not be measured therefore could not be greater than any quantity. So the definitions are quite controversial.
The history of infinity starts off in ancient Greece with the concept of infinity that came from the observations of time seeming to be without end, that space and time can be unendingly subdivided, and space is without bound. The first person to record this idea of infinity was Anaximander, a pre-Socratic Greek philosopher who lived in Miletus. He used the word apeiron which means infinite or limitless. The issue of infinitely small numbers even led to the discovery of calculus in the late 1600s and later on the common symbol for infinity, ∞, was invented by the English mathematician John Wallis in 1657. Now we are at the time where we still are figuring out what infinity completely encompasses.
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The Discussion of the Grand Hotel goes back to the German mathematician David Hilbert, who used the example of a hotel to demonstrate how the use of infinity works. For example in this discussion if you needed to make room for another guest in a hotel, you could move all the other guests down one room. The problem with this is the fact that you only have a finite amount of rooms, but in this discussion though you have an infinite amount of rooms. This means you can make room for any guest arriving by just moving the guests down one simultaneously. This relates to infinity by showing that if there are infinitely as many guests as rooms there is still always more room for one more guest.
In conclusion, infinity isn’t just an endless value but a value that can be used to show an unknown amount that can be expanded if needed but not measured. The definitions help show the many ways people have described this and the history has shown how infinity has been used. Infinity is also useful because it helped create certain maths like calculus. The discussions of infinity have also explained how it can work in real life. So in the end, this sums up what infinity really was.
