Whenever we think of maths, the recurring thought is associated with infinity. Infinity is the pop culture mascot for mathematics, and its opulent uncertainty makes it special yet misunderstood on many levels. Firstly, we don’t understand whether such a concept exists and simply trivialise it; secondly, we seem to think there is one value of infinity, and third, we think any answer which isn’t certain would simply be infinity. People think infinity is a weak point of maths, a point at which the fabric of maths breaks due to the flawed nature of the number system. These misconceptions are to be expected given the abstract nature of infinity, however, understanding the infinity Paradox which defines a certain set of rules which limit infinity can make us realise why such a concept is simply restricted by our capabilities and not maths itself.

Infinity in short is a boundless value, and ‘goes on forever such that it is the last known mathematical value. Infinity is the upper bound for all real numbers, but the difference in behaviour concerning how real numbers are classified is extreme. For example, if we take the real numbers between 1 and 2 it would be Infinity. The real numbers from 1 to 3 would also be Infinity. The infinity of the latter must be twice the infinity of the former because the interval between 1 and 3 is twice the interval between 1 and 2, but both are infinity. In the first case, “∞ = ∞”, corresponds to our logic. However, in the latter case, “2∞ = ∞”. Theoretically, multiplying infinity by a scalar should be infinite in itself, but here it behaves differently. The equation “2∞ = ∞” can be thought of as a simplification to “2 = 1” (this does not happen exactly because infinity cannot be divided by infinity, but for Hazel Grace’s argument, Let’s limit the to reduce comparisons (spacing size). This is illogical and does not follow the course of mathematical logic. Here we can see that mathematics deviates from the seemingly decisive nature that we are accustomed to throughout our lives, suggesting its room for development, and infinity’s wicked traits as a concept. 

Infinity has various sides to it, now that we have defined its existence in real numbers, here is a paradox related to sets and infinity we can examine for a better understanding. When Georg Cantor introduced ‘set theory’, the mathematics world was shaken as finally there was a method of defining numbers in groups, and quantifying infinity into relative terms, however, the breakthrough came as over the years Cantor also developed ‘Cantor’s Paradox’. He claimed that infinity has to exist as multiple different values and each time we assume a particular value there are many values of infinity higher than that. The lowest value of infinity was defined to be the set (1,2,3..), known as countable infinity and the highest value would be (1,1.000…,2,2.00….) a set of all real numbers almost impossible to quantify. 

This theorem is particularly interesting because it shows us the beauty of mathematics. Cantor was able to prove that maths has no bounds, it is truly limitless within logical limits, allowing us to prove things on relative terms without a definitive value. While this might seem like an overcomplication it proves how there is no particular end to the imagination of numbers, and while nobody may be able to write down a definitive value for infinity, its uncertainty still allows us to prove that a countable and uncountable infinity exists. Given the amazingly appreciative nature of infinity towards breaking maths, this paradox isn’t a paradox, however, there lies the beauty of maths. A perfect set of theorems that are broken by a single concept the ‘Infinity’