August 14, 2020

What does an infinity stand for? Infinity in mathematics signifies an absurdly large number, so large that you can’t write it off. The work around this is to just represent it symbolically. Writing it off would be an almost impossible task if you are serious about the true representation of the number without any approximation going on. Approximation can definitely ease it in a certain way. But if one can imagine an absurdly large number, one can also imagine the next number larger than that one. For eg: If the first absurdly large number is represented by ‘n’ (using approximations) instead of ‘infinity’, one can easily get to the next larger number ‘n+1’. And then repeat the process all over again for ‘n+1’ to finally get a number that is absurdly bigger than the original absurdly bigger number, finally making the use of ‘infinity’ inevitable. So, approximation doesn’t ease our task anyway. To get a better context of what an infinity is, consider this. Imagine there is a big Pizza to be divided among 10 people equally. Each individual will have one-tenth of the original pizza. Now imagine the pizza to be divided among only 5 people similarly. Since the number of people between which the pizza is to be divided gets smaller, the part of pizza that each receives gets bigger. Now, if the pizza is to be divided between only two people equally, each person will receive half of the pizza. Notice that as the number of people gets smaller, the size of the part of pizza that each gets, gets larger. Similarly, if its only 1 person, he can have the whole of pizza. Now understanding this process from an inductive viewpoint, a law can be formulated like this. ‘As the number of parts into which a certain thing is to be divided equally, gets smaller, each part will keep getting bigger.’ Taking this law to its singular limit, if a certain thing is to be divided into zero parts, the size of each part is such that it gets infinitely bigger. We reached to the classic paradox ‘can part be greater than the whole?’ The answer to this, is actually two-fold. First, how can you consider ‘zero’ as a valid number of parts of something? Well, this is the reason behind this impossible result above. The second answer is, ‘this is singularity baby. So, anything can happen. It has this name called ‘singularity’ for a reason. Zero is a powerful number. If you are to use it, you will also have to bear its consequences. And a lot of similar mumbojumbo of the sort, how can you dare to….’The lesson that we can learn from the paragraph above is that ‘zero’ is a weird number. You gotta be careful while you use it whether it be while signing the cheque or while concluding that you have made zero progress in life. Wherever you use it, it will have consequences. Consequences of its kind. To avoid this, mathematicians and physicists so frame the science that numbers can be anything (positive numbers, negative numbers, fractions, decimals) but zero, in a given mathematical structure. The purpose is clear: to restrict the infinities from happening. But then still, some infinity is emergent quantity. Like for example, the extent of our universe. This infinite extent of our universe isn’t the consequence of any zeros. But it is the emergent property of expanding universe. The question that we are asking here is, are all infinities equally big? And if there is a comparison of this sort, would it be fair to even compare without knowing how big an infinity actually is? The answer is simple and straightforward. We do not need to know explicitly how big something is to know if it is infinite or not. All we do is pair up these quantities with the sequence of natural numbers. This in the language of mathematics is called, One-One-Correspondence. If you can map, each of these quantities with some number sequentially (basically what I am doing here is explaining how to do counting), the quantity definitely is infinite because eventually, that quantity is gonna pair up with the infinity of natural numbers. Thus, they have the same size.Well! Then lets come specifically to the topic, how can some infinities be bigger than the other infinities? How many numbers are there between any two natural numbers? To make things simple, how many numbers are there between 1 and 2? Well! There are 1.1, 1.2, 1.3, ……..1.8, 1.9. But there are also 1.11, 1.12,……1.31, 1.32, ……1.91, 1.92,…… But not just these, there are also 1.000000000000001, 1.0000000000000000012,………….1.999999999999998, 1.99999999999999999 and infinitely many other numbers too like 1.1745638464783963084628. It turns out that, there are infinite number of numbers between 1 and 2. The question is… Is this infinity any bigger/smaller/equal to the set of natural numbers? When we try to put each of these numbers between 1 and 2 into a one-one-correspondence with the set of natural numbers starting from 1, we can inductively see that not all real numbers are paired up with the sequence of natural numbers. The formal proof of this was carried out by Cantor in his Diagonal Argument. This is the case of being uncountably infinite. We say that the set of real numbers are uncountably infinite. There simply are too many of real numbers to pair up with natural numbers. This is how some infinities (infinities of real numbers) are bigger than other infinities (infinities of natural numbers).#physics#science#infinity#projectmodulus

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