Counting beyond infinity

All have at some point in our lives used the word infinity without understanding what it truly encapsulates. Infinity contrary to popular belief is not a number rather it is an abstract concept that dwells in the “infinite” realm that is the universe.

For a little mathematical understanding infinity is not on the number line (a line representation of all numbers) per se, still it permeates the line. Infinity is not a number as such; however it does include all numbers. All this still seems philosophical because it is, there are no strict mathematical definitions for infinity and even if there were definitions defining infinity as a number it would be impossible to wrap our minds around it.

There are however two types of infinities:

  • Countable Infinity – Well one might think that if it was countable then it really wouldn’t be infinity. But the term countable is not in the literal sense. You may be able to count to say a “gazillion” (not sure if that even exists, just made it up) or something like that over the course of your time because you know when you need to stop counting. However consider counting all numbers starting from 1 (also called natural numbers) and keep on counting. It seems fairly obvious that one will reach “infinity” in a finite time even if the time is beyond the end of the universe (which it most probably is!). Thus, we observe that even countable infinity is not countable in the true sense.
  •  Uncountable infinity – Here is where complications start to arise. Let’s understand the number line a bit better first.

 

As you can see that in the number line every number occupies a fixed spot and can be figured out very easily. However if you come to think about it, what is the 1st number after 0 and before 1? You may say, well that is easy it is 0.1 or 0.01 or 0.001 et cetra…No matter what you say you will always be wrong because there will always be room for one more zero after the decimal place.

Thus the sought after number would be 0.000000…..000001 (You can put in as many zeros as possible, almost an infinite number of zeros). And thus, the pattern finally begins until we reach 1. Thus we observe that there exist an infinite many numbers between any two numbers not just 0 and 1.

Infinity was a great topic of interest for philosophers which thereby resulted in a lot of paradoxes pertaining to infinity.

Zeno’s Paradox: Say you have to walk two steps, easy right? But wait here comes the interesting part.

You can split the two steps into one step followed by another. Next split one step into half and continue the process. As you might have guessed we will land with an infinite number of steps.

No. of steps = 1+1/2+1/4+1/8+1/16……  to infinity (every succeeding term is half of the previous term). And finally to top it off any person cannot execute an infinite terms. Thus resulting in the paradox as to how many steps there are two or infinite.

Hilbert’s Hotel Paradox – This is one of the paradoxes which brings out the true nature of infinity. Imagine a hotel which has an infinite guests and there are an infinite number of rooms in the hotel and there is another guest that walks in to the hotel asking for a room. Then the person in the 1st room is shifted to the 2nd room, the 2nd to the 3rd, the 3rd to the 4th……. and so on. Thus, freeing up the 1st room in which the new guest can stay. thus the hotel still has all its rooms filled.

These are two of the most interesting paradoxes that do exist regarding infinity.

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