Counting Infinity

One of the first things we learn at primary school is counting numbers. Even small kids can count up to very large numbers. We all are familiar with counting finitely many objects. But what about counting infinitely many objects? If I would have two infinitely many sets to compare, how would I understand the concept of “larger” quantity? Or do they equal to each other? Well, they are not in general! But how do we know that? I will briefly explain it’s underlying concept ‘cardinality’, but don’t worry, it won’t be an article that only addresses to mathematicians. I will summarise what math tells us and how math interprets infinity.. In fact, I will not go into details with cardinality in order to preserve the concept plain and simple.

Counting finite objects is defined to understand the ‘larger’ quantity between them. However, because infinity is not really a number, we cannot just try counting infinitely many objects. So, what to do? I will write how math deals with it. You can skip this part and don’t worry I will interpret what math explain us.

— Math Mod On–

If we have two sets A,B, we say card A = card B if we can find a bijective correspondences between A and B. That means, for each object in A, there has to be a unique object in B that corresponds to it. For example, if I take A={a,b,c} and B={e,d,f} then we all know that both sets have 3 elements. So they have same amount of elements.

Similarly,  card A = card B because I can correspond a with e, b with d and c with f. Therefore, for finite sets this concept coincide with the concept of counting the objects!

Using this relation we can also decide which infinite set is ‘bigger’. Here are some quite shocking examples that you will not believe they are correct at your first read!

N={0,1,2,..}={set of all natural numbers}

Z={..,-2,-1,0,1,2,..}={set of all integers}

R={set of all real numbers}

(0,1)={x : x is a real number with 0<x<1}

we have:

card N = card Z 

card (0,1) > card Z

card R = card (0,1)

— Math Mode Off–

Cardinality says that if I cannot find a correspondences between objects in set, then one set has to have some elements that cannot match with any other object. Whichever set has left element that is not matched, has larger quantity. For instance, If we set A={a,b,c} and B={e,f}. Then card A > card B (what a surprise huh?) because when we match a with e and b with f, we left c unmatched! This example is compare two finite sets but similarly we can apply the same for infinite sets. Thats the idea of cardinality.

Can we find two infinite sets with different cardinality? If yes, how many different Infinite sets exists? Well, I already gave an example of two infinite sets one is ‘larger’ than the other (card R > card N). About how many different infinite set exists, ironically we cannot count it! In fact, there are more than infinitely many infinite cardinality sets!(Makes no sense, right?)

Well, because we have too many infinite sets mathematicians decide to split it into two parts: countable infinite sets and uncountable infinite sets. All engineers, physicists etc make their calculations in the countable infinite world. Because uncountable infinite sets are too complicated comparing with countable ones. Another ironic result is that there is only one countable infinite set! All the others have same cardinality with this one! Uncountable infinite sets have unaccountably many different cardinality results. Just check the following line of infinite numbers!

Screen Shot 2017-11-16 at 19.30.33 

0 – denotes the finite numbers

N – denotes the natural numbers which is countable infinite set

C – denotes the real numbers which is uncountable infinite set

and all the rests F,G and so on are uncountable infinite sets.

As you might realised line of infinite numbers are not like real number line. It has many jumps! 

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