Counting Beyond Infinity- Ninad Kumbhojkar

What is the largest number known to man? Some geeks will say Googol. Some will say Googol Plex. Googol is 10100. GoogolPlex is 10googol . But these numbers are minute compared to what we call as Infinity. Scientists have given some theories about what infinity can be and how to count past it. What are those? How do we count numbers beyond infinity? What are cardinal and ordinal numbers? Let’s understand them in a very simple way…

Some Basics!

There are basically two classes of natural numbers. One is ordinal and another class is cardinal. Cardinal is kind of a scalar quantity. Cardinal number is just the number of terms in a set. But ordinal numbers are somewhat different.  The given number is ordinal when, including cardinality of terms in a set, we have to consider the order in which the terms are arranged. So Ordinal numbers are kind of vector quantities.

So what is the estimation of the basic or the ‘smallest’ infinity?

Smallest infinity defined is א‎0. This is the first character of Hebrew alphabet, Aleph with subscript 0, normally dictated as Aleph null. If we take all the set of countable numbers like 1,2,3,4 and so on and put them all together, we will get the א‎0. א‎0 is a cardinal number. It shows only the number of terms in a given set. We can say that it is the smallest infinite cardinal number ever defined. How do we count beyond א‎0? This is what is counting beyond infinity!

א‎0: A cardinal!

We all have learnt some Mathematics. They teach us that infinity+1=infinity. Similarly, if we add one term to the set of terms whose cardinality is א‎0 we will still have א‎0 terms in the set. Lets understand this. Lets say, I have א‎0 amount of bananas in a box. I have numbered them from 1 to א‎0. Now I add one more banana to the box. But still the number of bananas will be same as it was before i.e. א‎0. Because we can now denote the added banana as 1 and rest of them from 2,3,4… And we know them actual quantity or the number corresponding to א‎0 has not been defined. Rather it will never be defined according to the axioms that we use in Mathematics today. The axiom is, א‎0 in an inaccessible number. There is nothing in the universe that corresponds the א‎0. Not even the size of this whole universe. So, the cardinality of any of the infinite set remains to be א‎0. The cardinality of the infinite set of odd numbers equals to the cardinality of infinite set of even numbers and it also equals to the cardinality of the set of all the Rational numbers. And all the cardinalities are א‎0. Even if we join to sets having cardinality א‎0, we will still get the set with the cardinality of א‎0. Confusing, isn’t it? This is the basis for the Continuum Hypotheses in the number theory.

ω: An ordinal!

So again for the set of bananas, I take into account, the order in with the bananas are places, then the number corresponding to the recently added banana will change now. This is because, we are now considering ordinal for the set. We cannot now say that the number corresponding to the latest banana is א‎0. For this, a new term comes into picture. This is ω. ω is the ordinal corresponding to the banana added to the set which contained already א‎0 bananas. Now as we keep on adding the terms, here bananas, to the set, ordinal keeps changing. It will follow the sequence ω+1, ω+2 and so on until it reaches ω+ω i.e. 2*ω. This will continue until the ordinal becomes ω*ω i.e. ω2. It will still keep on continuing till we get the ordinal ωω. Still this series has not completed. It will reach to the number which is ω raised to ω raised to ω raised to ω up to ω times. Ugh! Too confusing, right? On top of that, though we have these ordinals in the set, the cardinality of the set is still א‎0. Amazing!

When all such series of the terms containing ω end, thereafter we start with new set of numbers. Which is denoted by א‎1. We can continue such sequence until we reach א‎א‎‎, which is Aleph Subscript Aleph. We can just imagine gigantic amount of numbers it will be. But still we believe that there will be an infinity θ which is the highest among all of the infinities. א‎0 is the smallest of all infinites and θ is the largest of all of them.

This is just the tip of the iceberg of the theory of infinite numbers. It depends on some of the basic axioms of Mathematics. If in future, any of the physical result, violates those theorized axioms, the whole theory of infinity will change and we can’t predict what it will be. But it has the slightest of the slightest chances to predict the infinity physically in this universe. Everything we know about this universe is finite. Even the number of atoms in this whole universe can be predicted and the number will turn out to be finite. So, this is one theory that has been developed by man, completely on his own. There is no physical evidence for that. This shows how far this humankind has evolved. We just have to keep the good work going and with this intelligence, there is hardly any problem that human kind cannot solve!

– Ninad Kumbhojkar

Pics Courtesy- Google Images

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