Counting Beyond Infinity

Counting Beyond Infinity

“The conceptual jump from nothing to first infinity is same as the jump from first infinity to an inaccessible.”

~Jay Mehta

Googol Number = 10^100

Googolplex = 10^ (Googol Number) = 10^ (10^100)

Millimillion = 10^3003

Millimillionplex = 10^ (Millimillion)

#Fact: Biggest number in terms of surface area on The Earth is: 40
It covers more than 12000 square meters - made out of strategically planted trees in Russia. It is larger than Italian Markers on Signal Hill in Calgary.

Infinity is not the biggest or the largest number; it is defined as the set of unending numbers.

Georg Cantor's Diagonal Argument published in 1891 has a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began.

http://www.trottermath.net/personal/infinity.html

When a number refers to how many things there are it refers to the CARDINAL NUMBERS.

Natural numbers are cardinal numbers.

Aleph-Null (No) is the term used to define an unending amount of natural numbers - (COUNTABLE INFINITIES). It is said to form the smallest kind of infinity.

Aleph is the first alphabet in the Hebrew Language.

What does Aleph Null represent?

It represents a number of natural numbers, even numbers, odd numbers or rational number.

It is bigger than any finite amount.

To count past or beyond aleph-null we use the concept of super-task.

Super-Task: - An infinite sequence of the task that has no last task is called a super-task. If we draw a bunch of lines such that each line is a fraction smaller than the previous one and is spaced at a fraction near to the previous two lines. By doing so, we can fit an unending amount of lines in a finite amount of space in the finite amount of time. In other words, an infinite amount of task done within a finite amount of time is called a Super-Task. The Gabriel’s Cake Paradox is based on the same concept.

A true super-task requires doing infinitely many distinct actions, leading to becoming a product of our imagination.

If now we assign each line formed by the super-task we performed a whole number starting from 0 to an indefinite end. As, we can have one to one correspondence between a line and a number, both the sets belong to aleph-null – have the same cardinality.

What would happen if we add a line after the set aleph-null?

Well, it wouldn’t be, (No + 1) as unending numbers can’t be treated as finite numbers. We can’t just randomly add one to it. We will still be left with an aleph-null amount of lines. As we can call this new line as zero and start counting from one in the previous set of the lines.

 What if we add an aleph-null amount of lines in front of it?

We would still remain with an aleph-null amount of lines. As now we can have the first set of lines assigned to all the odd numbers and the second set of lines assigned with all the even numbers; making the complete set still a set of aleph-null number.

What if we label them with a particular name or number?

We introduce ordinal numbers to solve this problem. An ordinal number is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another. Any finite collection of objects can be put in order just by the process of counting: labeling the objects with distinct whole numbers. Ordinal numbers are thus the "labels" needed to arrange collections of objects in order.

The first finite-trans-ordinal number is “w” (omega) – it is the next label that we would use after the completion of infinite numbers of natural numbers.

After “w” we have “w + 1” followed by “w + 2” so on and so forth. An ordinal number doesn’t answer the question ‘how many things are there?’ rather they represent the arrangement of things i.e. the order type.

“w + 1” is not greater than “w” it just comes after omega. It doesn’t define that it is a larger number. #JustALabel.

We can have bigger infinities than aleph-null using the George Carter’s Diagonal Argument. We can use this argument to prove that number of real numbers is bigger than a number of natural numbers.

Power Set of Aleph-Null

The power set of a set is a set of all the different subsets you can make from it.

For example, power set of 1 and 2 is: {}, {1}, {2} and {1, 2};

We can observe from the above example that a power set contains 2 ^ n members if there are ‘n’ members in the original set.

Hence, a power set of all naturals would be given by 2 ^ (No);

~Jay Mehta

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Jay Mehta.

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