All infinities are not the same??

What do you mean when you say something is infinite?

Something that’s not finite, right? And what is finite?

Would you consider something finite to be whatever falls under the limits of your own experience? — but that’s rather anthropocentric, isn’t it? The universe can’t possibly be designed around humans.
(That could be two separate blog topics right there: the meaning of “finite,” and the appearance of design in the universe.)

But that’s not the topic of discussion today.

Would you say all infinities are the same size?

For example, is the infinity of all the points on an infinitely long stick the same as the infinity of counting numbers?

Well, it turns out — it’s not

All infinities are not equal.

ok, Let’s prove it.

Some important Terminologies

(can skip if you already know these)

• Bijective / one-to-one correspondence: A function is said to be bijective, iff every element in the domain can be mapped to a unique element in the codomain and every element in the codomain is mapped to exactly one element in the domain.
• Finite: A set S is said to be finite if it can be put into a one-to-one correspondence with {1,2,3,4,… n} for some n.
  (eg : S = {8 , 9 , 1 , 5 } is finite because it is bijective with the set {1 , 2 , 3 , 4})
• Countably infinite: An infinite set S is said to be countable if there is a bijection between S and N.

The Proof

Suppose we have a function

f : N -> [0,1]

which maps the natural numbers to real numbers between 0 to 1.

let’s suppose the data below:

Now, let’s take one diagonal digit from each row (i.e. first digit of f(1), the second digit of f(2), the third digit of f(3), and so on) — creating a new number.

0.8322..

increasing each digit by value of 1 gives us :

0.9433..

NOWWW,

will this number — 0.9433.. exist in the table above?

Spoiler alert — it will not,

Because you see .. this number differs from f(1) in it’s 1st digit, f(2) in it’s 2nd digit .. similarly f(n) in it’s nth digit.

That means our list wasn’t complete, i.e f can’t be onto, no matter how we try to list the real numbers between 0 and 1, there will always be some missing.

Hence, we cannot put all real numbers into a one-to-one correspondence with the natural numbers.

Hence, the infinity of the continuum > infinity of Natural Numbers

This version of proof is called the Cantor’s diagonal argument and it proves that the infinity of the real numbers is larger than the infinity of the natural numbers.

So , all infinities are not the same!

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❤️ SUPRIYA