7.5. Conclusion

So Cantor proved a very interesting result as well. He showed that you take any set A then the cardinality of that set is strictly less than the cardinality of its power set. So remember the notation P(A) denotes the power set of A. Where the power set is the set of all subsets of that set.

Cantor's theorem shows a fundamental property of sets related to their sizes, known as cardinality. If you have any given set A, the size of set A (its cardinality) is always smaller than the size of its power set, which is P(A). The power set includes every possible subset of A, and this means that for any finite set with n elements, the power set will contain 2^n subsets, which is always more than n.

Imagine you have a box of 3 different colored balls (red, blue, and green). You can form different groups of these balls (like keeping just the red one, or the red and blue together), leading to a total of 8 different groups (subsets). The number of these groups (8) is larger than just having 3 balls. Hence, Cantor’s theorem highlights the idea that even a small set can lead to a larger collection of possibilities.

And we can always prove that n is always strictly less than 2n. What if A is an infinite set can we conclude that this theorem is true that is even for infinite set and Cantor showed yes, so the proof is again is contradiction and we will run the diagonalization argument here as well.

This part discusses the application of Cantor's theorem to infinite sets. While it’s clear for finite sets, the same principle applies for infinite sets as well. Cantor demonstrated this using a method known as diagonalization, providing a contradiction to the assumption that an infinite set could have a cardinality equal to or greater than its power set.

Think about an infinite library that keeps getting new books every day. No matter how many books you already have, there will always be more combinations of books that you could create (new subsets) than the books currently contained in the library. This is analogous to how the power set of an infinite set is uncountably larger than the set itself.

So now what I am going to show is since I am assuming that the function f is surjective function and I have to arrive at a contradiction. I will show that this f actually does not exist and how do I show that f is does not exist?

In this argument, Cantor assumed that there exists a surjective function f from set A to its power set P(A). To prove it wrong, he constructed a new subset S that could not be an output of this function f, demonstrating that a surjective function could not exist due to this contradiction.

Imagine a situation like a party where everyone must bring a dish. If you assume everyone can bring every dish possible (surjectively), but then realize there’s a unique dish (like a secret recipe one) that no one brings, it shows the assumption can't be true – just like the function f in Cantor's proof.

So what is the implication of the Cantor’s theorem it has a very beautiful implication. So this is the statement of the cantor’s theorem.

Cantor's theorem implies that there are different sizes of infinity. For example, the set of natural numbers has a countable infinity represented by א₀, while the power set of natural numbers is an uncountable infinity, showing that not all infinities are the same size.

Consider how there are different types of infinite collections. For instance, you can count the number of people in a room (countable infinity – you can list them all), but the number of points on a line (uncountable infinity) is so vast that you can't name or list them all. This comparison helps visualize how Cantor demonstrated that infinities can continue indefinitely, creating a hierarchy of different types of infinity.

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