9.5.3. Union of Sets S

In question number 2 we have to prove that there is no infinite set A whose cardinality is strictly less than א. That means its cardinality is strictly less than the cardinality of set of positive integers.

This chunk discusses the concept of cardinality, particularly in relation to infinite sets. The statement asserts that if you have any infinite set A, its size (or cardinality) cannot be less than a certain value denoted as א (aleph-null), which is the cardinality of the set of positive integers. This is significant as it introduces the idea of 'smallest infinity' in set theory, indicating that some infinities can be larger than others.

Imagine a box of chocolates. If you have a finite number of chocolates, you can count them easily. Once you start making infinite boxes that each hold an infinite number of chocolates (but less than a full truckload), you still cannot have more chocolates than the total truckload which represents the cardinality of א. Just like in real life, where we can create infinitely large collections, in math, sizes of these collections can also be compared.

The first claim is that if you have any set A whose cardinality is less than equal to the cardinality of set of positive integers then you can always find a subset of the set of positive integers which has the same cardinality as your set A.

Claim 1 states that any set A that is not larger than the set of positive integers (denoted by Z+) can be equated in size to a subset of Z+. This means that if A is infinite but its size is considered 'smaller' than that of the integers, there exists some subset of the integers that can be 'matched' with the elements of A. This is key in understanding how different infinite sets compare in size.

Think of a classroom with an infinite number of desks (representing the positive integers) and a handful of students (representing set A). Each student can take a desk, and there will always be enough desks for them, even if not all desks are occupied. As long as A does not exceed the number of desks available in Z+, it can find its place within that infinite set.

Claim 2 is that any subset of the set of positive integers is either finite or has the same cardinality as א.

Claim 2 asserts that any subset of Z+ either would be finite or would have a size (or cardinality) that is equivalent to א (aleph-null), meaning it has an infinite size. This helps establish a clear understanding of the nature of subsets: they either remain countably finite or become countably infinite. It emphasizes that when dealing with the infinite subset of integers, one cannot form a set that is 'in between' finite and infinite.

Consider a bookcase filled with books (Z+) where each shelf represents a finite count of books laid out in order. A specific shelf will either have a few books (finite) or be filled with an endless number of books (like the infinite collection of the same author). If you were to take only certain shelves (subsets), you find they either hold a few books or a whole series of endless books—never something in between.

Now by applying claim 1 on that set A we also know that there exist some subset B of the set of positive integers whose cardinality is the same as the cardinality of your A set.

Here, the proof uses contradiction to show that assuming the existence of an infinite set A with cardinality less than that of Z+ leads to an inconsistency. By assuming there exists a subset B of integers that can match A's size, it can be shown that if A is infinite, so must B be, hence leading to a violation of the assumption that A is less than Z+. It highlights a crucial principle in mathematical proofs involving infinite sets.

Imagine you claimed to have a limited number of students in a class (infinite set A), but then you realize this means there are more students to suit every desk in the classroom than you thought. It shows the impossibility of ‘more’ students fitting into the limited school property when you have to assign desks. Therefore, some contradictions force you to reconsider that perhaps your initial numbers weren’t accurate.

In each step in each iteration the arbitrarily element which I am picking from the left over set if I list it out, if I list down all those elements, then that sequence of the elements will be or the set of those elements which I am picking in each iteration from the left over set will of course be a subset of my original set.

This concept revolves around the idea of creating a sequence by iteratively selecting elements from set A, continually demonstrating that what remains is also infinite. By selecting elements recursively, one can form a new infinite subset. This not only illustrates the infinite nature of A but also provides a mechanism to assign cardinality to the whole set including A.

Imagine a candy factory where every time you pick a piece of candy (element from A), the factory produces more. As you take candy, more are constantly created. The moment you think you have taken all candies (removed the last of A), the factory has infinitely supplied you with another batch, assuring you always have an infinite number to choose from, thus keeping the candy supply (set A) endlessly available.

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