9.1.3. Question 3

In question number 3 we are asked to show that if you are given an infinite set then it does not matter whether A is countable or uncountable, you can always find the subset of A which is countably infinite. So of course if my set A itself is countably infinite then the subset would be the set itself but in the statement I am asking you to prove this even if the set A is uncountable.

In this first part, we establish the task which is showing that every infinite set, regardless of whether it is countably or uncountably infinite, contains a countably infinite subset. To understand this, we start by noting that if the set A is already countably infinite, then the subset we are looking for is simply A itself. However, our proof must extend beyond this case to cover even uncountable sets like the set of all real numbers. For an uncountable set, we will show how to find a countably infinite subset of that set.

Imagine you have a large box filled with an infinite number of coins. If this box is organized such that some coins are grouped and labeled according to their types or values (some groups are very large and some are less), there will always be a way to go through these coins and take a few from each type to create a small, organized collection that still represents an infinite count of values (countably infinite).

Since A is an infinite set, it will have at least one element arbitrarily I pick that element I call it as a1. Now my claim is that if I remove that element a from the set A, the remaining set is still an infinite set...

We start with our infinite set A and remove an arbitrary element (let's call it a1). The important aspect here is that even after removing a1, the set remains infinite. If the remaining set, named A – a1, were finite, that would imply that A itself was finite because removing just one element from a finite set would result in another finite set, contradicting our assumption that A is infinite. Hence, removing one element from an infinite set does not affect its overall size; it is still infinite.

Think of a large classroom filled with students. If one student (representing a1) leaves for a moment, the class is still filled with many students. Whether it has ten or a hundred, as long as there are multiple students grouped together, the essence of the crowd remains. Even more students can keep coming in, further emphasizing that the group can be infinite.

Now I again apply the same argument. I focus on the left over set namely the set which I obtained by removing element a1...what I can say now is that, in each step in each iteration the arbitrarily element which I am picking from the left over set...

The process continues as we repeatedly remove elements from the infinite set. After removing a1, we take another arbitrary element (let's call this a2) from the remaining set A – a1, and repeat our argument. Each time we remove an element, the remaining set still contains an infinite number of elements. By following this process and documenting each element we remove, we form a new subset composed entirely of these removed elements. This iterative process will always yield infinitely many elements, resulting in a countably infinite set.

Imagine you are trying to create a long chain out of infinite links. Each time you take out a link and add it to your chain, the number of links left in the original box remains enormous, regardless of how many links you take out. By removing them one by one, you're still linking endless chains.

So that shows that you can always find out an infinite subset from the set A whose cardinality is א.

After iterating through the removal of elements and forming our growing countable subset from A, we conclude that this process leads us to construct an infinite subset whose cardinality is the same as the smallest infinite cardinal number, denoted as א (aleph-null). Therefore, regardless of whether our original set A was countably or uncountably infinite, we have effectively demonstrated the existence of a countably infinite subset.

Using the previous analogy of links, regardless of how many links (elements of A) you started with, as long the procedure of extracting links continues, you are bound to gather a full sized chain (countably infinite set). Like collecting marbles, you don’t lose your collection even if you subtract a few; your total remains substantial.

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