9.1.2. Question 2

In question number 2 we have to prove that there is no infinite set A whose cardinality is strictly less than א0. That means its cardinality is strictly less than the cardinality of set of positive integers. So in some sense we want to prove here that א0 is the smallest infinity here.

This chunk introduces the problem we need to solve. We are trying to demonstrate that there is no infinite set A that is smaller in size (cardinality) than the set of positive integers. In mathematical terms, this means showing that the cardinality of A cannot be less than the smallest infinite cardinality, denoted as א0 (aleph null). This is significant because it establishes a foundation for understanding the nature of infinite sets and their sizes.

Imagine an infinite collection of books, where each unique book represents a whole number. If we say there are fewer unique books (infinite set A) than the books representing all positive integers, we encounter a contradiction because we would imply a collection that can’t exist. Just like you can't count an infinite number of items in a finite way, there’s no way to have an infinite set smaller than the infinite set of all positive integers.

To prove this statement we will use 2 claims and we will assume for the moment that these two claims are correct and later on we will focus our attention on proving these two claims. 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.

This chunk sets the stage for the proof by introducing two claims that will form the backbone of our argument. The first claim states that if an infinite set A has a cardinality less than or equal to the cardinality of positive integers, then it is possible to find a subset of positive integers that has the same number of elements as set A. This is crucial because it provides a linkage between any 'smaller' infinite set and the well-known countable set of positive integers.

Consider a large party (set A) that can only have a limited number of guests (positive integers). If we say some guests are absent, then we can always form a smaller group of attendees who mirror some of the remaining guests. No matter how we rearrange the party, if it keeps being larger than the count of original guests, at least one smaller group will share the same size as those who stayed.

Claim 2 is that any subset of the set of positive integers is either finite or has the same cardinality as א0, you cannot have any other third category of subset of the set of positive integers.

The second claim reinforces our understanding of the nature of subsets of the set of positive integers. It asserts that any subset we take from the positive integers can either be a finite set (like having three integers: {1, 2, 3}) or an infinite set that is countably infinite (like {1, 2, 3,...} following an infinite pattern). This claim ensures that when dealing with the cardinalities of infinite sets, they either align with known finite or countably infinite structures.

Imagine every person at a concert is numbered 1, 2, 3, and so forth. If some people leave (making it finite) or if a never-ending line of fans keeps joining (making it countably infinite), we know for sure that the counts do not create a paradox. There can’t be a third type of count that isn’t either finite or accounted for by the continuous numbering already established with the ones attending.

Now the proof boils down to how exactly we prove these two claims; they are very simple. Assuming we have an infinite set A whose cardinality is strictly less than א0, we will apply claim 1 on that set A to show that there exists some subset B of positive integers with the same cardinality as set A.

This chunk discusses utilizing our earlier claims in a proof by contradiction. The idea here is to assume we have an infinite set A with a cardinality smaller than א0. When we apply the first claim, it suggests there's a subset B from the positive integers that matches A's cardinality. But if we apply the second claim, it states that these sets can only be finite or countably infinite, bringing us back to the realization that if A is infinite, it must match the infinite structure of positive integers itself, leading to a contradiction.

Picture our earlier party analogy but this time, we suspect someone has invited fewer guests than the total. If we find exactly the same number of chairs available for guests as the count of people present, we can’t have an empty room filled with infinite guests—every chair occupied only matches to certain people, which eventually leads to the same total count or fewer based on the number of original invitees.

But we also know that the cardinality of B is same as cardinality of A means that the cardinality of A is also א0 which is a contradiction.

In concluding the proof, herein lies the contradiction we initially set out to find. We discover that our assumption—that there exists an infinite set A with cardinality less than א0—leads us to conclude that A must actually possess the same cardinality as א0. Given the nature of our initial assumption proved false, we have successfully shown that our claim holds true: there can be no infinite set A whose size is smaller than the infinite count of positive integers.

Returning to our concert, once we discover that the number of attendees we assumed was limited reflects the actual infinite guests present through total seats and attendance, it would confirm the very count we thought marginal could not be 'less than’ to begin with. Just like realizing our assumptions about invites mismatched the true scenario, our mathematical conclusion accurately debunks the idea of a smaller infinite set.

--