# Very Quick Note on Lessons 11-15

Lessons 11-15 are noticeably more difficult than the lessons that precede it, and those that follow it. As usual, they will not require pen and paper, nor do they involve any ideas that we haven’t already introduced. They might, however, require a bit more time staring at the computer screen and thinking, and/or rereading certain passages a couple of times. While the results are indeed worth it, they are also not necessary for the ideas that will follow.

In lessons 11-14, we establish that in some sense “infinity type 1 times infinity type 1 is still infinity type 1”, and then go on to see that there is in fact an “infinity type 2” which is in a clear way “greater than” infinity type 1. Finally, we show that there is an **infinitely high** tower of greater and greater infinities. In lesson 15 we use functions to prove the general pattern that was proposed to exist in lesson 3 that a set with N elements has (“2 raised to the power of N”, see lesson 3) distinct subsets. Thus, lessons 11-14 are meant to show a remarkable result that follows from the new form of counting that we’ve introduced, and lesson 15 is meant mainly for completeness of the mathematics that has been introduced.

As mentioned, none of these lessons are impossible. I would recommend that at some point in the reader’s life he or she takes the time to fully understand them, because their results and the methods of reasoning used to attain those results are truly remarkable. However, if the reader is not in the mood to put in a slightly elevated level of effort (which is totally reasonable), then he or she can go on to lesson 16 without missing anything at all. I.e., the ideas from the lessons prior to lesson 11 do continue smoothly into lesson 16.

If the reader is interested in plowing forward to uncover some of these beautiful results, then by all means, head on to lesson 11!

(Think of this as one of those “choose your own adventure” stories…)

On to lesson 11

On to lesson 16

Back to lesson 10

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