Let me try again. Think of Step 4 like this:

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STEP 4 REDUX RELOADED

Imagine a powerful argument. Let's call it "The Decider." "The Decider" is a series of logical steps that allows us to go from A to B, where A is a certain set of assumptions or facts that we know to be true, and B is a conclusion that we can demonstrate must be true if A is true. We can use the metaphor of a machine. “The Decider” is a machine that takes material in on one side, and spits out a product on the other. The input material is facts or assumptions, and the output is new facts or assumptions that logically follow from the input. For now, let's not worry about how "The Decider" works. Let's just imagine that there is in fact some way to fabricate such a logical apparatus that works the way we need it to work.

Now, suppose that the kind of fact “The Decider” ingests is a very special kind of fact. If you tell it “the sky is blue,” the metaphorical gears within “The Decider” grind to a halt. It doesn’t know what to do with this kind of fact. Instead, it requires a fact of the following form:

IN: “Within any group of k people, everyone has the same age.”

So for example, you can tell “The Decider”:

IN: “Within any group of 1 person, everyone has the same age.”

And “The Decider” will grind away until it spits out a shiny new assertion that must be true if your input was true. In fact, let’s suppose that the shiny new assertion “The Decider” would produce in this case would be:

OUT: “Within any group of 2 people, everyone has the same age.”

You could tell “The Decider,”

IN: “Within any group of 9 people, everyone has the same age.”

and it would work, too.

OUT: “Within any group of 10 people, everyone has the same age.”

In fact, “The Decider” follows a pattern. Whatever value k is in the input, the number in the output will be k+1. In other words, for any k that is a positive whole number, “The Decider” can turn “within any group of k people, everyone has the same age” into “within any group of k+1 people, everyone has the same age.”

If “The Decider” actually existed, if there was a general pattern of reasoning that could be used to go from A to B in this way, we would accomplish the task set out for us in Step 3. Namely, we would prove that, “whenever S(n) is true for one number (say n=k), it is also true for the next number (that is, n=k+1).”

END STEP 4 REDUX RELOADED

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Steps 5 through 13 then proceed to describe how “The Decider” actually works. Or at least they try to. That is, they describe a general pattern of reasoning that purportedly can be used to show that if “within any group of 9 people, everyone has the same age,” then it must also be true that “within any group of 10 people, everyone has the same age.” It can also show that if it is true that “within any group of 2 people, everyone has the same age,” then it must also be true that “within any group of 3 people, everyone has the same age.” In fact, Steps 5 through 13 endeavor to create a logical argument so powerful that it can take any statement of the form “within any group of k people, everyone has the same age” and demonstrate that if that is true, it’s also true that “within any group of k+1 people, everyone has the same age.” This is “The Decider” I describe above.

Spoiler

The argument ultimately falls down because “The Decider” has a fatal flaw. When k=2, the logical machinery chokes. It implicitly relies on k being greater than or equal to 3.

Is that helpful at all?