Once they've got a nearly full set of rules, Keltham remarks that the last puzzle piece for identifying the numbers, as well as they can ever be identified in a certain sense he's not going into right now, is one he really doesn't expect them to get.
And then he drops the induction axiom schema on them.
After clearing up any misapprehensions about that as best he can, Keltham is ready to move on to his next point.
"We're running through things a lot faster than I went through them as a kid, and I'm probably accidentally leaving out important ideas along the way - all of this would take more like a month, if you were eight years old and doing the exercises, even if you were doing nothing else. But you may recall that some time earlier, I posed some puzzles about asking for examples of necessary truths, and why they were ever good for anything, and what it means to say that one equals one is a necessary truth - what you ought to expect to see happen as a result - especially given that a necessary truth should still end up being true no matter what happens to you, if it could happen inside any illusion depicted in full detail."
"We now have a language in which I can give some of my own answers there. But before then, does anybody want to take a renewed shot at saying what it is we're talking about, and what we should expect to see happen, if we say that one equals one is a necessary truth?"